andatoshiki/toshiki-notebook
1
0
Fork 0
mirror of https://github.com/andatoshiki/toshiki-notebook.git synced 2026-06-06 09:16:45 +00:00
Code Issues Actions Packages Projects Releases Wiki Activity
62f33c1348
toshiki-notebook/academic/physics/ipho-formulas-jpn/9.html

43 lines
144 KiB
HTML
Raw Blame History

This file contains invisible Unicode characters

This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

<!DOCTYPE html>
<html lang="en-US" dir="ltr">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<title>Formulas for IPhO 日本語版: Section 9 | Toshiki's Note</title>
<meta name="description" content="Toshiki's web notebook served via Vitepress!">
<link rel="preload stylesheet" href="/assets/style.bd240186.css" as="style">
<script type="module" src="/assets/app.6020b1dd.js"></script>
<link rel="preload" href="/assets/inter-roman-latin.2ed14f66.woff2" as="font" type="font/woff2" crossorigin="">
<link rel="modulepreload" href="/assets/chunks/framework.b7580407.js">
<link rel="modulepreload" href="/assets/chunks/theme.7de1b093.js">
<link rel="modulepreload" href="/assets/chunks/commonjsHelpers.725317a4.js">
<link rel="modulepreload" href="/assets/chunks/PageInfo.vue_vue_type_script_setup_true_lang.250b3e56.js">
<link rel="modulepreload" href="/assets/academic_physics_ipho-formulas-jpn_9.md.da4d857a.lean.js">
<link rel="stylesheet" href="https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css">
<link rel="stylesheet" href="https://cdnjs.toshiki.dev/ajax/libs/font-awesome/6.3.0/css/all.min.css">
<link rel="icon" href="https://r2.toshiki.dev/cdn/toshiki-notebook-favicon/favicon.ico">
<meta name="author" content="Anda Toshiki">
<meta name="keywords" content="Toshiki, Anda Toshiki, andatoshiki, GitHub, GitHub action, Vitepress, Vite, Notebook, Knowledge base, Programming, Programming Notes, Academic, Personal, Notebook, Productivity, Journal, Note-taking, Markdown, Notepad, Organization, Tutorial">
<meta name="google-site-verification" content="lm7PNJiYSPEx1dMast1Xptc0Vk0cU06o-daZSsIgr2I">
<meta name="HandheldFriendly" content="True">
<meta name="MobileOptimized" content="320">
<meta name="theme-color" content="#3c8772">
<meta property="og:type" content="website">
<meta property="og:locale" content="en-US">
<meta property="og:title" content="Toshiki&#39;s Note">
<meta property="og:description" content="Toshiki&#39;s web notebook served via Vitepress!">
<meta property="og:site" content="https://note.toshiki.dev">
<meta property="og:site_name" content="Toshiki&#39;s Note">
<meta property="og:image" content="https://note.toshiki.dev/og-cover.png">
<script>function siteruntime(){window.setTimeout("siteruntime()",1e3),X=new Date("8/24/2021 10:28:00"),Y=new Date,T=Y.getTime()-X.getTime(),M=24*60*60*1e3,a=T/M,A=Math.floor(a),b=(a-A)*24,B=Math.floor(b),c=(b-B)*60,C=Math.floor((b-B)*60),D=Math.floor((c-C)*60),siteruntime_span.innerHTML="This site has been running for: "+A+" day(s) "+B+" hour(s) "+C+" minute(s) "+D+" second(s)"}siteruntime();</script>
<script async defer data-website-id="" src=""></script>
<script id="check-dark-mode">(()=>{const e=localStorage.getItem("vitepress-theme-appearance")||"auto",a=window.matchMedia("(prefers-color-scheme: dark)").matches;(!e||e==="auto"?a:e==="dark")&&document.documentElement.classList.add("dark")})();</script>
<script id="check-mac-os">document.documentElement.classList.toggle("mac",/Mac|iPhone|iPod|iPad/i.test(navigator.platform));</script>
</head>
<body>
<div id="app"><div class="Layout" data-v-bb676018><!--[--><!--]--><!--[--><span tabindex="-1" data-v-197f0fa6></span><a href="#VPContent" class="VPSkipLink visually-hidden" data-v-197f0fa6> Skip to content </a><!--]--><!----><header class="VPNav" data-v-bb676018 data-v-f92d2e4c><div class="VPNavBar has-sidebar" data-v-f92d2e4c data-v-e4fc9f8b><div class="container" data-v-e4fc9f8b><div class="title" data-v-e4fc9f8b><div class="VPNavBarTitle has-sidebar" data-v-e4fc9f8b data-v-d919d4a8><a class="title" href="/" data-v-d919d4a8><!--[--><!--]--><!--[--><img class="VPImage logo" src="/logos/logo.png" alt data-v-26d78c9a><!--]--><!--[-->Toshiki&#39;s Note<!--]--><!--[--><!--]--></a></div></div><div class="content" data-v-e4fc9f8b><div class="curtain" data-v-e4fc9f8b></div><div class="content-body" data-v-e4fc9f8b><!--[--><!--]--><div class="VPNavBarSearch search" data-v-e4fc9f8b><!--[--><!----><div id="docsearch"><button type="button" class="DocSearch DocSearch-Button" aria-label="Search"><span class="DocSearch-Button-Container"><svg class="DocSearch-Search-Icon" width="20" height="20" viewBox="0 0 20 20" aria-label="search icon"><path d="M14.386 14.386l4.0877 4.0877-4.0877-4.0877c-2.9418 2.9419-7.7115 2.9419-10.6533 0-2.9419-2.9418-2.9419-7.7115 0-10.6533 2.9418-2.9419 7.7115-2.9419 10.6533 0 2.9419 2.9418 2.9419 7.7115 0 10.6533z" stroke="currentColor" fill="none" fill-rule="evenodd" stroke-linecap="round" stroke-linejoin="round"></path></svg><span class="DocSearch-Button-Placeholder">Search</span></span><span class="DocSearch-Button-Keys"><kbd class="DocSearch-Button-Key"></kbd><kbd class="DocSearch-Button-Key">K</kbd></span></button></div><!--]--></div><nav aria-labelledby="main-nav-aria-label" class="VPNavBarMenu menu" data-v-e4fc9f8b data-v-368cfbc9><span id="main-nav-aria-label" class="visually-hidden" data-v-368cfbc9>Main Navigation</span><!--[--><!--[--><a class="VPLink link VPNavBarMenuLink" href="/development/" tabindex="0" data-v-368cfbc9 data-v-0eb8cc79><!--[--><span data-v-0eb8cc79>Development</span><!--]--></a><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup active" data-v-368cfbc9 data-v-419e6f76><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-419e6f76><span class="text" data-v-419e6f76><!----><span data-v-419e6f76>Academic</span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-419e6f76><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-419e6f76><div class="VPMenu" data-v-419e6f76 data-v-3795579e><div class="items" data-v-3795579e><!--[--><!--[--><div class="VPMenuGroup" data-v-3795579e data-v-bdc27f57><p class="title" data-v-bdc27f57>K-12</p><!--[--><!--[--><div class="VPMenuLink" data-v-bdc27f57 data-v-df1d8bd6><a class="VPLink link" href="/academic/chemistry/index" data-v-df1d8bd6><!--[-->Chemistry<!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-bdc27f57 data-v-df1d8bd6><a class="VPLink link" href="/discrete-math/index" data-v-df1d8bd6><!--[-->Discrete Math.<!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-bdc27f57 data-v-df1d8bd6><a class="VPLink link" href="/academic/literature/index" data-v-df1d8bd6><!--[-->Literature<!--]--></a></div><!--]--><!--]--></div><!--]--><!--[--><div class="VPMenuGroup" data-v-3795579e data-v-bdc27f57><p class="title" data-v-bdc27f57>Tools</p><!--[--><!--[--><div class="VPMenuLink" data-v-bdc27f57 data-v-df1d8bd6><a class="VPLink link active" href="/academic/physics/ipho-formulas-jpn/1" data-v-df1d8bd6><!--[-->Formulas for IPhO JPN.<!--]--></a></div><!--]--><!--]--></div><!--]--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><span class="VPLink" data-v-df1d8bd6><!--[--><!--]--></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><span class="VPLink" data-v-df1d8bd6><!--[--><!--]--></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><span class="VPLink" data-v-df1d8bd6><!--[--><!--]--></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><span class="VPLink" data-v-df1d8bd6><!--[--><!--]--></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><span class="VPLink" data-v-df1d8bd6><!--[--><!--]--></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><span class="VPLink" data-v-df1d8bd6><!--[--><!--]--></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><span class="VPLink" data-v-df1d8bd6><!--[--><!--]--></span></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup" data-v-368cfbc9 data-v-419e6f76><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-419e6f76><span class="text" data-v-419e6f76><!----><span data-v-419e6f76>Application</span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-419e6f76><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-419e6f76><div class="VPMenu" data-v-419e6f76 data-v-3795579e><div class="items" data-v-3795579e><!--[--><!--[--><div class="VPMenuGroup" data-v-3795579e data-v-bdc27f57><p class="title" data-v-bdc27f57>Personal projects</p><!--[--><!--[--><div class="VPMenuLink" data-v-bdc27f57 data-v-df1d8bd6><a class="VPLink link" href="/application/markdown-it-katex/how-to-use" data-v-df1d8bd6><!--[-->markdown-it-katex<!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-bdc27f57 data-v-df1d8bd6><a class="VPLink link" href="/application/vitepress-plugin-shiki-twoslash/index" data-v-df1d8bd6><!--[-->vitepress-plugin-shiki-twoslash<!--]--></a></div><!--]--><!--]--></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup" data-v-368cfbc9 data-v-419e6f76><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-419e6f76><span class="text" data-v-419e6f76><!----><span data-v-419e6f76>Save</span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-419e6f76><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-419e6f76><div class="VPMenu" data-v-419e6f76 data-v-3795579e><div class="items" data-v-3795579e><!--[--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><a class="VPLink link" href="/save/reading/index" data-v-df1d8bd6><!--[-->Reading<!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-3795579e data-v-df1d8bd6><a class="VPLink link" href="/academic/vocabulary/index" data-v-df1d8bd6><!--[-->Vocabulary<!--]--></a></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--]--></nav><!----><div class="VPNavBarAppearance appearance" data-v-e4fc9f8b data-v-82003167><button class="VPSwitch VPSwitchAppearance" type="button" role="switch" title="toggle dark mode" aria-checked="false" data-v-82003167 data-v-f31c77b6 data-v-ad8fa6cf><span class="check" data-v-ad8fa6cf><span class="icon" data-v-ad8fa6cf><!--[--><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="sun" data-v-f31c77b6><path d="M12,18c-3.3,0-6-2.7-6-6s2.7-6,6-6s6,2.7,6,6S15.3,18,12,18zM12,8c-2.2,0-4,1.8-4,4c0,2.2,1.8,4,4,4c2.2,0,4-1.8,4-4C16,9.8,14.2,8,12,8z"></path><path d="M12,4c-0.6,0-1-0.4-1-1V1c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,3.6,12.6,4,12,4z"></path><path d="M12,24c-0.6,0-1-0.4-1-1v-2c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,23.6,12.6,24,12,24z"></path><path d="M5.6,6.6c-0.3,0-0.5-0.1-0.7-0.3L3.5,4.9c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C6.2,6.5,5.9,6.6,5.6,6.6z"></path><path d="M19.8,20.8c-0.3,0-0.5-0.1-0.7-0.3l-1.4-1.4c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C20.3,20.7,20,20.8,19.8,20.8z"></path><path d="M3,13H1c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S3.6,13,3,13z"></path><path d="M23,13h-2c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S23.6,13,23,13z"></path><path d="M4.2,20.8c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C4.7,20.7,4.5,20.8,4.2,20.8z"></path><path d="M18.4,6.6c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C18.9,6.5,18.6,6.6,18.4,6.6z"></path></svg><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="moon" data-v-f31c77b6><path d="M12.1,22c-0.3,0-0.6,0-0.9,0c-5.5-0.5-9.5-5.4-9-10.9c0.4-4.8,4.2-8.6,9-9c0.4,0,0.8,0.2,1,0.5c0.2,0.3,0.2,0.8-0.1,1.1c-2,2.7-1.4,6.4,1.3,8.4c2.1,1.6,5,1.6,7.1,0c0.3-0.2,0.7-0.3,1.1-0.1c0.3,0.2,0.5,0.6,0.5,1c-0.2,2.7-1.5,5.1-3.6,6.8C16.6,21.2,14.4,22,12.1,22zM9.3,4.4c-2.9,1-5,3.6-5.2,6.8c-0.4,4.4,2.8,8.3,7.2,8.7c2.1,0.2,4.2-0.4,5.8-1.8c1.1-0.9,1.9-2.1,2.4-3.4c-2.5,0.9-5.3,0.5-7.5-1.1C9.2,11.4,8.1,7.7,9.3,4.4z"></path></svg><!--]--></span></span></button></div><div class="VPSocialLinks VPNavBarSocialLinks social-links" data-v-e4fc9f8b data-v-3c6c0985 data-v-aea42604><!--[--><a class="VPSocialLink no-icon" href="https://github.com/andatoshiki" aria-label="github" target="_blank" rel="noopener" data-v-aea42604 data-v-d53a05fb><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>GitHub</title><path d="M12 .297c-6.63 0-12 5.373-12 12 0 5.303 3.438 9.8 8.205 11.385.6.113.82-.258.82-.577 0-.285-.01-1.04-.015-2.04-3.338.724-4.042-1.61-4.042-1.61C4.422 18.07 3.633 17.7 3.633 17.7c-1.087-.744.084-.729.084-.729 1.205.084 1.838 1.236 1.838 1.236 1.07 1.835 2.809 1.305 3.495.998.108-.776.417-1.305.76-1.605-2.665-.3-5.466-1.332-5.466-5.93 0-1.31.465-2.38 1.235-3.22-.135-.303-.54-1.523.105-3.176 0 0 1.005-.322 3.3 1.23.96-.267 1.98-.399 3-.405 1.02.006 2.04.138 3 .405 2.28-1.552 3.285-1.23 3.285-1.23.645 1.653.24 2.873.12 3.176.765.84 1.23 1.91 1.23 3.22 0 4.61-2.805 5.625-5.475 5.92.42.36.81 1.096.81 2.22 0 1.606-.015 2.896-.015 3.286 0 .315.21.69.825.57C20.565 22.092 24 17.592 24 12.297c0-6.627-5.373-12-12-12"/></svg></a><a class="VPSocialLink no-icon" href="https://twitter.com/andatoshiki" aria-label="twitter" target="_blank" rel="noopener" data-v-aea42604 data-v-d53a05fb><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>Twitter</title><path d="M21.543 7.104c.015.211.015.423.015.636 0 6.507-4.954 14.01-14.01 14.01v-.003A13.94 13.94 0 0 1 0 19.539a9.88 9.88 0 0 0 7.287-2.041 4.93 4.93 0 0 1-4.6-3.42 4.916 4.916 0 0 0 2.223-.084A4.926 4.926 0 0 1 .96 9.167v-.062a4.887 4.887 0 0 0 2.235.616A4.928 4.928 0 0 1 1.67 3.148 13.98 13.98 0 0 0 11.82 8.292a4.929 4.929 0 0 1 8.39-4.49 9.868 9.868 0 0 0 3.128-1.196 4.941 4.941 0 0 1-2.165 2.724A9.828 9.828 0 0 0 24 4.555a10.019 10.019 0 0 1-2.457 2.549z"/></svg></a><!--]--></div><div class="VPFlyout VPNavBarExtra extra" data-v-e4fc9f8b data-v-edb32f2a data-v-419e6f76><button type="button" class="button" aria-haspopup="true" aria-expanded="false" aria-label="extra navigation" data-v-419e6f76><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="icon" data-v-419e6f76><circle cx="12" cy="12" r="2"></circle><circle cx="19" cy="12" r="2"></circle><circle cx="5" cy="12" r="2"></circle></svg></button><div class="menu" data-v-419e6f76><div class="VPMenu" data-v-419e6f76 data-v-3795579e><!----><!--[--><!--[--><!----><div class="group" data-v-edb32f2a><div class="item appearance" data-v-edb32f2a><p class="label" data-v-edb32f2a>Appearance</p><div class="appearance-action" data-v-edb32f2a><button class="VPSwitch VPSwitchAppearance" type="button" role="switch" title="toggle dark mode" aria-checked="false" data-v-edb32f2a data-v-f31c77b6 data-v-ad8fa6cf><span class="check" data-v-ad8fa6cf><span class="icon" data-v-ad8fa6cf><!--[--><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="sun" data-v-f31c77b6><path d="M12,18c-3.3,0-6-2.7-6-6s2.7-6,6-6s6,2.7,6,6S15.3,18,12,18zM12,8c-2.2,0-4,1.8-4,4c0,2.2,1.8,4,4,4c2.2,0,4-1.8,4-4C16,9.8,14.2,8,12,8z"></path><path d="M12,4c-0.6,0-1-0.4-1-1V1c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,3.6,12.6,4,12,4z"></path><path d="M12,24c-0.6,0-1-0.4-1-1v-2c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,23.6,12.6,24,12,24z"></path><path d="M5.6,6.6c-0.3,0-0.5-0.1-0.7-0.3L3.5,4.9c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C6.2,6.5,5.9,6.6,5.6,6.6z"></path><path d="M19.8,20.8c-0.3,0-0.5-0.1-0.7-0.3l-1.4-1.4c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C20.3,20.7,20,20.8,19.8,20.8z"></path><path d="M3,13H1c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S3.6,13,3,13z"></path><path d="M23,13h-2c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S23.6,13,23,13z"></path><path d="M4.2,20.8c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C4.7,20.7,4.5,20.8,4.2,20.8z"></path><path d="M18.4,6.6c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C18.9,6.5,18.6,6.6,18.4,6.6z"></path></svg><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="moon" data-v-f31c77b6><path d="M12.1,22c-0.3,0-0.6,0-0.9,0c-5.5-0.5-9.5-5.4-9-10.9c0.4-4.8,4.2-8.6,9-9c0.4,0,0.8,0.2,1,0.5c0.2,0.3,0.2,0.8-0.1,1.1c-2,2.7-1.4,6.4,1.3,8.4c2.1,1.6,5,1.6,7.1,0c0.3-0.2,0.7-0.3,1.1-0.1c0.3,0.2,0.5,0.6,0.5,1c-0.2,2.7-1.5,5.1-3.6,6.8C16.6,21.2,14.4,22,12.1,22zM9.3,4.4c-2.9,1-5,3.6-5.2,6.8c-0.4,4.4,2.8,8.3,7.2,8.7c2.1,0.2,4.2-0.4,5.8-1.8c1.1-0.9,1.9-2.1,2.4-3.4c-2.5,0.9-5.3,0.5-7.5-1.1C9.2,11.4,8.1,7.7,9.3,4.4z"></path></svg><!--]--></span></span></button></div></div></div><div class="group" data-v-edb32f2a><div class="item social-links" data-v-edb32f2a><div class="VPSocialLinks social-links-list" data-v-edb32f2a data-v-aea42604><!--[--><a class="VPSocialLink no-icon" href="https://github.com/andatoshiki" aria-label="github" target="_blank" rel="noopener" data-v-aea42604 data-v-d53a05fb><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>GitHub</title><path d="M12 .297c-6.63 0-12 5.373-12 12 0 5.303 3.438 9.8 8.205 11.385.6.113.82-.258.82-.577 0-.285-.01-1.04-.015-2.04-3.338.724-4.042-1.61-4.042-1.61C4.422 18.07 3.633 17.7 3.633 17.7c-1.087-.744.084-.729.084-.729 1.205.084 1.838 1.236 1.838 1.236 1.07 1.835 2.809 1.305 3.495.998.108-.776.417-1.305.76-1.605-2.665-.3-5.466-1.332-5.466-5.93 0-1.31.465-2.38 1.235-3.22-.135-.303-.54-1.523.105-3.176 0 0 1.005-.322 3.3 1.23.96-.267 1.98-.399 3-.405 1.02.006 2.04.138 3 .405 2.28-1.552 3.285-1.23 3.285-1.23.645 1.653.24 2.873.12 3.176.765.84 1.23 1.91 1.23 3.22 0 4.61-2.805 5.625-5.475 5.92.42.36.81 1.096.81 2.22 0 1.606-.015 2.896-.015 3.286 0 .315.21.69.825.57C20.565 22.092 24 17.592 24 12.297c0-6.627-5.373-12-12-12"/></svg></a><a class="VPSocialLink no-icon" href="https://twitter.com/andatoshiki" aria-label="twitter" target="_blank" rel="noopener" data-v-aea42604 data-v-d53a05fb><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>Twitter</title><path d="M21.543 7.104c.015.211.015.423.015.636 0 6.507-4.954 14.01-14.01 14.01v-.003A13.94 13.94 0 0 1 0 19.539a9.88 9.88 0 0 0 7.287-2.041 4.93 4.93 0 0 1-4.6-3.42 4.916 4.916 0 0 0 2.223-.084A4.926 4.926 0 0 1 .96 9.167v-.062a4.887 4.887 0 0 0 2.235.616A4.928 4.928 0 0 1 1.67 3.148 13.98 13.98 0 0 0 11.82 8.292a4.929 4.929 0 0 1 8.39-4.49 9.868 9.868 0 0 0 3.128-1.196 4.941 4.941 0 0 1-2.165 2.724A9.828 9.828 0 0 0 24 4.555a10.019 10.019 0 0 1-2.457 2.549z"/></svg></a><!--]--></div></div></div><!--]--><!--]--></div></div></div><!--[--><!--]--><button type="button" class="VPNavBarHamburger hamburger" aria-label="mobile navigation" aria-expanded="false" aria-controls="VPNavScreen" data-v-e4fc9f8b data-v-305e35d4><span class="container" data-v-305e35d4><span class="top" data-v-305e35d4></span><span class="middle" data-v-305e35d4></span><span class="bottom" data-v-305e35d4></span></span></button></div></div></div></div><!----></header><div class="VPLocalNav reached-top" data-v-bb676018 data-v-6e4e13c1><button class="menu" aria-expanded="false" aria-controls="VPSidebarNav" data-v-6e4e13c1><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="menu-icon" data-v-6e4e13c1><path d="M17,11H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h14c0.6,0,1,0.4,1,1S17.6,11,17,11z"></path><path d="M21,7H3C2.4,7,2,6.6,2,6s0.4-1,1-1h18c0.6,0,1,0.4,1,1S21.6,7,21,7z"></path><path d="M21,15H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h18c0.6,0,1,0.4,1,1S21.6,15,21,15z"></path><path d="M17,19H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h14c0.6,0,1,0.4,1,1S17.6,19,17,19z"></path></svg><span class="menu-text" data-v-6e4e13c1>Menu</span></button><div class="VPLocalNavOutlineDropdown" style="--vp-vh:0px;" data-v-6e4e13c1 data-v-55e6f1d6><button data-v-55e6f1d6>Return to top</button><!----></div></div><aside class="VPSidebar" data-v-bb676018 data-v-9b5436d7><div class="curtain" data-v-9b5436d7></div><nav class="nav" id="VPSidebarNav" aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-9b5436d7><span class="visually-hidden" id="sidebar-aria-label" data-v-9b5436d7> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-9b5436d7><section class="VPSidebarItem level-0 collapsible has-active" data-v-9b5436d7 data-v-c543663c><div class="item" role="button" tabindex="0" data-v-c543663c><div class="indicator" data-v-c543663c></div><h2 class="text" data-v-c543663c>IPhO Formulas: JP Ver.</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-c543663c><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-c543663c><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-c543663c><!--[--><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-c543663c><!--[--><p class="text" data-v-c543663c>1: 数学</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-c543663c><!--[--><p class="text" data-v-c543663c>2: 一般的な推奨事</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-c543663c><!--[--><p class="text" data-v-c543663c>3: 運動学</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-c543663c><!--[--><p class="text" data-v-c543663c>4: 力学</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-c543663c><!--[--><p class="text" data-v-c543663c>5: 振動と波</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-c543663c><!--[--><p class="text" data-v-c543663c>6: 幾何光学，測光</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-c543663c><!--[--><p class="text" data-v-c543663c>7: 波動光学</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-c543663c><!--[--><p class="text" data-v-c543663c>8: 電気回路</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-c543663c><!--[--><p class="text" data-v-c543663c>9: 電磁気学</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-c543663c><!--[--><p class="text" data-v-c543663c>10: 熱力</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-c543663c><!--[--><p class="text" data-v-c543663c>11: 量子力学</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-c543663c><!--[--><p class="text" data-v-c543663c>12: Keplerの法則</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-c543663c data-v-c543663c><div class="item" data-v-c543663c><div class="indicator" data-v-c543663c></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-c543663c><!--[--><p class="text" data-v-c543663c>13: 相対性理論</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-bb676018 data-v-7787f055><div class="VPDoc has-sidebar has-aside" data-v-7787f055 data-v-eb54b406><!--[--><!--]--><div class="container" data-v-eb54b406><div class="aside" data-v-eb54b406><div class="aside-curtain" data-v-eb54b406></div><div class="aside-container" data-v-eb54b406><div class="aside-content" data-v-eb54b406><div class="VPDocAside" data-v-eb54b406 data-v-d3dab3d7><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" role="navigation" data-v-d3dab3d7 data-v-98b3e62f><div class="content" data-v-98b3e62f><div class="outline-marker" data-v-98b3e62f></div><div class="outline-title" role="heading" aria-level="2" data-v-98b3e62f>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-98b3e62f><span class="visually-hidden" id="doc-outline-aria-label" data-v-98b3e62f> Table of Contents for current page </span><ul class="root" data-v-98b3e62f data-v-1772882b><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-d3dab3d7></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://jsd.toshiki.dev/gh/andatoshiki/toshiki-notebook@master/assets/logo/sponsor/telegram.png"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-eb54b406><div class="content-container" data-v-eb54b406><!--[--><!--]--><!----><main class="main" data-v-eb54b406><div style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_9" data-v-eb54b406><div><h1 id="formulas-for-ipho-日本語版-section-9" tabindex="-1">Formulas for IPhO 日本語版: Section 9 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-9" aria-label="Permalink to &quot;Formulas for IPhO 日本語版: Section 9&quot;">​</a></h1><div><section class="border-b-1 border-[var(--vp-c-divider)] w-full border-b-solid mt-[24px] pb-[12px] flex gap-[12px] mb-[12px] flex-wrap max-w-[85%]"><div class="flex gap-[4px] items-center"><svg style="display:inline-block;" viewBox="0 0 16 16" width="1.2em" height="1.2em"><path fill="currentColor" d="M8 16A8 8 0 1 1 8 0a8 8 0 0 1 0 16Zm.847-8.145a2.502 2.502 0 1 0-1.694 0C5.471 8.261 4 9.775 4 11c0 .395.145.995 1 .995h6c.855 0 1-.6 1-.995c0-1.224-1.47-2.74-3.153-3.145Z"></path></svg> Author:<span>Anda Toshiki</span></div><!----><div class="flex gap-[4px] items-center"><svg style="display:inline-block;" viewBox="0 0 15 15" width="1.2em" height="1.2em"><path fill="currentColor" fill-rule="evenodd" d="M1.903 7.297c0 3.044 2.207 5.118 4.686 5.547a.521.521 0 1 1-.178 1.027C3.5 13.367.861 10.913.861 7.297c0-1.537.699-2.745 1.515-3.663c.585-.658 1.254-1.193 1.792-1.602H2.532a.5.5 0 0 1 0-1h3a.5.5 0 0 1 .5.5v3a.5.5 0 0 1-1 0V2.686l-.001.002c-.572.43-1.27.957-1.875 1.638c-.715.804-1.253 1.776-1.253 2.97Zm11.108.406c0-3.012-2.16-5.073-4.607-5.533a.521.521 0 1 1 .192-1.024c2.874.54 5.457 2.98 5.457 6.557c0 1.537-.699 2.744-1.515 3.663c-.585.658-1.254 1.193-1.792 1.602h1.636a.5.5 0 1 1 0 1h-3a.5.5 0 0 1-.5-.5v-3a.5.5 0 1 1 1 0v1.845h.002c.571-.432 1.27-.958 1.874-1.64c.715-.803 1.253-1.775 1.253-2.97Z" clip-rule="evenodd"></path></svg> Updated:<span>4 minutes ago</span></div><div class="flex gap-[4px] items-center"><svg style="display:inline-block;" viewBox="0 0 16 16" width="1.2em" height="1.2em"><path fill="currentColor" d="M9.293 0H4a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h8a2 2 0 0 0 2-2V4.707A1 1 0 0 0 13.707 4L10 .293A1 1 0 0 0 9.293 0zM9.5 3.5v-2l3 3h-2a1 1 0 0 1-1-1zM5.485 6.879l1.036 4.144l.997-3.655a.5.5 0 0 1 .964 0l.997 3.655l1.036-4.144a.5.5 0 0 1 .97.242l-1.5 6a.5.5 0 0 1-.967.01L8 9.402l-1.018 3.73a.5.5 0 0 1-.967-.01l-1.5-6a.5.5 0 1 1 .97-.242z"></path></svg> Words:<span>950</span></div><div class="flex gap-[4px] items-center"><svg style="display:inline-block;" viewBox="0 0 20 20" width="1.2em" height="1.2em"><path fill="currentColor" d="M10 0a10 10 0 1 0 10 10A10 10 0 0 0 10 0zm2.5 14.5L9 11V4h2v6l3 3z"></path></svg> Reading:<span>4 min</span></div></section></div><h2 id="_9-電磁気学" tabindex="-1">9: 電磁気学 <a class="header-anchor" href="#_9-電磁気学" aria-label="Permalink to &quot;9: 電磁気学&quot;">​</a></h2><h3 id="_9-1-coulomb-の法則" tabindex="-1">9.1: Coulomb の法則 <a class="header-anchor" href="#_9-1-coulomb-の法則" aria-label="Permalink to &quot;9.1: Coulomb の法則&quot;">​</a></h3><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><mi>k</mi><msub><mi>q</mi><mn>1</mn></msub><msub><mi>q</mi><mn>2</mn></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>U</mi><mo>=</mo><mi>k</mi><msub><mi>q</mi><mn>1</mn></msub><msub><mi>q</mi><mn>2</mn></msub><mi mathvariant="normal">/</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">F=k q_1 q_2 / r^2, U=k q_1 q_2 / r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> で, Kepler の法則が 使える (<a href="./12">Section 12 参照</a>).</li></ol><h3 id="_9-2-gauss-の法則" tabindex="-1">9.2: Gauss の法則 <a class="header-anchor" href="#_9-2-gauss-の法則" aria-label="Permalink to &quot;9.2: Gauss の法則&quot;">​</a></h3><ol start="2"><li><p>Gauss の法則 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∮</mo><mi mathvariant="bold-italic">B</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\oint \boldsymbol{B} \cdot \mathrm{d} \boldsymbol{S}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>,</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∮</mo><mi>ε</mi><mi mathvariant="bold-italic">E</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mi>Q</mi><mo separator="true">,</mo><mo>∮</mo><mi mathvariant="bold-italic">g</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">S</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>π</mi><mi>G</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">\oint \varepsilon \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{S}=Q, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{S}=-4 \pi G M </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ε</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord mathnormal">Q</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">g</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.10903em;">GM</span></span></span></span></span></p></li></ol><h3 id="_9-3-循環定理" tabindex="-1">9.3: 循環定理 <a class="header-anchor" href="#_9-3-循環定理" aria-label="Permalink to &quot;9.3: 循環定理&quot;">​</a></h3><ol start="3"><li><p>循環定理 :</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∮</mo><mi mathvariant="bold-italic">E</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mo>=</mo><mover accent="true"><mi mathvariant="normal">Φ</mi><mo>˙</mo></mover><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>∮</mo><mfrac><mrow><mi mathvariant="bold-italic">B</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi></mrow><mi>μ</mi></mfrac><mo>=</mo><mi>I</mi><mo separator="true">,</mo><mo>∮</mo><mi mathvariant="bold-italic">g</mi><mo>⋅</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\oint \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{l}=0(=\dot{\Phi}), \oint \frac{\boldsymbol{B} \cdot \mathrm{d} \boldsymbol{l}}{\mu}=I, \oint \boldsymbol{g} \cdot \mathrm{d} \boldsymbol{l}=0 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mopen">(</span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">Φ</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">g</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li></ol><h3 id="_9-4-電流素片により生じる磁束密度" tabindex="-1">9.4: 電流素片により生じる磁束密度 <a class="header-anchor" href="#_9-4-電流素片により生じる磁束密度" aria-label="Permalink to &quot;9.4: 電流素片により生じる磁束密度&quot;">​</a></h3><ol start="4"><li>電流素片により生じる磁束密度 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">B</mi><mo>=</mo><mfrac><mrow><mi>μ</mi><mi>I</mi></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">l</mi><mo>×</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\mathrm{d} \boldsymbol{B}=\frac{\mu I}{4 \pi} \frac{\mathrm{d} \boldsymbol{l} \times \boldsymbol{e}_r}{r^2} . </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.0088em;">l</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">.</span></span></span></span></span></p> したがって電流 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span> が流れる円形回路の中心では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 I}{2 r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_9-5-ローレンツ力" tabindex="-1">9.5: ローレンツ力 <a class="header-anchor" href="#_9-5-ローレンツ力" aria-label="Permalink to &quot;9.5: ローレンツ力&quot;">​</a></h3><ol start="5"><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi>e</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">E</mi><mo>+</mo><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="bold-italic">B</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi mathvariant="bold-italic">I</mi><mo>×</mo><mi mathvariant="bold-italic">B</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}=e(\boldsymbol{E}+\boldsymbol{v} \times \boldsymbol{B}), \boldsymbol{F}=\boldsymbol{I} \times \boldsymbol{B} l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7694em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.07778em;">I</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span>.</li></ol><h3 id="_9-6-gauss-の定理と循環定理より" tabindex="-1">9.6: Gauss の定理と循環定理より <a class="header-anchor" href="#_9-6-gauss-の定理と循環定理より" aria-label="Permalink to &quot;9.6: Gauss の定理と循環定理より&quot;">​</a></h3><ol start="6"><li>Gauss の定理と循環定理より：帯電した導線について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mfrac><mi>σ</mi><mrow><mn>2</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">E=\frac{\sigma}{2 \pi \varepsilon_0 r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1405em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span><span class="mord mtight"><span class="mord mathnormal mtight">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 電流が流れる導線について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>I</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>r</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 I}{2 \pi r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. 帯電した面について <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mfrac><mi>σ</mi><mrow><mn>2</mn><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">E=\frac{\sigma}{2 \varepsilon_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1405em;vertical-align:-0.4451em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight">ε</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4451em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 電流が流れる面につい て <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>i</mi></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">B=\frac{\mu_0 i}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2528em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9078em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. 一様に帯電した球殼（又は無限に長い円 筒）の内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, 軸に沿って表面に電流が流れる 円筒の内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">B=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> で一様に帯電, 又は一様 な電流 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">i</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6933em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">i</span></span></span></span></span></span> が流れる, 球 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi></mrow><annotation encoding="application/x-tex">(d=3) /</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mord">/</span></span></span></span> 円柱 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi></mrow><annotation encoding="application/x-tex">(d-2) /</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mord">/</span></span></span></span> 平面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d=1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> の内部で,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">E</mi><mo>=</mo><mfrac><mi>ρ</mi><mrow><mi>ε</mi><mi>d</mi></mrow></mfrac><mi mathvariant="bold-italic">r</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">B</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>μ</mi><mi>d</mi></mrow></mfrac><mi mathvariant="bold-italic">i</mi><mo>×</mo><mi mathvariant="bold-italic">r</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{E}=\frac{\rho}{\varepsilon d} \boldsymbol{r}, \boldsymbol{B}=\frac{1}{\mu d} \boldsymbol{i} \times \boldsymbol{r} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ε</span><span class="mord mathnormal">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ρ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2019em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="mord mathnormal">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol">i</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span></span></span></span></span></p></li></ol><h3 id="_9-7-長いソレノイド" tabindex="-1">9.7: 長いソレノイド <a class="header-anchor" href="#_9-7-長いソレノイド" aria-label="Permalink to &quot;9.7: 長いソレノイド&quot;">​</a></h3><ol start="7"><li>長いソレノイド: 内部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mi>μ</mi><mi>n</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">B=\mu n I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>, 外部で <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">B=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. 磁束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>=</mo><mi>N</mi><mi>B</mi><mi>S</mi><mrow><mo fence="true">(</mo><mi>n</mi><mo>=</mo><mfrac><mi>N</mi><mi>l</mi></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\Phi=N B S\left(n=\frac{N}{l}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2223em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">NBS</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span>. インダクタンス <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">L=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">/</mi><mi>I</mi><mo>=</mo><mi>μ</mi><msup><mi>n</mi><mn>2</mn></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">\Phi / I=\mu n^2 V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ/</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>. 短いソレノイド <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><msub><mi>B</mi><mi mathvariant="normal">∥</mi></msub><mo>=</mo><mfrac><mrow><mi>μ</mi><mi>n</mi><mi>I</mi><mi mathvariant="normal">Ω</mi></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">: B_{\|}=\frac{\mu n I \Omega}{4 \pi}(\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0385em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∥</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2694em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9244em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span><span class="mord mtight">Ω</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord">Ω</span></span></span></span> は 立体角).</li></ol><h3 id="_9-8-磁場を小型コイルや衝撃検流計で測定する" tabindex="-1">9.8: 磁場を小型コイルや衝撃検流計で測定する <a class="header-anchor" href="#_9-8-磁場を小型コイルや衝撃検流計で測定する" aria-label="Permalink to &quot;9.8: 磁場を小型コイルや衝撃検流計で測定する&quot;">​</a></h3><ol start="9"><li>磁場を小型コイルや衝撃検流計で測定する： <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">q=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mfrac><mi>V</mi><mi>R</mi></mfrac><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>=</mo><mi>N</mi><mi>S</mi><mi mathvariant="normal">Δ</mi><mi>B</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">\int \frac{V}{R} \mathrm{~d} t=N S \Delta B / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">NS</span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>.</li></ol><h3 id="_9-9-静電場のエネルギー" tabindex="-1">9.9: 静電場のエネルギー <a class="header-anchor" href="#_9-9-静電場のエネルギー" aria-label="Permalink to &quot;9.9: 静電場のエネルギー&quot;">​</a></h3><ol start="9"><li>静電場のエネルギー：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>U</mi><mo>=</mo><mi>k</mi><munder><mo>∑</mo><mrow><mi>i</mi><mo>&lt;</mo><mi>j</mi></mrow></munder><mfrac><mrow><msub><mi>q</mi><mi>i</mi></msub><msub><mi>q</mi><mi>j</mi></msub></mrow><msub><mi>r</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>q</mi><mo separator="true">,</mo><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>q</mi><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">U=k \sum_{i&lt;j} \frac{q_i q_j}{r_{i j}}=\frac{1}{2} \int \phi(\boldsymbol{r}) \mathrm{d} q, \mathrm{~d} q=\rho(\boldsymbol{r}) \mathrm{d} V </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5213em;vertical-align:-1.4138em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">&lt;</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2222em;vertical-align:-0.8622em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ρ</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span></span></p></li></ol><h3 id="_9-10-一様に帯電した球面や円筒面の各部分の間に働く力" tabindex="-1">9.10: 一様に帯電した球面や円筒面の各部分の間に働く力 <a class="header-anchor" href="#_9-10-一様に帯電した球面や円筒面の各部分の間に働く力" aria-label="Permalink to &quot;9.10: 一様に帯電した球面や円筒面の各部分の間に働く力&quot;">​</a></h3><ol start="10"><li>一様に帯電した球面や円筒面の各部分の間に働く力 : 帯電による力を静水圧による力に置き換える.</li></ol><h3 id="_9-11-全ての電荷" tabindex="-1">9.11: 全ての電荷 <a class="header-anchor" href="#_9-11-全ての電荷" aria-label="Permalink to &quot;9.11: 全ての電荷&quot;">​</a></h3><ol start="12"><li>全ての電荷が距離 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> にある場合（例えば，不均一に帯 電した球やリングの中心） <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mi>ϕ</mi><mo>=</mo><mi>k</mi><mi>Q</mi><mi mathvariant="normal">/</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\phi \phi=k Q / r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">Q</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span></li></ol><h3 id="_9-12-外部電荷" tabindex="-1">9.12: 外部電荷 <a class="header-anchor" href="#_9-12-外部電荷" aria-label="Permalink to &quot;9.12: 外部電荷&quot;">​</a></h3><ol start="13"><li>外部電荷によって引き起こされる正味の電荷（又は電 位）を求めるには, 電荷を「出現」させて問題を対称的 にし，重ね合わせの原理を用いる.</li></ol><h3 id="_9-14-導体" tabindex="-1">9.14: 導体 <a class="header-anchor" href="#_9-14-導体" aria-label="Permalink to &quot;9.14: 導体&quot;">​</a></h3><ol start="14"><li>導体は電荷や電場を遮蔽する．例えば，中空の球体の 内部の電荷分布は外から見えない（あたかも <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span> という 電荷を持った導電性の球があるように見える).</li></ol><h3 id="_9-15-静電容量" tabindex="-1">9.15: 静電容量 <a class="header-anchor" href="#_9-15-静電容量" aria-label="Permalink to &quot;9.15: 静電容量&quot;">​</a></h3><ol start="15"><li>静電容量: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mo>=</mo><mi>ε</mi><mi>S</mi><mi mathvariant="normal">/</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">C=\varepsilon S / d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">εS</span><span class="mord">/</span><span class="mord mathnormal">d</span></span></span></span> (平板), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi>π</mi><mi>ε</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">4 \pi \varepsilon r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal">ε</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> (球), <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi><mi>ε</mi><mi>l</mi><mo stretchy="false">(</mo><mi>ln</mi><mo>⁡</mo><mi>R</mi><mi mathvariant="normal">/</mi><mi>r</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2 \pi \varepsilon l(\ln R / r)^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.01968em;">εl</span><span class="mopen">(</span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> (同軸円筒).</li></ol><h3 id="_9-16-双極子モーメント" tabindex="-1">9.16: 双極子モーメント <a class="header-anchor" href="#_9-16-双極子モーメント" aria-label="Permalink to &quot;9.16: 双極子モーメント&quot;">​</a></h3><ol start="16"><li><p>双極子モーメント：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="bold-italic">p</mi><mi>e</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>q</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>=</mo><mi>q</mi><mi mathvariant="bold-italic">d</mi><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">p</mi><mi>μ</mi></msub><mo>=</mo><mi>I</mi><mi mathvariant="bold-italic">S</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{p}_e=\sum q_i \boldsymbol{r}_i=q \boldsymbol{d}, \boldsymbol{p}_\mu=I \boldsymbol{S} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6886em;vertical-align:-0.2441em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.6em;vertical-align:-0.55em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.3802em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord"><span class="mord"><span class="mord boldsymbol">d</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05382em;">S</span></span></span></span></span></span></span></p></li></ol><h3 id="_9-17-双極子場" tabindex="-1">9.17: 双極子場 <a class="header-anchor" href="#_9-17-双極子場" aria-label="Permalink to &quot;9.17: 双極子場&quot;">​</a></h3><ol start="17"><li>双極子場 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo>=</mo><mi>k</mi><mi mathvariant="bold-italic">p</mi><mo>⋅</mo><msub><mi mathvariant="bold-italic">e</mi><mi>r</mi></msub><mi mathvariant="normal">/</mi><msup><mi>r</mi><mn>2</mn></msup><mo separator="true">,</mo><mi>E</mi><mo separator="true">,</mo><mi>B</mi><mo>∝</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\phi=k \boldsymbol{p} \cdot \boldsymbol{e}_r / r^2, E, B \propto r^{-3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></li></ol><h3 id="_9-18-双極子に働く力" tabindex="-1">9.18: 双極子に働く力 <a class="header-anchor" href="#_9-18-双極子に働く力" aria-label="Permalink to &quot;9.18: 双極子に働く力&quot;">​</a></h3><ol start="19"><li>双極子に働く力 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><msup><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">p</mi><mi>e</mi></msub><mo>⋅</mo><mi mathvariant="bold-italic">E</mi><mo fence="true">)</mo></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><mi>F</mi><mo>=</mo><msup><mrow><mo fence="true">(</mo><msub><mi mathvariant="bold-italic">p</mi><mi>μ</mi></msub><mo>⋅</mo><mi mathvariant="bold-italic">B</mi><mo fence="true">)</mo></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">F=\left(\boldsymbol{p}_e \cdot \boldsymbol{E}\right)^{\prime}, F=\left(\boldsymbol{p}_\mu \cdot \boldsymbol{B}\right)^{\prime}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1418em;vertical-align:-0.25em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.05451em;">E</span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8918em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.372em;vertical-align:-0.3802em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">p</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.0573em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3802em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9918em;"><span style="top:-3.3029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> [訳 者注 : ここの微分はむしろ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">grad</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\operatorname{grad}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mord mathrm">grad</span></span></span></span></span> である]. 2 つの双極 子間の相互作用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo><mi>F</mi><mo>∝</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow><annotation encoding="application/x-tex">: F \propto r^{-4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span>.</li></ol><h3 id="_9-19-磁気双極子としての点電荷" tabindex="-1">9.19: 磁気双極子としての点電荷 <a class="header-anchor" href="#_9-19-磁気双極子としての点電荷" aria-label="Permalink to &quot;9.19: 磁気双極子としての点電荷&quot;">​</a></h3><ol start="19"><li>磁気双極子としての点電荷 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>μ</mi></msub><mo>∝</mo><mi mathvariant="normal">Φ</mi><mo>∝</mo><msubsup><mi>v</mi><mo lspace="0em" rspace="0em">⊥</mo><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">p_\mu \propto \Phi \propto v_{\perp}^2 / B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0972em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">⊥</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> は断熱 不変量 (<a href="./4#_4-22-断熱不変量">Section 4: #22</a> 参照).</li></ol><h3 id="_9-20-鏡像法" tabindex="-1">9.20: 鏡像法 <a class="header-anchor" href="#_9-20-鏡像法" aria-label="Permalink to &quot;9.20: 鏡像法&quot;">​</a></h3><ol start="20"><li>鏡像法 : 接地された（磁石の場合は超電導の）平面が鏡 の役割をする. 接地された（又は孤立した）球体の場 は, 球体の内部にある 1 つ（又は 2 つ）の架空の電荷 のつくる場として求められる. 平面導波管（金属板の 間のスリット）内の場は, 電磁平面波の重ね合わせと して求められる.</li></ol><h3 id="_9-21-一様-電-場中の球-円柱-の分極" tabindex="-1">9.21: 一様（電）場中の球 (円柱) の分極 <a class="header-anchor" href="#_9-21-一様-電-場中の球-円柱-の分極" aria-label="Permalink to &quot;9.21: 一様（電）場中の球 (円柱) の分極&quot;">​</a></h3><ol start="21"><li>一様（電）場中の球 (円柱) の分極 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>+</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">(+\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">+</span><span class="mord mathnormal">ρ</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">-\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord mathnormal">ρ</span></span></span></span> に一 様に帯電した球 (円柱) の重ね合わせで， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>∝</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">d \propto E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span>.</li></ol><h3 id="_9-22-渦電流" tabindex="-1">9.22: 渦電流 <a class="header-anchor" href="#_9-22-渦電流" aria-label="Permalink to &quot;9.22: 渦電流&quot;">​</a></h3><ol start="22"><li>渦電流: 電流損失密度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><msup><mi>B</mi><mn>2</mn></msup><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mi>ρ</mi><mi mathvariant="normal">.</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">\approx B^2 v^2 / \rho .1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">ρ</span><span class="mord">.1</span></span></span></span> 回の通過で与え られる運動量 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mi>τ</mi><mo>≈</mo><msup><mi>B</mi><mn>2</mn></msup><msup><mi>a</mi><mn>3</mn></msup><mi>d</mi><mi mathvariant="normal">/</mi><mi>ρ</mi></mrow><annotation encoding="application/x-tex">F \tau \approx B^2 a^3 d / \rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord">/</span><span class="mord mathnormal">ρ</span></span></span></span> (ここで <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> は厚さ, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> は大きさ).</li></ol></div></div></main><footer class="VPDocFooter" data-v-eb54b406 data-v-d801cd4d><!--[--><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><!--]--><div class="edit-info" data-v-d801cd4d><div class="edit-link" data-v-d801cd4d><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/9.md" target="_blank" rel="noreferrer" data-v-d801cd4d><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" aria-label="edit icon" data-v-d801cd4d><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--></a></div><div class="last-updated" data-v-d801cd4d><p class="VPLastUpdated" data-v-d801cd4d data-v-43ed3de5>Last updated: <time datetime="2023-09-22T10:12:43.000Z" data-v-43ed3de5></time></p></div></div><nav class="prev-next" data-v-d801cd4d><div class="pager" data-v-d801cd4d><a class="pager-link prev" href="/academic/physics/ipho-formulas-jpn/8" data-v-d801cd4d><span class="desc" data-v-d801cd4d>Previous page</span><span class="title" data-v-d801cd4d>8: 電気回路</span></a></div><div class="pager" data-v-d801cd4d><a class="pager-link next" href="/academic/physics/ipho-formulas-jpn/10" data-v-d801cd4d><span class="desc" data-v-d801cd4d>Next page</span><span class="title" data-v-d801cd4d>10: 熱力</span></a></div></nav></footer><!--[--><!--[--><!--[--><div id="comment-container"></div><!--]--><!--]--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-bb676018 data-v-855b7db4><div class="container" data-v-855b7db4><p class="message" data-v-855b7db4>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and <i class="coffee fa fa-coffee fa-xs" aria-hidden="true"></i> by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code></p><p class="copyright" data-v-855b7db4>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a>, <a href="https://github.com/lolilab">LoliLab</a> and <a href="https://github.com/toshikidev">Toshiki Dev</a> present <br /><span id="siteruntime_span"></span></p></div></footer><!--[--><!--]--></div></div>
<script>window.__VP_HASH_MAP__=JSON.parse("{\"academic_chemistry_problems_03-02-3.md\":\"5c40792d\",\"academic_literature_index.md\":\"72bce40a\",\"academic_chemistry_problems_03-02-2.md\":\"26afa053\",\"academic_chemistry_problems_02-20.md\":\"7015a0e8\",\"academic_literature_writing_methods-of-development.md\":\"b70ad7bf\",\"academic_chemistry_notes_12-5.md\":\"4c698a37\",\"academic_chemistry_problems_03-02-1.md\":\"054898d2\",\"academic_physics_ipho-formulas-jpn_12.md\":\"b01af32f\",\"academic_chemistry_index.md\":\"ca0457b9\",\"academic_physics_ipho-formulas-jpn_1.md\":\"fa61cf48\",\"application_vitepress-plugin-shiki-twoslash_api_queries.md\":\"9bd11a96\",\"academic_physics_ipho-formulas-jpn_11.md\":\"5f6411f3\",\"academic_physics_ipho-formulas-jpn_10.md\":\"5252ef94\",\"academic_physics_ipho-formulas-jpn_6.md\":\"bc519d11\",\"academic_physics_ipho-formulas-jpn_3.md\":\"aadd6536\",\"academic_physics_index.md\":\"62b4f030\",\"academic_physics_ipho-formulas-jpn_13.md\":\"5d106b26\",\"academic_physics_ipho-formulas-jpn_2.md\":\"f9a6af54\",\"academic_physics_ipho-formulas-jpn_4.md\":\"67c0b4f9\",\"academic_physics_ipho-formulas-jpn_7.md\":\"353d535e\",\"development_aws_handson-qabot.md\":\"3c125433\",\"academic_physics_ipho-formulas-jpn_8.md\":\"f1df91b7\",\"academic_physics_ipho-formulas-jpn_9.md\":\"da4d857a\",\"academic_vocabulary_2023_02_2023-02-27.md\":\"06541670\",\"academic_vocabulary_index.md\":\"f4afb44a\",\"application_markdown-it-katex_how-to-use.md\":\"e6ab0018\",\"application_markdown-it-katex_tips.md\":\"4fa4c33c\",\"application_vitepress-plugin-shiki-twoslash_api_annotations.md\":\"105e6b7f\",\"application_vitepress-plugin-shiki-twoslash_api_cutting.md\":\"b6b7a28c\",\"application_vitepress-plugin-shiki-twoslash_api_emit.md\":\"8d757f81\",\"application_vitepress-plugin-shiki-twoslash_api_errors.md\":\"8032d46f\",\"application_vitepress-plugin-shiki-twoslash_api_includes.md\":\"d69b2a67\",\"application_vitepress-plugin-shiki-twoslash_api_logging.md\":\"d3641bab\",\"application_vitepress-plugin-shiki-twoslash_api_multi-file.md\":\"aea0802f\",\"application_vitepress-plugin-shiki-twoslash_config_flags.md\":\"a683705b\",\"application_vitepress-plugin-shiki-twoslash_config_reference.md\":\"42f9e642\",\"application_vitepress-plugin-shiki-twoslash_guide_custom-theme.md\":\"58ed95fc\",\"application_vitepress-plugin-shiki-twoslash_guide_markdown-extensions.md\":\"6731e737\",\"application_vitepress-plugin-shiki-twoslash_index.md\":\"7fc07d32\",\"development_aws_acknowledgement.md\":\"daba60ba\",\"development_aws_appendix.md\":\"568c3e80\",\"development_aws_author.md\":\"025b6bde\",\"development_aws_aws-get-started.md\":\"e1a56953\",\"development_aws_closing.md\":\"cf9dad64\",\"development_aws_cloud.md\":\"e679b136\",\"development_aws_docker-system.md\":\"4275fd21\",\"development_aws_handson-bashoutter.md\":\"d59bc53b\",\"development_aws_handson-ec2.md\":\"2a763e5b\",\"development_aws_handson-jupyter.md\":\"7f479319\",\"development_aws_index.md\":\"1f9e71a6\",\"development_aws_license.md\":\"5f519b50\",\"academic_physics_ipho-formulas-jpn_5.md\":\"6c32aa10\",\"development_aws_assignments.md\":\"35fe9703\",\"application_vitepress-plugin-shiki-twoslash_api_types.md\":\"9dbcbe1c\",\"development_aws_aws-batch.md\":\"de2d649a\",\"development_aws_main.md\":\"58c079df\",\"development_aws_scientific-computing.md\":\"1b192c5b\",\"development_aws_serverless.md\":\"715b8a8d\",\"development_aws_webserver.md\":\"441ce29f\",\"development_file-naming-convention.md\":\"ff928233\",\"development_rclone-for-r2.md\":\"72ca2dcc\",\"getting-started.md\":\"54a05997\",\"index.md\":\"bcdaa8d8\",\"javascript_notes_1_1-1.md\":\"444635ea\",\"javascript_notes_1_1-2.md\":\"cddd09d6\",\"jp_index.md\":\"5b353799\",\"roadmap.md\":\"d668d571\",\"save_reading_index.md\":\"d900b2c8\",\"save_reading_outliers_1.md\":\"64c11331\",\"save_reading_outliers_2.md\":\"74187620\",\"save_reading_outliers_3.md\":\"0cab1bb8\",\"save_reading_outliers_4.md\":\"d65868f7\",\"development_aws_handson-serverless.md\":\"330af1e8\",\"application_markdown-it-katex_support-function.md\":\"009fe258\",\"application_markdown-it-katex_support-table.md\":\"52b042b8\"}");window.__VP_SITE_DATA__=JSON.parse("{\"lang\":\"en-US\",\"dir\":\"ltr\",\"title\":\"Toshiki's Note\",\"description\":\"Toshiki's web notebook served via Vitepress!\",\"base\":\"/\",\"head\":[],\"appearance\":true,\"themeConfig\":{\"nav\":[{\"text\":\"Development\",\"link\":\"/development/\"},{\"text\":\"Academic\",\"items\":[{\"text\":\"K-12\",\"items\":[{\"text\":\"Chemistry\",\"link\":\"/academic/chemistry/index\",\"activeMatch\":\"/academic/chemistry/\"},{\"text\":\"Discrete Math.\",\"link\":\"/discrete-math/index\",\"activeMatch\":\"/categories/fragments/\"},{\"text\":\"Literature\",\"link\":\"/academic/literature/index\",\"activeMatch\":\"/academic/literature/\"}]},{\"text\":\"Tools\",\"items\":[{\"text\":\"Formulas for IPhO JPN.\",\"link\":\"/academic/physics/ipho-formulas-jpn/1\",\"activeMatch\":\"/academic/physics/ipho-formulas-jpn/\"}]},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"},{\"text\":\"\",\"link\":\"\",\"activeMatch\":\"\"}],\"activeMatch\":\"/academic/\"},{\"text\":\"Application\",\"items\":[{\"text\":\"Personal projects\",\"items\":[{\"text\":\"markdown-it-katex\",\"link\":\"/application/markdown-it-katex/how-to-use\",\"activeMatch\":\"/application/markdown-it-katex/\"},{\"text\":\"vitepress-plugin-shiki-twoslash\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/index\",\"activeMatch\":\"/application/vitepress-plugin-shiki-twoslash/index\"}]}],\"activeMatch\":\"/save/\"},{\"text\":\"Save\",\"items\":[{\"text\":\"Reading\",\"link\":\"/save/reading/index\",\"activeMatch\":\"/save/reading/\"},{\"text\":\"Vocabulary\",\"link\":\"/academic/vocabulary/index\",\"activeMatch\":\"/academic/vocabulary/\"}],\"activeMatch\":\"/save/\"}],\"sidebar\":{\"/development/\":[{\"text\":\"Notes & Issues\",\"collapsed\":false,\"items\":[{\"text\":\"File Naming Convention\",\"link\":\"/development/file-naming-convention\"},{\"text\":\"RClone for R2\",\"link\":\"/development/rclone-for-r2\"}]},{\"text\":\"コードで学ぶAWS入門\",\"collapsed\":false,\"items\":[{\"text\":\"背景\",\"link\":\"/development/aws/index\"},{\"text\":\"はじめに！\",\"link\":\"/development/aws/main\"},{\"text\":\"クラウド概論\",\"link\":\"/development/aws/cloud.md\"},{\"text\":\"AWS 入門\",\"link\":\"/development/aws/aws-get-started\"},{\"text\":\"Hands-on 1: 初めての EC2 インスタンスを起動する\",\"link\":\"/development/aws/handson-ec2.md\"},{\"text\":\"クラウドで行う科学計算・機械学習\",\"link\":\"/development/aws/scientific-computing.md\"},{\"text\":\"Hands-on 2: AWS でディープラーニングを実践\",\"link\":\"/development/aws/handson-ec2.md\"},{\"text\":\"Docker 入門\",\"link\":\"/development/aws/docker-system\"},{\"text\":\"Hands-on 3: AWS で自動質問回答ボットを走らせる\",\"link\":\"/development/aws/handson-qabot\"},{\"text\":\"Hands-on 4: AWS Batch を使って機械学習のハイパーパラメータサーチを並列化する\",\"link\":\"/development/aws/aws-batch\"},{\"text\":\"Web サービスの作り方\",\"link\":\"/development/aws/webserver\"},{\"text\":\"Serverless architecture\",\"link\":\"/development/aws/serverless\"},{\"text\":\"Hands-on 5: サーバーレス入門\",\"link\":\"/development/aws/handson-serverless\"},{\"text\":\"Hands-on 6: Bashoutter\",\"link\":\"/development/aws/handson-bashoutter\"},{\"text\":\"まとめ\",\"link\":\"/development/aws/closing\"},{\"text\":\"ppendix: 環境構築\",\"link\":\"/development/aws/appendix\"},{\"text\":\"謝辞\",\"link\":\"/development/aws/acknowledgement\"}]}],\"/academic/chemistry/\":[{\"text\":\"Textbook\",\"collapsed\":true,\"items\":[{\"text\":\"12-5: Reaction Mechanism\",\"link\":\"/academic/chemistry/notes/12-5\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"}]},{\"text\":\"Kinetics\",\"collapsed\":false,\"items\":[{\"text\":\"Rate determining steps\",\"link\":\"/academic/chemistry/notes/kinetics/rate-determining-step\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"}]},{\"text\":\"Problems & Solutions\",\"collapsed\":true,\"items\":[{\"text\":\"Problem: 02-20\",\"link\":\"/academic/chemistry/problems/02-20\"},{\"text\":\"Problem: 03-02-1\",\"link\":\"/academic/chemistry/problems/03-02-1\"},{\"text\":\"Problem: 03-02-2\",\"link\":\"/academic/chemistry/problems/03-02-2\"},{\"text\":\"Problem: 03-02-3\",\"link\":\"/academic/chemistry/problems/03-02-3\"}]}],\"/academic/physics\":[{\"text\":\"IPhO Formulas: JP Ver.\",\"collapsed\":false,\"items\":[{\"text\":\"1: 数学\",\"link\":\"/academic/physics/ipho-formulas-jpn/1\"},{\"text\":\"2: 一般的な推奨事\",\"link\":\"/academic/physics/ipho-formulas-jpn/2\"},{\"text\":\"3: 運動学\",\"link\":\"/academic/physics/ipho-formulas-jpn/3\"},{\"text\":\"4: 力学\",\"link\":\"/academic/physics/ipho-formulas-jpn/4\"},{\"text\":\"5: 振動と波\",\"link\":\"/academic/physics/ipho-formulas-jpn/5\"},{\"text\":\"6: 幾何光学，測光\",\"link\":\"/academic/physics/ipho-formulas-jpn/6\"},{\"text\":\"7: 波動光学\",\"link\":\"/academic/physics/ipho-formulas-jpn/7\"},{\"text\":\"8: 電気回路\",\"link\":\"/academic/physics/ipho-formulas-jpn/8\"},{\"text\":\"9: 電磁気学\",\"link\":\"/academic/physics/ipho-formulas-jpn/9\"},{\"text\":\"10: 熱力\",\"link\":\"/academic/physics/ipho-formulas-jpn/10\"},{\"text\":\"11: 量子力学\",\"link\":\"/academic/physics/ipho-formulas-jpn/11\"},{\"text\":\"12: Keplerの法則\",\"link\":\"/academic/physics/ipho-formulas-jpn/12\"},{\"text\":\"13: 相対性理論\",\"link\":\"/academic/physics/ipho-formulas-jpn/13\"}]}],\"/academic/vocabulary/\":[{\"text\":\"Vocabulary\",\"collapsed\":true,\"items\":[{\"text\":\"2023-02-27\",\"link\":\"/academic/vocabulary/2023/02/2023-02-27\"}]}],\"/academic/literature/\":[{\"text\":\"Writing Resources\",\"collapsed\":true,\"items\":[{\"text\":\"Patterns of Organization and Methods of Development\",\"link\":\"/academic/literature/writing/methods-of-development\"}]}],\"/javascript/\":[{\"text\":\"1: Basic JavaScript-Value, Variables, and Control Flow\",\"collapsed\":true,\"items\":[{\"text\":\"1-1: Numbers\",\"link\":\"/javascript/notes/1/1-1\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"},{\"text\":\"\",\"link\":\"\"}]}],\"/save/reading/\":[{\"text\":\"Outliers\",\"collapsed\":true,\"items\":[{\"text\":\"Introduction & Chapter 1: The Roseto Mystery\",\"link\":\"/save/reading/outliers/1\"},{\"text\":\"Chapter 2: The 10,000-Hour Rule\",\"link\":\"/save/reading/outliers/2\"},{\"text\":\"Chapter 3: The Trouble with Geniuses, Part 1\",\"link\":\"/save/reading/outliers/3\"},{\"text\":\"Chapter 4: The Trouble with Geniuses, Part 2\",\"link\":\"/save/reading/outliers/4\"}]}],\"/application/markdown-it-katex/\":[{\"text\":\"markdown-it-katex\",\"collapsed\":false,\"items\":[{\"text\":\"1: How to use?\",\"link\":\"/application/markdown-it-katex/how-to-use\"},{\"text\":\"2: KaTeX supported functions\",\"link\":\"/application/markdown-it-katex/support-function\"},{\"text\":\"3: KaTeX support tables\",\"link\":\"/application/markdown-it-katex/support-table\"},{\"text\":\"4: Tips\",\"link\":\"/application/markdown-it-katex/tips\"}]}],\"/application/vitepress-plugin-shiki-twoslash/\":[{\"text\":\"Guide\",\"collapsed\":false,\"items\":[{\"text\":\"Getting Started\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/\"},{\"text\":\"Markdown Extensions\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/guide/markdown-extensions\"},{\"text\":\"Using a Custom Theme\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/guide/custom-theme\"}]},{\"text\":\"Features\",\"collapsed\":false,\"items\":[{\"text\":\"Queries\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/queries\"},{\"text\":\"Errors\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/errors\"},{\"text\":\"Emit\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/emit\"},{\"text\":\"Cutting\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/cutting\"},{\"text\":\"Multi-file\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/multi-file\"},{\"text\":\"@types\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/types\"},{\"text\":\"Meta Annotations\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/annotations\"},{\"text\":\"Logging\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/logging\"},{\"text\":\"Includes\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/api/includes\"}]},{\"text\":\"Config\",\"collapsed\":false,\"items\":[{\"text\":\"Reference\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/config/reference\"},{\"text\":\"Compiler Flags\",\"link\":\"/application/vitepress-plugin-shiki-twoslash/config/flags\"}]}]},\"footer\":{\"copyright\":\"Copyright © 2023-2023 <a href=\\\"https://github.com/andatoshiki\\\">Anda Toshiki</a>, <a href=\\\"https://github.com/lolilab\\\">LoliLab</a> and <a href=\\\"https://github.com/toshikidev\\\">Toshiki Dev</a> present <br /><span id=\\\"siteruntime_span\\\"></span>\",\"message\":\"Wrote with <i class=\\\"heart fa fa-heart fa-xs fa-beat\\\"></i> and <i class=\\\"coffee fa fa-coffee fa-xs\\\" aria-hidden=\\\"true\\\"></i> by <a href=\\\"https://toshiki.dev\\\">Anda Toshiki</a> at <code>root@andatoshiki:/~</code>\"},\"logo\":\"/logos/logo.png\",\"outline\":\"deep\",\"outlineTitle\":\"TOC\",\"outlineBadges\":false,\"lastUpdatedText\":\"Last updated\",\"algolia\":{\"appId\":\"G9IUR45K98\",\"apiKey\":\"8528cc91281d8112b28f508317a96dd3\",\"indexName\":\"toshiki-notebook\"},\"editLink\":{\"pattern\":\"https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/:path\",\"text\":\"Edit this page on GitHub\"},\"socialLinks\":[{\"icon\":\"github\",\"link\":\"https://github.com/andatoshiki\"},{\"icon\":\"twitter\",\"link\":\"https://twitter.com/andatoshiki\"}]},\"locales\":{\"/\":{\"label\":\"English\",\"lang\":\"en-US\"},\"/jp/\":{\"label\":\"Japanese\",\"title\":\"Vue Test Utils\",\"lang\":\"jp-JP\",\"description\":\"La documentation officielle de Vue Test Utils\"}},\"scrollOffset\":90,\"cleanUrls\":true}");</script>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink