{"id":4460,"date":"2022-12-12T12:40:28","date_gmt":"2022-12-12T03:40:28","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=4460"},"modified":"2024-12-23T10:42:31","modified_gmt":"2024-12-23T01:42:31","slug":"%e9%9d%a2%e5%af%be%e7%a7%b0%e3%81%aa%e9%9b%bb%e8%8d%b7%e5%af%86%e5%ba%a6%e3%81%8c%e3%81%a4%e3%81%8f%e3%82%8b%e9%9b%bb%e5%a0%b4","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4460\/","title":{"rendered":"\u9762\u5bfe\u79f0\u306a\u96fb\u8377\u5bc6\u5ea6\u304c\u3064\u304f\u308b\u96fb\u5834"},"content":{"rendered":"

\u96fb\u78c1\u6c17\u5b66\u306e\u7df4\u7fd2\u554f\u984c\u3002\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u3092\u4f7f\u3063\u3066\uff0c\u9762\u5bfe\u79f0\u306a\u96fb\u8377\u5bc6\u5ea6\u304c\u3064\u304f\u308b\u96fb\u5834\u3092\u6c42\u3081\u308b\u3002<\/p>\n

<\/p>\n

\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u3092\u4f7f\u3063\u305f\u6c42\u3081\u65b9<\/h3>\n

$x = 0$ \u306e $yz$ \u5e73\u9762\u306b\u5bfe\u3057\u3066\u9762\u5bfe\u79f0\u306a\u96fb\u8377\u5bc6\u5ea6\u306f $\\rho(|x|)$ \u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u308b\u3002\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f<\/p>\n

$$\\nabla^2 \\phi = – \\frac{1}{\\varepsilon_0} \\rho(|x|)$$<\/p>\n

\u304b\u3089\uff0c\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb $\\phi$ \u3082\u307e\u305f\u9762\u5bfe\u79f0 $\\phi = \\phi(|x|)$ \u3068\u307f\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3060\u308d\u3046\u3002\u3053\u306e\u3068\u304d\uff0c$\\xi \\equiv \\sqrt{x^2} = |x|$ \u3068\u3057\u3066<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{E} &=& – \\nabla \\phi(\\xi) = – \\frac{d\\phi(\\xi)}{d\\xi} \\nabla \\xi\\\\
\n&\\equiv& E_{\\xi} (\\xi) \\nabla \\xi\\\\
\nE_x &=& E_{\\xi}(\\xi) \\frac{x}{|x|} \\\\
\nE_y &=& 0 \\\\
\nE_z &=& 0
\n\\end{eqnarray}<\/p>\n

\"\"<\/p>\n

\u6b21\u306b\uff0c\u56f3\u306e\u3088\u3046\u306a\u76f4\u65b9\u4f53\u3092 $V$ \u3068\u3057\u3066\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u3092\u4f53\u7a4d\u7a4d\u5206\u3059\u308b\u3068<\/p>\n

$$\\iiint_V (\\nabla\\cdot\\boldsymbol{E}) dV = \\frac{1}{\\varepsilon_0} \\iiint_V \\rho dV$$<\/p>\n

\u53f3\u8fba\u306e\u7a4d\u5206\u306f\uff0c\u3053\u306e\u4f53\u7a4d\u5185\u306e\u5168\u96fb\u8377 $Q_{\\xi}$ \u3067\u66f8\u3051\u308b\u3002<\/p>\n

$$\\iiint_V \\rho dV = \\int_{-x}^{x} \\rho(|x|) dx \\equiv Q_{\\xi}$$<\/p>\n

\u5de6\u8fba\u306f\u30ac\u30a6\u30b9\u306e\u5b9a\u7406\u306b\u3088\u308a\uff0c\u9ec4\u8272\u304f\u5857\u3089\u308c\u305f\u90e8\u5206\u3067\u306e\u9762\u7a4d\u7a4d\u5206\u306b\u5e30\u7740\u3057\u3066\u2026<\/p>\n

$$\\iiint_V (\\nabla\\cdot\\boldsymbol{E}) dV = \\int_S \\boldsymbol{E}\\cdot\\boldsymbol{n} dS = 2 E_{\\xi}$$<\/p>\n

\u4e21\u8fba\u3092\u7b49\u3057\u3044\u3068\u304a\u3044\u3066<\/p>\n

\\begin{eqnarray}
\n2 E_{\\xi} &=& \\frac{Q_{\\xi}}{\\varepsilon_0} \\\\
\nE_{\\xi} &=& \\frac{Q_{\\xi}}{2 \\varepsilon_0} \\\\
\n\\therefore\\ \\ E_x &=& \\frac{Q_{\\xi}}{2 \\varepsilon_0} \\frac{x}{|x|}
\n\\end{eqnarray}<\/p>\n

\n

\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u306b\u983c\u3089\u305a\u306b<\/p>\n

$$
\n\\boldsymbol{E} = \\frac{1}{4\\pi\\varepsilon_0}
\n\\iiint \\frac{\\rho(|x\u2019|) \\left(\\boldsymbol{r}-\\boldsymbol{r}\u2019\\right)}{|\\boldsymbol{r}-\\boldsymbol{r}\u2019|^3} dV\u2019
\n$$<\/p>\n

\u3092\u76f4\u63a5\u7a4d\u5206\u3057\u3066\u3082\u540c\u3058\u7b54\u3048\u304c\u6c42\u3081\u3089\u308c\u308b\u3002\u3053\u3061\u3089<\/a><\/strong><\/span>\u306b\u66f8\u3044\u3066\u3042\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u96fb\u78c1\u6c17\u5b66\u306e\u7df4\u7fd2\u554f\u984c\u3002\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u3092\u4f7f\u3063\u3066\uff0c\u9762\u5bfe\u79f0\u306a\u96fb\u8377\u5bc6\u5ea6\u304c\u3064\u304f\u308b\u96fb\u5834\u3092\u6c42\u3081\u308b\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[19],"tags":[],"class_list":["post-4460","post","type-post","status-publish","format-standard","hentry","category-19","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4460","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=4460"}],"version-history":[{"count":10,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4460\/revisions"}],"predecessor-version":[{"id":9967,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4460\/revisions\/9967"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=4460"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=4460"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=4460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}