{"id":2669,"date":"2022-03-28T11:30:22","date_gmt":"2022-03-28T02:30:22","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2669"},"modified":"2024-07-08T15:06:36","modified_gmt":"2024-07-08T06:06:36","slug":"%e9%9d%99%e9%9b%bb%e5%a0%b4%ef%bc%9a%e7%82%b9%e9%9b%bb%e8%8d%b7%e3%81%ae%e9%9b%bb%e8%8d%b7%e5%af%86%e5%ba%a6%e3%81%a8%e3%83%87%e3%82%a3%e3%83%a9%e3%83%83%e3%82%af%e3%81%ae%e3%83%87%e3%83%ab%e3%82%bf","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e9%9d%99%e9%9b%bb%e5%a0%b4%ef%bc%9a%e7%82%b9%e9%9b%bb%e8%8d%b7%e3%81%ae%e9%9b%bb%e8%8d%b7%e5%af%86%e5%ba%a6%e3%81%a8%e3%83%87%e3%82%a3%e3%83%a9%e3%83%83%e3%82%af%e3%81%ae%e3%83%87%e3%83%ab%e3%82%bf\/","title":{"rendered":"\u9759\u96fb\u5834\uff1a\u70b9\u96fb\u8377\u306e\u96fb\u8377\u5bc6\u5ea6\u3068\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570"},"content":{"rendered":"

<\/p>\n

\u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377\u304c\u3064\u304f\u308b\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u89e3<\/h3>\n

\u3059\u3067\u306b\u300c\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306e\u89e3\u300d\u306e\u30da\u30fc\u30b8\u3067\uff0c\u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377 \\(q\\) \u304c\u3064\u304f\u308b\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f\uff0c\uff08\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u304b\u306a\u304f\u3066\u3082\uff09
\n$$ \\phi = \\frac{q}{4\\pi \\varepsilon_0} \\frac{1}{r}$$
\n\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u3068\u66f8\u3044\u305f\u3002<\/p>\n

 <\/p>\n

\u3053\u306e\u89e3\u306b\u5b9f\u969b\u306b\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50 \\(\\nabla^2\\) \u3092\u4f5c\u7528\u3055\u305b\u305f\u8a08\u7b97\u3092\u3057\u3066\u307f\u308b\u3068\uff0c
\n\\begin{eqnarray}
\n\\phi &=& \\frac{1}{4\\pi\\varepsilon_0} \\frac{1}{r}\\\\
\n\\nabla \\phi &=& – \\frac{1}{4\\pi\\varepsilon_0} \\frac{\\boldsymbol{r}}{r^3}\\\\
\n\\nabla^2 \\phi &=& \\nabla \\cdot (\\nabla\\phi) \\\\
\n&=& – \\frac{1}{4\\pi\\varepsilon_0}\\left(
\n\\frac{\\partial}{\\partial x} \\frac{x}{r^3} + \\frac{\\partial}{\\partial y} \\frac{y}{r^3} + \\frac{\\partial}{\\partial z} \\frac{z}{r^3}
\n\\right)
\n\\end{eqnarray}
\n\\(x\\) \u306b\u95a2\u3059\u308b\u504f\u5fae\u5206\u306e\u9805\u306f
\n\\begin{eqnarray}
\n\\frac{\\partial}{\\partial x} \\frac{x}{r^3} &=& \\frac{1}{r^3} + x \\frac{\\partial}{\\partial x} \\left(x^2 + y^2 + z^2\\right)^{-\\frac{3}{2}} \\\\
\n&=& \\frac{1}{r^3} + x\\cdot\\left(-\\frac{3}{2}\\right) \\left(x^2 + y^2 + z^2\\right)^{-\\frac{5}{2}}\\cdot 2 x\\\\
\n&=& \\frac{1}{r^3} – 3 \\frac{x^2}{r^5}
\n\\end{eqnarray}
\n\\(y, z\\) \u306b\u95a2\u3059\u308b\u504f\u5fae\u5206\u306e\u9805\u3082\u540c\u69d8\u306b\u8a08\u7b97\u3057\u3066
\n$$\\nabla^2 \\phi = – \\frac{1}{4\\pi\\varepsilon_0} \\left(3 \\frac{1}{r^3} – 3 \\frac{x^2 + y^2 + z^2}{r^5}\\right) = 0\\ \\ \\mbox{!!}$$
\n\u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6 \\(\\rho\\) \u306b\u5bfe\u3059\u308b\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f
\n$$ \\nabla^2 \\phi = – \\frac{\\rho}{\\varepsilon_0}$$\u306e\u89e3\u3092\u6c42\u3081\u305f\u306f\u305a\u306a\u306e\u306b\uff0c\u5b9f\u969b\u306b\u306f<\/p>\n

$$\\nabla^2 \\phi =0$$\u306e\u89e3\u306b\u306a\u3063\u3066\u3044\u308b\uff01\u3053\u308c\u306f\u3069\u3046\u3044\u3046\u3053\u3068\u306a\u306e\u3060\u308d\u3046\u304b\uff1f\u70b9\u96fb\u8377\u306e\u96fb\u8377\u5bc6\u5ea6 \\(\\rho\\) \u3068\u306f \\(\\rho = 0\\) \u306e\u3053\u3068\u306a\u306e\u3060\u308d\u3046\u304b\uff1f<\/p>\n

 <\/p>\n

\u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u3068\u306f\uff1f<\/h3>\n

\u4ee5\u4e0a\u306e\u8a71\u3092\u3082\u3046\u5c11\u3057\u6574\u7406\u3059\u308b\u3002<\/p>\n

\u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377 \\(q\\) \u304c\u3064\u304f\u308b\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f\uff0c\uff08\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u304b\u306a\u304f\u3066\u3082\uff09
\n$$ \\phi = \\frac{q}{4\\pi \\varepsilon_0} \\frac{1}{r}$$\u3067\u3042\u308b\u3068\u66f8\u3044\u305f\u3002\u5b9f\u306f\u3053\u3053\u306b\uff08\u81ea\u660e\u3067\u3042\u308b\u306e\u3067\u66f8\u304b\u306a\u304b\u3063\u305f\u304c\u91cd\u8981\u306a\uff09\u3042\u308b\u6761\u4ef6\u304c\u5b58\u5728\u3059\u308b\u3002\u305d\u308c\u306f<\/p>\n

\u300c\u3053\u306e\u89e3\u306f \\(r \\neq 0\\) \u306e\u5834\u5408\u306e\u89e3\u3067\u3042\u308b\u3053\u3068\u300d<\/p>\n

\u5f53\u305f\u308a\u524d\u306e\u3053\u3068\u3060\u304c\uff0c\\(r = 0\\) \u3067\u306f\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u306e\u3067\uff0c\\(r = 0\\) \u306e\u70b9\u306f\u9664\u5916\u3057\u305f\u89e3\u3067\u3042\u308b\u3002\u300c \u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377<\/strong><\/span>\u300d\u3068\u306f\uff0c\u5927\u304d\u3055\u304c\u7121\u8996\u3067\u304d\u308b\u3088\u3046\u306a\uff0c\\(r = 0\\) \u306e 1\u70b9\u306b\u306e\u307f\u5b58\u5728\u3059\u308b\u96fb\u8377<\/strong><\/span>\u306e\u3053\u3068\u3067\u3042\u308b\u304b\u3089\uff0c\u3053\u306e\u70b9\u96fb\u8377\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u306f\u539f\u70b9\u4ee5\u5916\u3067\u306f \\(\\rho = 0\\) \u3067\u3042\u308b\u3002<\/p>\n

\u3067\u306f\uff0c\u307e\u3055\u306b \\(r = 0\\) \u306e\u539f\u70b9\u306b\u70b9\u96fb\u8377\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u3068\u306f\u3069\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u306e\u304b\u3002\u305d\u306e\u305f\u3081\u306b\uff0c\uff08\u3068\u3066\u3082\u666e\u901a\u3068\u306f\u601d\u3048\u306a\u3044\u95a2\u6570\u306a\u306e\u3067\uff09\u300c\u8d85\u95a2\u6570<\/strong><\/span>\u300d\u306e\u4e00\u7a2e\u3067\u3042\u308b\u300c\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570<\/strong><\/span>\u300d\u306e\u8aac\u660e\u3092\u3057\u3066\u304a\u304f\u3002<\/p>\n

1\u6b21\u5143\u306e\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570<\/h3>\n

\uff081\u5909\u6570 \\(x\\) \u306e\u307f\u306e\u95a2\u6570\u3068\u3044\u3046\u610f\u5473\u3067\uff09\u300c1\u6b21\u5143\u306e<\/strong><\/span>\u300d\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570<\/strong><\/span> \\(\\delta(x)\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002
\n$$ \\delta(x) = 0 \\quad\\mbox{for} \\ x \\neq 0$$
\n$$\\int_{-\\infty}^{\\infty} f(x) \\delta(x) \\,dx = f(0)$$
\n\u307e\u305f\u306f\uff0c\u4e00\u822c\u306b
\n$$\\int_{-\\infty}^{\\infty} f(x) \\delta(x -x_1) \\,dx = f(x_1)$$
\n\u7279\u306b $$\\int_{-\\infty}^{\\infty} \\delta(x) \\,dx = 1$$<\/p>\n

\\(x = 0\\)\u300c\u4ee5\u5916\u306e<\/strong><\/u><\/span>\u300d\u5168\u3066\u306e\u70b9\u3067 \\(\\delta(x) = 0\\) \u3067\u3042\u308a\uff0c\u3057\u304b\u3082\u5168\u533a\u9593\u3067\u7a4d\u5206\u3057\u305f\u3089 \\(1\\) \u3068\u3044\u3046\u6709\u9650\u306e\u5024\u304c\u51fa\u308b\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u666e\u901a\u306b\u8003\u3048\u305f\u3089 \\(\\delta(0) \\rightarrow \\infty \\mbox{??}\\) \u3068\u306a\u308b\u3088\u3046\u306a\u4e0d\u601d\u8b70\u306a\u95a2\u6570\u3002<\/p>\n

3\u6b21\u5143\u306e\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570<\/h3>\n

\uff08\\(x, y, z\\) \u306e3\u5909\u6570\u306e\u95a2\u6570\u3068\u3044\u3046\u610f\u5473\u3067\uff09\u300c3\u6b21\u5143\u306e<\/strong><\/span>\u300d\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570 \\(\\delta^3(x)\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u3002
\n$$\\delta^3(\\boldsymbol{r}) \\equiv \\delta(x) \\delta(y) \\delta(z)$$
\n$$\\delta^3(\\boldsymbol{r}) = 0 \\quad\\mbox{for} \\ \\boldsymbol{r}\u00a0 \\neq \\boldsymbol{0}$$<\/p>\n

\\begin{eqnarray}\\iiint_{-\\infty}^{\\infty} f(\\boldsymbol{r}) \\delta^3(\\boldsymbol{r})\\,dV &=&\u00a0 f(\\boldsymbol{0})
\n\\end{eqnarray}<\/p>\n

\u307e\u305f\u306f\uff0c\u4e00\u822c\u306b<\/p>\n

\\begin{eqnarray}\\iiint_{-\\infty}^{\\infty} \\boldsymbol{f}(\\boldsymbol{r}) \\delta^3(\\boldsymbol{r}-\\boldsymbol{r}_1)\\,dV &=&\u00a0 \\boldsymbol{f}(\\boldsymbol{r}_1)
\n\\end{eqnarray}<\/p>\n

 <\/p>\n

\u70b9\u96fb\u8377\u306e\u96fb\u8377\u5bc6\u5ea6\u3092\u30c7\u30eb\u30bf\u95a2\u6570\u3092\u4f7f\u3063\u3066\u8868\u3059<\/h3>\n

\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570\u3092\u4f7f\u3046\u3068\uff0c\u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377 \\(q\\) \u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u306f
\n$$ \\rho = q \\delta^3(\\boldsymbol{r})$$ \u3068\u306a\u308b\u3002\u3064\u307e\u308a\uff0c\u3053\u306e\u70b9\u96fb\u8377\u304c\u3064\u304f\u308b\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f
\n$$\\nabla^2 \\phi = – \\frac{1}{\\varepsilon_0} q \\delta^3(\\boldsymbol{r})$$\u306e\u89e3\u3067\u3042\u308a\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002
\n$$ \\phi = \\frac{q}{4\\pi\\varepsilon_0} \\frac{1}{r}$$<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2561,"menu_order":22,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2669","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t