<\/p>\n
$$ \\nabla^2 \\phi = – \\frac{\\rho}{\\varepsilon_0}, \\quad \\boldsymbol{E} = – \\nabla \\phi $$
\n\u307e\u305a\uff0c\u96fb\u8377\u5206\u5e03 \\(\\rho(\\boldsymbol{r})\\) \u306b\u3088\u3063\u3066\u3064\u304f\u3089\u308c\u308b\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092\uff0c\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306b\u3088\u3063\u3066\u6c42\u3081\uff0c\u6c42\u3081\u305f\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u52fe\u914d\u3092\u3068\u3063\u3066\u30de\u30a4\u30ca\u30b9\u3092\u3064\u3051\u308b\u3068\uff0c\u96fb\u5834\u304c\u6c42\u307e\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\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
\u306a\u305c\u304b\u3068\u3044\u3046\u3068\uff0c\u3059\u3067\u306b\u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377\u304c\u3064\u304f\u308b\u96fb\u5834 \\(\\boldsymbol{E}\\) \u306f
\n$$ \\boldsymbol{E}\u00a0 =\\frac{q}{4\\pi \\varepsilon_0}\\frac{\\boldsymbol{r}}{r^3}$$\u3067\u3042\u308b\u3053\u3068\u304c\u30af\u30fc\u30ed\u30f3\u306e\u6cd5\u5247\u304b\u3089\u308f\u304b\u3063\u3066\u3044\u308b\u304b\u3089\u3060\u306d\u3002<\/p>\n
\u4e00\u65b9\uff0c\u4e0a\u8a18\u306e\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u52fe\u914d\u306b\u8ca0\u53f7\u3092\u3064\u3051\u305f\u91cf\u3092\u8a08\u7b97\u3059\u308b\u3068
\n$$- \\nabla\\phi = – \\frac{q}{4\\pi \\varepsilon_0} \\nabla \\frac{1}{r} = \\frac{q}{4\\pi \\varepsilon_0}\\frac{\\boldsymbol{r}}{r^3} = \\boldsymbol{E}$$\u3068\u306a\u308b\u304b\u3089\u3002<\/p>\n
\u5177\u4f53\u7684\u306b \\(\\displaystyle \\nabla\\frac{1}{r}\\) \u306e \\(x\\) \u6210\u5206\u3092\u8a08\u7b97\u3059\u308b\u3068\uff0c
\n\\begin{eqnarray}
\n\\frac{\\partial}{\\partial x} \\frac{1}{r} &=& \\frac{\\partial}{\\partial x} \\left(x^2 + y^2 + z^2 \\right)^{-\\frac{1}{2}}\\\\
\n&=& -\\frac{1}{2} \\left(x^2 + y^2 + z^2 \\right)^{-\\frac{3}{2}} \\cdot 2 x \\\\
\n&=& – \\frac{x}{r^3}
\n\\end{eqnarray}\\(y\\) \u6210\u5206\uff0c\\(z\\) \u6210\u5206\u3082\u540c\u69d8\u306b\u8a08\u7b97\u3067\u304d\u3066
\n$$ \\nabla \\frac{1}{r} = – \\frac{\\boldsymbol{r}}{r^3}$$<\/p>\n
\\(\\boldsymbol{r} = \\boldsymbol{r}_1\\) \u306b\u304a\u3044\u305f\u70b9\u96fb\u8377 \\(q_1\\) \u304c\u3064\u304f\u308b\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f\uff0c\u70b9\u96fb\u8377\u307e\u3067\u306e\u8ddd\u96e2\u304c $r \\ \\rightarrow \\ |\\boldsymbol{r} – \\boldsymbol{r}_1|$ \u3068\u5909\u308f\u308b\u306e\u3067\uff0c
\n$$ \\phi(\\boldsymbol{r}) = \\frac{1}{4\\pi \\varepsilon_0} \\frac{q_1}{|\\boldsymbol{r} – \\boldsymbol{r}_1|}$$<\/p>\n
\u4f4d\u7f6e \\(\\boldsymbol{r}_i\\) \u306b\u305d\u308c\u305e\u308c\u304a\u3044\u305f\u8907\u6570\u306e\u70b9\u96fb\u8377 \\(q_i\\) \uff08\\(i = 1, 2, 3, \\dots\\)\uff09\u304c\u3064\u304f\u308b\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f\uff0c\u91cd\u306d\u5408\u308f\u305b\u306e\u539f\u7406\u304b\u3089
\n$$\\phi(\\boldsymbol{r}) = \\frac{1}{4\\pi\\varepsilon_0} \\sum_{i} \\frac{q_i}{|\\boldsymbol{r} – \\boldsymbol{r}_i|}$$<\/p>\n
\u9023\u7d9a\u7684\u306a\u96fb\u8377\u5206\u5e03 \\(\\rho(\\boldsymbol{r})\\) \u306e\u5834\u5408\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7f6e\u304d\u63db\u3048\u3092\u3057\u3066
\n\\begin{eqnarray}
\n\\boldsymbol{r}_i &\\rightarrow& \\boldsymbol{r}’ \\\\
\nq_i &\\rightarrow& \\rho(\\boldsymbol{r}_i ) dV_i \\rightarrow\\rho(\\boldsymbol{r}’ ) dV_i\\\\
\n\\sum_{i} dV_i&\\rightarrow& \\iiint \\, dV’
\n\\end{eqnarray}
\n\\begin{eqnarray}
\n\\phi(\\boldsymbol{r}) &=& \\frac{1}{4\\pi\\varepsilon_0} \\sum_{i} \\frac{q_i}{|\\boldsymbol{r} – \\boldsymbol{r}_i|}\\\\
\n&\\Downarrow& \\\\
\n\\phi(\\boldsymbol{r}) &=& \\frac{1}{4\\pi\\varepsilon_0} \\iiint \\frac{\\rho(\\boldsymbol{r}’)}{|\\boldsymbol{r} – \\boldsymbol{r}’|} dV’ \\\\
\n&=& \\frac{1}{4\\pi} \\iiint \\frac{\\rho(\\boldsymbol{r}’)}{\\varepsilon_0}\\frac{1}{|\\boldsymbol{r} – \\boldsymbol{r}’|} dV’
\n\\end{eqnarray}
\n\u3053\u308c\u304c\uff0c\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f
\n$$\\nabla^2\\phi = – \\frac{\\rho}{\\varepsilon_0}$$\u306e\u5b8c\u5168\u306a\u89e3\u3067\u3042\u308b\u3002<\/p>\n
\u96fb\u78c1\u6c17\u5b66\u306b\u304a\u3044\u3066\u306f\uff0c\u3053\u308c\u307e\u3067\u307e\u3068\u3081\u305f\u9759\u96fb\u5834\u4ee5\u5916\u306b\u3082\uff08\u30af\u30fc\u30ed\u30f3\u30b2\u30fc\u30b8\u6761\u4ef6\u4e0b\u3067\u306e\u9759\u78c1\u5834\u306e\u5834\u5408\u306b\u3082\uff09\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u304c\u51fa\u3066\u304f\u308b\u3002\u4e00\u822c\u306b\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306f
\n$$\\nabla^2\u00a0 u(\\boldsymbol{r}) = – f(\\boldsymbol{r})$$\u306e\u5f62\u306b\u66f8\u304b\u308c\u308b\u304c\uff0c\u65b9\u7a0b\u5f0f\u306e\u5f62\u304c\u540c\u3058\u306a\u3089\u89e3\u306e\u5f62\u3082\u540c\u3058\u3067\u3042\u308a\uff0c\u305f\u3060\u3061\u306b
\n$$ u(\\boldsymbol{r}) = \\frac{1}{4\\pi} \\iiint \\frac{f(\\boldsymbol{r}’)}{|\\boldsymbol{r} – \\boldsymbol{r}’|} dV’$$\u306e\u5f62\u306b\u89e3\u3051\u308b\u3002\u3053\u308c\u306f\u3059\u3050\u5f8c\u3067\uff08\u9759\u78c1\u5834\u306e\u9805\u3067\uff09\u4f7f\u3046\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3088\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2561,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2665","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