{"id":2673,"date":"2022-03-28T11:39:48","date_gmt":"2022-03-28T02:39:48","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2673"},"modified":"2025-12-12T16:53:38","modified_gmt":"2025-12-12T07:53:38","slug":"%e9%9d%99%e9%9b%bb%e5%a0%b4%ef%bc%9a%e9%9b%bb%e8%8d%b7%e5%af%86%e5%ba%a6%e3%81%8b%e3%82%89%e7%9b%b4%e6%8e%a5%e9%9b%bb%e5%a0%b4%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b","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%e9%9b%bb%e8%8d%b7%e5%af%86%e5%ba%a6%e3%81%8b%e3%82%89%e7%9b%b4%e6%8e%a5%e9%9b%bb%e5%a0%b4%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/","title":{"rendered":"\u9759\u96fb\u5834\uff1a\u96fb\u8377\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u96fb\u5834\u3092\u6c42\u3081\u308b"},"content":{"rendered":"

\u30ac\u30a6\u30b9\u306e\u6cd5\u5247\u3092\u4f7f\u308f\u305a\u306b\uff0c\u96fb\u8377\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u96fb\u5834\u3092\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n

<\/p>\n

\u96fb\u8377\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u96fb\u5834\u3092\u6c42\u3081\u308b\u5f0f<\/h3>\n

\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f<\/p>\n

$$\\nabla^2\\phi = -\\frac{\\rho}{\\varepsilon_0}$$\u306e\u5b8c\u5168\u306a\u89e3<\/p>\n

$$\\phi(\\boldsymbol{r}) = \\frac{1}{4\\pi \\varepsilon_0} \\iiint \\frac{\\rho(\\boldsymbol{r}’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|} dV’$$<\/p>\n

\u304b\u3089\u3042\u3089\u305f\u3081\u3066\u70b9\u96fb\u8377\u304c\u3064\u304f\u308b\u96fb\u5834\u3092\u6c42\u3081\u3066\u307f\u308b\u3002\u96fb\u5834\u306f<\/p>\n

\\begin{eqnarray}
\\boldsymbol{E} &=& -\\nabla\\phi \\\\
&=&-\\frac{1}{4\\pi \\varepsilon_0} \\iiint {\\rho(\\boldsymbol{r}’)}\\nabla\\left(\\frac{1}{|\\boldsymbol{r} -\\boldsymbol{r}’|} \\right)dV’ \\\\
&=& \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\boldsymbol{r}’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’
\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\u4ee5\u4e0b\u306e\u7d50\u679c\u3092\u4f7f\u3063\u305f\u3002<\/p>\n

$$ -\\nabla\\left(\\frac{1}{|\\boldsymbol{r} -\\boldsymbol{r}’|} \\right) = \\frac{ (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}$$<\/p>\n

\u4ee5\u4e0b\u3067\u793a\u3059\u3088\u3046\u306b\uff0c\u70b9\u96fb\u8377\u306e\u5834\u5408\u306b\u306f\u30af\u30fc\u30ed\u30f3\u306e\u6cd5\u5247\u306b\u5e30\u7740\u3059\u308b\u3053\u3068\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\\boldsymbol{E} &=&\u00a0 \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\boldsymbol{r}’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’
\\end{eqnarray}<\/p>\n

\u306f\u96fb\u8377\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u96fb\u5834\u3092\u6c42\u3081\u308b\u5f0f\u3067\u3042\u308a\uff0c\u9023\u7d9a\u7684\u306a\u96fb\u8377\u5bc6\u5ea6 \\(\\rho\\) \u306e\u5834\u5408\u306e\u4e00\u822c\u5316\u3055\u308c\u305f\u30af\u30fc\u30ed\u30f3\u306e\u6cd5\u5247\u3068\u547c\u3076\u3053\u3068\u304c\u3067\u304d\u308b\u3002\uff08\u9759\u78c1\u5834\u306e\u5834\u5408\u306b\u306f\u30d3\u30aa\u30fb\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u3068\u3044\u3046\u540d\u524d\u304c\u3064\u3044\u3066\u3044\u308b\u304c\uff0c\u9759\u96fb\u5834\u306e\u5834\u5408\u306e\u3053\u306e\u5f0f\u306b\u306f\u540d\u524d\u304c\u3064\u3044\u3066\u3044\u306a\u3044\u3088\u3046\u3060\u3002\uff09<\/p>\n

\u4ee5\u4e0b\u3067\u306f\uff0c\u3044\u304f\u3064\u304b\u306e\u5834\u5408\u306e\u9759\u96fb\u5834\u3092\uff0c\u4e0a\u8a18\u306e\u5f0f\uff08\u4e00\u822c\u5316\u3055\u308c\u305f\u30af\u30fc\u30ed\u30f3\u306e\u6cd5\u5247\uff09\u304b\u3089\u5168\u3066\u7d71\u4e00\u7684\u306b\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n

\u70b9\u96fb\u8377\u306e\u96fb\u8377\u5bc6\u5ea6\u3068\u96fb\u5834<\/h3>\n

\u539f\u70b9\u306b\u304a\u3044\u305f\u70b9\u96fb\u8377\u306e\u96fb\u8377\u5bc6\u5ea6\u306f\uff0c\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n

$$\\rho(\\boldsymbol{r}) = q\\,\\delta^3(\\boldsymbol{r})$$<\/p>\n

\u3053\u308c\u3092\u96fb\u5834\u306e\u5f0f\u306b\u5165\u308c\u3066\u3084\u308b\u3068<\/p>\n

\\begin{eqnarray}
\\boldsymbol{E}&=& \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\boldsymbol{r}’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’ \\\\
&=& \\frac{q}{4\\pi \\varepsilon_0} \\iiint\\frac{\\delta^3(\\boldsymbol{r}’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’\\\\
&=& \\frac{q}{4\\pi \\varepsilon_0} \\frac{ (\\boldsymbol{r} -\\boldsymbol{0}) }{|\\boldsymbol{r} -\\boldsymbol{0}|^3} \\\\
&=& \\frac{q}{4\\pi \\varepsilon_0} \\frac{ \\boldsymbol{r}\u00a0 }{r^3}
\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c3\u6b21\u5143\u306e\u30c7\u30a3\u30e9\u30c3\u30af\u306e\u30c7\u30eb\u30bf\u95a2\u6570 $\\delta^3(\\boldsymbol{r})$ \u306e\u6027\u8cea<\/p>\n

\\begin{eqnarray}
\n\\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

\u3092\u4f7f\u3044\uff0c$\\boldsymbol{r}_1 = \\boldsymbol{0}$ \u3068\u3057\u305f\u3002<\/p>\n

\"\"<\/a><\/p>\n

\"\"<\/a><\/p>\n

2\u3064\u306e\u70b9\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834<\/h3>\n

\u4f4d\u7f6e $\\boldsymbol{r}_1$ \u306b\u96fb\u8377 $q_1$\uff0c$\\boldsymbol{r}_2$ \u306b\u96fb\u8377 $q_2$ \u306e2\u3064\u306e\u96fb\u8377\u304c\u3042\u308b\u3068\u3059\u308b\u3068\uff0c\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$\\rho(\\boldsymbol{r}) = q_1 \\delta^3(\\boldsymbol{r} -\\boldsymbol{r}_1) + q_2 \\delta^3(\\boldsymbol{r} -\\boldsymbol{r}_2)$$<\/p>\n

\u306e\u3088\u3046\u306b\u66f8\u3051\uff0c\u96fb\u5834\u306f\uff08\u91cd\u306d\u5408\u308f\u305b\u306e\u539f\u7406<\/strong><\/span>\u304b\u3089\u3082\u308f\u304b\u308b\u3088\u3046\u306b\uff09<\/p>\n

$$\\boldsymbol{E} = \\frac{q_1}{4\\pi \\varepsilon_0} \\frac{ \\boldsymbol{r}\u00a0 -\\boldsymbol{r}_1}{|\\boldsymbol{r}\u00a0 -\\boldsymbol{r}_1|^3} + \\frac{q_2}{4\\pi \\varepsilon_0} \\frac{ \\boldsymbol{r}\u00a0 -\\boldsymbol{r}_2}{|\\boldsymbol{r}\u00a0 -\\boldsymbol{r}_2|^3}$$<\/p>\n

\"\"<\/a><\/p>\n

\"\"<\/a><\/h4>\n

\u91cd\u306d\u5408\u308f\u305b\u306e\u539f\u7406\u3068\u306f<\/h4>\n

\u3053\u3053\u3067\u3044\u3046\u300c\u91cd\u306d\u5408\u308f\u305b\u306e\u539f\u7406<\/strong><\/span>\u300d\u3068\u306f\uff0c\u96fb\u8377\u5bc6\u5ea6 $\\rho$ \u304c 2 \u3064\u306e\u96fb\u8377\u5bc6\u5ea6 $\\rho_1, \\rho_2$ \u306e\u91cd\u306d\u5408\u308f\u305b\uff08\u8db3\u3057\u7b97\uff09\u3067<\/p>\n

$$\\rho = \\rho_1 + \\rho_2$$<\/p>\n

\u3068\u66f8\u3051\u308b\u3068\u304d\u306b\uff0c$\\rho$ \u306b\u3088\u3063\u3066\u3064\u304f\u3089\u308c\u308b\u96fb\u5834 $\\boldsymbol{E}$ \u306f\uff0c$\\rho_1, \\rho_2$ \u304c\u305d\u308c\u305e\u308c\u72ec\u7acb\u306b\u3064\u304f\u308b\u96fb\u5834 $\\boldsymbol{E}_1, \\boldsymbol{E}_2$ \u306e\u91cd\u306d\u5408\u308f\u305b\uff08\u8db3\u3057\u7b97\uff09\u3067<\/p>\n

$$\\boldsymbol{E} = \\boldsymbol{E}_1 + \\boldsymbol{E}_2$$<\/p>\n

\u3068\u66f8\u3051\u308b\u3053\u3068\u3092\u3044\u3046\u3002\u8a3c\u660e\u306f\u7c21\u5358\u3067<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{E}_1 &=&\u00a0 \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho_1\\, (\\boldsymbol{r} -\\boldsymbol{r}\u2019) }{|\\boldsymbol{r} -\\boldsymbol{r}\u2019|^3}dV\u2019 \\\\
\n\\boldsymbol{E}_2 &=&\u00a0 \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho_2\\,(\\boldsymbol{r} -\\boldsymbol{r}\u2019) }{|\\boldsymbol{r} -\\boldsymbol{r}\u2019|^3}dV\u2019 \\\\
\n\\end{eqnarray}<\/p>\n

\u3067\u3042\u308c\u3070\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{E} &=&\u00a0 \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho\\, (\\boldsymbol{r} -\\boldsymbol{r}\u2019) }{|\\boldsymbol{r} -\\boldsymbol{r}\u2019|^3}dV\u2019 \\\\
\n&=&\u00a0 \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\left(\\rho_1 + \\rho_2\\right)\\,(\\boldsymbol{r} -\\boldsymbol{r}\u2019) }{|\\boldsymbol{r} -\\boldsymbol{r}\u2019|^3}dV\u2019 \\\\
\n&=& \\boldsymbol{E}_1 + \\boldsymbol{E}_2
\n\\end{eqnarray}<\/p>\n

\u96fb\u6c17\u53cc\u6975\u5b50\u306b\u3088\u308b\u96fb\u5834<\/h3>\n

\u6b63\u96fb\u8377 $q > 0$ \u3068\u5927\u304d\u3055\u306e\u7b49\u3057\u3044\u8ca0\u96fb\u8377 $ -q$ \u306e2\u3064\u306e\u96fb\u8377\u304c\u7121\u9650\u5c0f\u9593\u9694 $\\boldsymbol{d}$ \u3067\u5bfe\u306b\u306a\u3063\u3066\u3044\u308b\u72b6\u614b\u3092\u300c\u96fb\u6c17\u53cc\u6975\u5b50<\/strong><\/span>\u300d\u3068\u3044\u3046\u3002<\/p>\n

\"\"<\/p>\n

\u96fb\u6c17\u53cc\u6975\u5b50\u306b\u3088\u308b\u9060\u65b9\u306e\u96fb\u5834\u306f\uff0c\u6b63\u96fb\u8377 $q_1 = +q$ \u306e\u4f4d\u7f6e\u3092 $\\boldsymbol{r}_1=\\frac{\\boldsymbol{d}}{2}$\uff0c\u8ca0\u96fb\u8377 $q_2 = -q$ \u306e\u4f4d\u7f6e\u3092 $\\boldsymbol{r}_2=-\\frac{\\boldsymbol{d}}{2}$ \u3068\u3057\u3066<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{E} &=& \\frac{q}{4\\pi \\varepsilon_0} \\frac{ \\boldsymbol{r}\u00a0 -\\frac{1}{2}\\boldsymbol{d}}{|\\boldsymbol{r}\u00a0 -\\frac{1}{2}\\boldsymbol{d}|^3} –
\n\\frac{q}{4\\pi \\varepsilon_0} \\frac{ \\boldsymbol{r}\u00a0 + \\frac{1}{2}\\boldsymbol{d}}{| \\boldsymbol{r}\u00a0 + \\frac{1}{2}\\boldsymbol{d}|^3}
\n\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\uff0c$|\\boldsymbol{d}| \\ll r \\equiv \\sqrt{\\boldsymbol{r}\\cdot\\boldsymbol{r}} $ \u3067\u3042\u308b\u304b\u3089 $\\displaystyle\\frac{|\\boldsymbol{d}|^2}{r^2}$ \u306e\u9805\u3092\u7121\u8996\u3059\u308b\u8fd1\u4f3c\u3092\u304a\u3053\u306a\u3063\u3066\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n

\u307e\u305a\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{1}{|\\boldsymbol{r}\u00a0 \\mp \\frac{1}{2}\\boldsymbol{d}|^3} &=&
\n\\left\\{\\left( \\boldsymbol{r}\u00a0 \\mp \\frac{1}{2}\\boldsymbol{d}\\right)\\cdot\\left( \\boldsymbol{r}\u00a0 \\mp \\frac{1}{2}\\boldsymbol{d}\\right)\\right\\}^{-\\frac{3}{2}} \\\\
\n&=&
\n\\left( r^2 \\mp \\boldsymbol{r}\\cdot \\boldsymbol{d} + \\frac{1}{4} |\\boldsymbol{d}|^2
\n\\right)^{-\\frac{3}{2}} \\\\
\n&=&
\n\\left\\{ r^2\\left( 1\\mp \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{d}}{r^2} + \\frac{1}{4} \\frac{|\\boldsymbol{d}|^2}{r^2}
\n\\right)\\right\\}^{-\\frac{3}{2}} \\\\
\n&\\simeq& \\left\\{ r^2\\left( 1\\mp \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{d}}{r^2}
\n\\right)\\right\\}^{-\\frac{3}{2}} \\\\
\n&=&\\left( r^2\\right)^{-\\frac{3}{2}}
\n\\left( 1\\mp \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{d}}{r^2}
\n\\right)^{-\\frac{3}{2}} \\\\
\n&\\simeq& \\frac{1}{r^3} \\left(1\\pm \\frac{3}{2} \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{d} }{r^2} \\right)
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u3092\u4f7f\u3046\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{E} &=& \\frac{q}{4\\pi \\varepsilon_0} \\frac{ \\boldsymbol{r}\u00a0 -\\frac{1}{2}\\boldsymbol{d}}{|\\boldsymbol{r}\u00a0 -\\frac{1}{2}\\boldsymbol{d}|^3} –
\n\\frac{q}{4\\pi \\varepsilon_0} \\frac{ \\boldsymbol{r}\u00a0 + \\frac{1}{2}\\boldsymbol{d}}{| \\boldsymbol{r}\u00a0 + \\frac{1}{2}\\boldsymbol{d}|^3} \\\\
\n&\\simeq& \\frac{q}{4\\pi\\varepsilon_0} \\left\\{
\n\\left(\\boldsymbol{r}\u00a0 -\\frac{1}{2}\\boldsymbol{d} \\right)\\left(\\frac{1}{r^3} + \\frac{3}{2} \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{d} }{r^5} \\right) –
\n\\left(\\boldsymbol{r} + \\frac{1}{2}\\boldsymbol{d} \\right)\\left(\\frac{1}{r^3} -\\frac{3}{2} \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{d} }{r^5} \\right)
\n\\right\\} \\\\
\n&\\simeq& \\frac{q}{4\\pi\\varepsilon_0}
\n\\left\\{ 3 \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{d} }{r^5} \\boldsymbol{r} -\\frac{\\boldsymbol{d}}{r^3}\\right\\}
\n\\end{eqnarray}<\/p>\n

\u96fb\u6c17\u53cc\u6975\u30e2\u30fc\u30e1\u30f3\u30c8 $\\boldsymbol{p}$ \u3092<\/p>\n

$$\\boldsymbol{p} \\equiv q \\boldsymbol{d}$$
\n\u3068\u5b9a\u7fa9\u3059\u308b\u3068\uff0c\u3053\u306e\u96fb\u6c17\u53cc\u6975\u5b50\u304c\u3064\u304f\u308b\u9060\u65b9\u306e\u96fb\u5834\u306f<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{E}
\n&=& \\frac{1}{4\\pi\\varepsilon_0}
\n\\left\\{ 3 \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{p} }{r^5} \\boldsymbol{r} -\\frac{\\boldsymbol{p}}{r^3}\\right\\}
\n\\end{eqnarray}<\/p>\n

\u96fb\u6c17\u53cc\u6975\u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u5411\u304d\u3092 $z$ \u8ef8\u3068\u3057\uff0c$\\boldsymbol{p} = (0, 0, p)$ \u3068\u3057\u3066\u96fb\u5834 $\\boldsymbol{E}$ \u3092\u5404\u6210\u5206\u3054\u3068\u306b\u8868\u3059\u3068<\/p>\n

\\begin{eqnarray}
\nE_x &=& \\frac{ p}{4\\pi\\varepsilon_0}\\frac{3}{r^5} zx \\\\
\nE_y &=& \\frac{ p}{4\\pi\\varepsilon_0}\\frac{3}{r^5} zy \\\\
\nE_z &=& \\frac{ p}{4\\pi\\varepsilon_0}\\left\\{ \\frac{3}{r^5} z^2 -\\frac{1}{r^3} \\right\\}
\n\\end{eqnarray}<\/p>\n

\"\"<\/a><\/p>\n

\u4e00\u69d8\u306a\u7dda\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834<\/h3>\n

\"\"<\/a><\/p>\n

\\(z\\) \u8ef8\u4e0a\u306e\u4e00\u69d8\u306a\u7dda\u96fb\u8377\u5bc6\u5ea6 \\(\\lambda (\\mbox{C}\/\\mbox{m})\\) \u306f\uff0c\\(z\\) \u8ef8\u4e0a\u3064\u307e\u308a \\(x = 0, y = 0\\) \u306b\u3057\u304b\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u304b\u3089\uff0c\u96fb\u8377\u5bc6\u5ea6\u3068\u3057\u3066\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n

$$\\rho(\\boldsymbol{r}) = \\lambda \\delta(x) \\delta(y)$$<\/p>\n

\u3053\u308c\u3092\u96fb\u5834\u306e\u5f0f\u306b\u5165\u308c\u3066\u3084\u308b\u3068<\/p>\n

\\begin{eqnarray}
\\boldsymbol{E}&=& \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\boldsymbol{r}’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’ \\\\
&=& \\frac{\\lambda}{4\\pi \\varepsilon_0} \\iiint\\frac{\\delta(x’) \\delta(y’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’
\\end{eqnarray}<\/p>\n

\u5404\u6210\u5206\u3054\u3068\u306b\u8a08\u7b97\u3059\u308b\u3068\uff0c\\(x\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
E_x&=& \\frac{\\lambda}{4\\pi \\varepsilon_0} \\iiint\\frac{\\delta(x’) \\delta(y’) (x -x’) }{\\left((x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right)^{3\/2}}dx’ dy’ dz’ \\\\
&=& \\frac{\\lambda x}{4\\pi \\varepsilon_0} \\int_{-\\infty}^{\\infty}
\\frac{1}{\\left(x^2 + y^2 + (z-z’)^2\\right)^{3\/2}} dz’\\\\
&=& \\frac{\\lambda x }{4\\pi \\varepsilon_0} \\frac{2}{x^2 + y^2} \\\\
\n&=& \\frac{\\lambda\u00a0 }{2\\pi \\varepsilon_0} \\frac{x}{x^2 + y^2}
\\end{eqnarray}<\/p>\n

\\(y\\) \u6210\u5206\u3082\u540c\u69d8\u306b\uff0c<\/p>\n

\\begin{eqnarray}
E_y&=& \\frac{\\lambda}{4\\pi \\varepsilon_0} \\iiint\\frac{\\delta(x’) \\delta(y’) (y -y’) }{\\left((x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right)^{3\/2}}dx’ dy’ dz’ \\\\
&=& \\frac{\\lambda y}{4\\pi \\varepsilon_0} \\int_{-\\infty}^{\\infty}
\\frac{1}{\\left(x^2 + y^2 + (z-z’)^2\\right)^{3\/2}} dz’\\\\
&=& \\frac{\\lambda y}{4\\pi \\varepsilon_0} \\frac{2}{x^2 + y^2} \\\\
\n&=& \\frac{\\lambda }{2\\pi \\varepsilon_0} \\frac{y}{x^2 + y^2}
\\end{eqnarray}<\/p>\n

\\(z\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
E_z&=& \\frac{\\lambda}{4\\pi \\varepsilon_0} \\iiint\\frac{\\delta(x’) \\delta(y’) (z -z’) }{\\left((x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right)^{3\/2}}dx’ dy’ dz’ \\\\
&=& \\frac{\\lambda}{4\\pi \\varepsilon_0} \\int_{-\\infty}^{\\infty}
\\frac{z -z’}{\\left(x^2 + y^2 + (z-z’)^2\\right)^{3\/2}} dz’\\\\
&=&0
\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u3002<\/p>\n

\"\"\\(z\\) \u8ef8\u306b\u76f4\u4ea4\u3059\u308b\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3092 \\(\\boldsymbol{\\varrho} \\equiv (x, y, 0), \\ {\\varrho}^2 \\equiv \\boldsymbol{\\varrho} \\cdot\\boldsymbol{\\varrho} \\) \u3068\u3057\u3066\u307e\u3068\u3081\u308b\u3068<\/p>\n

$$\\boldsymbol{E} = \\frac{\\lambda}{2\\pi\\varepsilon_0} \\frac{\\boldsymbol{\\varrho}}{{\\varrho}^2}$$<\/p>\n

\u4ee5\u4e0a\u306e\u8a08\u7b97\u3067\u4f7f\u3063\u305f\u7a4d\u5206\u306f<\/p>\n

$$\\int_{-\\infty}^{\\infty}
\\frac{1}{\\left(x^2 + y^2 + (z-z’)^2\\right)^{3\/2}} dz’ =
\n\\int_{-\\infty}^{\\infty}
\\frac{1}{\\left(x^2 + y^2 + Z^2\\right)^{3\/2}} dZ =\u00a0 \\frac{2}{x^2 + y^2}$$<\/p>\n

\u304a\u3088\u3073<\/p>\n

$$\\int_{-\\infty}^{\\infty}
\\frac{z -z’}{\\left(x^2 + y^2 + (z-z’)^2\\right)^{3\/2}} dz’ = \\int_{-\\infty}^{\\infty}
\\frac{-Z}{\\left(x^2 + y^2 + Z^2\\right)^{3\/2}} dZ =0$$\uff08\u88ab\u7a4d\u5206\u95a2\u6570\u304c \\(Z \\equiv z’ -z\\) \u306b\u3064\u3044\u3066\u5947\u95a2\u6570\u3060\u304b\u3089\u7c21\u5358\u306b\u30bc\u30ed\uff01\uff09<\/p>\n

\u8ef8\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u96fb\u5834<\/h3>\n

\\(z\\) \u8ef8\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u8ef8\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$\\rho(\\boldsymbol{r}) = \\rho(\\sqrt{x^2 + y^2})\u00a0 $$<\/p>\n

\u3068\u66f8\u3051\u308b\u3002\\(z\\) \u306b\u306f\u4f9d\u5b58\u3057\u306a\u3044\u306e\u3067\uff0c\\(z\\) \u8ef8\u65b9\u5411\u306b\u306f\u4e00\u69d8\u3067\u3042\u308b\u3002\u7a4d\u5206\u5909\u6570 $x’, y’, z’$ \u3092\u5186\u7b52\u5ea7\u6a19 $\\varrho’, \\phi’, z’$ \u3067\u66f8\u304f\u3068\uff08\u672c\u6765\u3067\u3042\u308c\u3070 $z$ \u8ef8\u304b\u3089\u306e\u8ddd\u96e2 $\\sqrt{(x’)^2 + (y’)^2}$ \u306f\u6975\u5ea7\u6a19\u7cfb\u306b\u304a\u3051\u308b\u52d5\u5f84\u5ea7\u6a19 $r’$\u00a0 \u3068\u533a\u5225\u3059\u308b\u305f\u3081\u306b $\\rho’$\u00a0 \u306a\u3069\u3068\u66f8\u304d\u305f\u3044\u3068\u3053\u308d\u3060\u304b\uff0c\u3053\u308c\u3060\u3068\u96fb\u8377\u5bc6\u5ea6\u3068\u304b\u3076\u3063\u3066\u3057\u307e\u3046\u306e\u3067\u4ee5\u4e0b\u3067\u306f $\\varrho’ \\equiv\\sqrt{(x’)^2 + (y’)^2}$ \u3068\u3057\u3066\u3044\u308b\u3002\uff09<\/p>\n

\\begin{eqnarray}
\nx’ &=& \\varrho’ \\cos \\phi’ \\\\
\ny’ &=& \\varrho’ \\sin \\phi’\\\\
\nz’ &=& z’ \\\\
\ndV’ &=& \\varrho’ d\\varrho’\\,d\\phi’ \\,dz’
\n\\end{eqnarray}<\/p>\n

\u300c3\u6b21\u5143\u5186\u7b52\u5ea7\u6a19\u7cfb\u306e\u4f53\u7a4d\u8981\u7d20<\/a>\u300d\u3092\u53c2\u7167\u306e\u3053\u3068\u3002<\/p>\n

\u5404\u6210\u5206\u3054\u3068\u306b\u8a08\u7b97\u3059\u308b\u3068\uff0c\\(x\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
E_x&=&\\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\varrho’) (x -\\varrho’ \\cos\\phi’) }{\\left\\{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2 + (z-z’)^2\\right\\}^{3\/2}}dV’ \\\\
&=&\\frac{1}{4\\pi \\varepsilon_0}
\n\\!\\!\\int\\!\\! \\varrho’ d\\varrho’ \\rho(\\varrho’)
\n\\int\\!\\! d\\phi’ (x -\\varrho’ \\cos\\phi’)
\n\\!\\!\\int_{-\\infty}^{\\infty} \\frac{dz’ }{\\left\\{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2 + (z-z’)^2\\right\\}^{3\/2}} \\\\
&=& \\frac{1}{4\\pi \\varepsilon_0} \\!\\!\\int\\!\\! \\varrho’ d\\varrho’ \\rho(\\varrho’)\\int_{0}^{2\\pi}\\!\\! d\\phi’ (x -\\varrho’ \\cos\\phi’)\\frac{2\u00a0 }{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2}\\\\
\n&=& \\frac{1}{2\\pi \\varepsilon_0} \\!\\!\\int\\!\\! \\varrho’ d\\varrho’ \\rho(\\varrho’)\\int_{0}^{2\\pi}\\!\\! d\\phi’\\frac{ (x -\\varrho’ \\cos\\phi’) }{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2}\\\\
\n&=& \\frac{1}{2\\pi \\varepsilon_0} \\!\\!\\int_{0}^{\\infty}\\!\\! \\varrho’ d\\varrho’ \\rho(\\varrho’)\\, \\frac{2\\pi\u00a0 x}{x^2 + y^2} H(\\sqrt{x^2 + y^2} -\\varrho’) \\\\
\n&=& \\frac{1}{2\\pi \\varepsilon_0} \\frac{x}{x^2 + y^2} \\!\\!\\int_{0}^{\\sqrt{x^2 + y^2} }\\!\\! 2\\pi \\rho(\\varrho’) \\varrho’ d\\varrho’ \\\\
\n&=& \\frac{Q_{\\varrho}}{2\\pi \\varepsilon_0} \\frac{x}{\\varrho^2}
\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c$H(x)$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30d8\u30d3\u30b5\u30a4\u30c9\u306e\u968e\u6bb5\u95a2\u6570<\/p>\n

$$H(x) = \\left\\{
\n\\begin{array}{ll}
\n1 & (x > 0)\\\\ \\ \\\\
\n0 & (x < 0)
\n\\end{array}
\n\\right.
\n$$<\/p>\n

\u3067\u3042\u308a\uff0c\\(\\displaystyle Q_{\\varrho} \\equiv \\int_{0}^{{\\varrho}}\\!\\! 2\\pi \\rho(\\varrho’) \\varrho’ d\\varrho’\\) \u306f \\(z\\) \u8ef8\u3092\u4e2d\u5fc3\u3068\u3057\uff0c\u5e95\u9762\u7a4d\u304c \\(\\pi {\\varrho}^2\\) \u306e\u5358\u4f4d\u9ad8\u3055\u306e\u5186\u67f1\u5185\u306e\u5168\u96fb\u8377\u3067\u3042\u308b\u3002\u307e\u305f\uff0c\\({\\varrho} = \\sqrt{x^2 + y^2}\\) \u3067\u3042\u308b\u3002<\/p>\n

\\(y\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
E_y&=&\\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\varrho’) (y -\\varrho’ \\sin\\phi’) }{\\left\\{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2 + (z-z’)^2\\right\\}^{3\/2}}dV’ \\\\
&=&\\frac{1}{4\\pi \\varepsilon_0} \\!\\!\\int\\!\\! \\varrho’ d\\varrho’\\rho(\\varrho’) \\int\\!\\! d\\phi’ (y -\\varrho’ \\sin\\phi’) \\!\\!\\int_{-\\infty}^{\\infty} \\frac{ dz’ }{\\left\\{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2 + (z-z’)^2\\right\\}^{3\/2}} \\\\
&=& \\frac{1}{4\\pi \\varepsilon_0} \\!\\!\\int\\!\\! \\varrho’ d\\varrho’ \\rho(\\varrho’) \\int_{0}^{2\\pi}\\!\\! d\\phi’ (y -\\varrho’ \\sin\\phi’)\\frac{2}{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2}\\\\
\n&=& \\frac{1}{2\\pi \\varepsilon_0} \\!\\!\\int\\!\\! \\varrho’ d\\varrho’\\rho(\\varrho’) \\int_{0}^{2\\pi}\\!\\! d\\phi’ \\frac{ (y -\\varrho’ \\sin\\phi’)}{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2}\\\\
\n&=& \\frac{1}{2\\pi \\varepsilon_0} \\!\\!\\int_{0}^{\\infty}\\!\\! \\varrho’ d\\varrho’ \\rho(\\varrho’) \\, \\frac{2\\pi y}{x^2 + y^2} H(\\sqrt{x^2 + y^2} -\\varrho’) \\\\
\n&=& \\frac{1}{2\\pi \\varepsilon_0} \\frac{y}{x^2 + y^2} \\!\\!\\int_{0}^{\\sqrt{x^2 + y^2} }\\!\\! 2\\pi \\rho(\\varrho’) \\varrho’ d\\varrho’ \\\\
\n&=& \\frac{Q_{\\varrho}}{2\\pi \\varepsilon_0} \\frac{y}{\\varrho^2}
\\end{eqnarray}<\/p>\n

\\(z\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
E_z&=&\\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\varrho’) (z -z’) }{\\left\\{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2 + (z-z’)^2\\right\\}^{3\/2}}dV’ \\\\
&=&\\frac{1}{4\\pi \\varepsilon_0} \\!\\!\\int\\!\\! \\varrho’ d\\varrho’ \\rho(\\varrho’) \\int\\!\\! d\\phi’ \\!\\!\\int_{-\\infty}^{\\infty} \\frac{(z -z’) \\,dz’ }{\\left\\{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2 + (z-z’)^2\\right\\}^{3\/2}} \\\\
&=& 0
\\end{eqnarray}<\/p>\n

\u30d9\u30af\u30c8\u30eb\u5f0f\u306b\u3057\u3066\u307e\u3068\u3081\u308b\u3068<\/p>\n

$$\\boldsymbol{E} = \\frac{Q_{\\varrho}}{2\\pi \\varepsilon_0} \\frac{\\boldsymbol{\\varrho}}{{\\varrho}^2}, \\qquad \\boldsymbol{\\varrho} = (x, y, 0)$$<\/p>\n

\u8ef8\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3067\u3042\u308c\u3070\uff0c\u305d\u306e\u5206\u5e03\u306e\u3057\u304b\u305f\u306b\u3088\u3089\u305a\uff0c\u5186\u67f1\u5185\u306e\u5168\u96fb\u8377 \\(Q_{\\varrho}\\) \u3067\u96fb\u5834\u304c\u6c7a\u307e\u308b\u3053\u3068\u306b\u522e\u76ee\u305b\u3088\u3002\u7279\u306b \\(z\\) \u8ef8\u4e0a\u306e\u7dda\u96fb\u8377\u5bc6\u5ea6\u306e\u5834\u5408\u306f \\(Q_{\\varrho} \\rightarrow \\lambda\\) \u3068\u3059\u308b\u3053\u3068\u3067\u7d50\u679c\u3092\u518d\u73fe\u3059\u308b\u3002<\/p>\n

\u3053\u3053\u3067\u4f7f\u3063\u305f\u65b0\u305f\u306a\u7a4d\u5206\u306f<\/p>\n

$$\\int_0^{2\\pi} d\\phi’ \\frac{x-\\varrho’ \\cos\\phi’}{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2} = \\frac{2\\pi x}{x^2 + y^2} H(\\sqrt{x^2 + y^2} -\\varrho’)$$<\/p>\n

$$\\int_0^{2\\pi} d\\phi’ \\frac{y-\\varrho’ \\sin\\phi’}{(x-\\varrho’ \\cos \\phi’)^2 + (y-\\varrho’ \\sin \\phi’)^2} = \\frac{2\\pi y}{x^2 + y^2} H(\\sqrt{x^2 + y^2} -\\varrho’)$$<\/p>\n

\u4e00\u69d8\u306a\u9762\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834<\/h3>\n

\"\"<\/a><\/p>\n

\\(yz\\) \u5e73\u9762\u4e0a\u306e\u4e00\u69d8\u306a\u9762\u96fb\u8377\u5bc6\u5ea6 \\(\\sigma (\\mbox{C}\/\\mbox{m}^2)\\) \u306f\uff0c \\(x = 0\\) \u306b\u3057\u304b\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u304b\u3089\uff0c\u96fb\u8377\u5bc6\u5ea6\u3068\u3057\u3066\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3067\u3042\u308d\u3046\u3002<\/p>\n

$$\\rho(\\boldsymbol{r}) = \\sigma \\delta(x)$$<\/p>\n

\u3053\u308c\u3092\u96fb\u5834\u306e\u5f0f\u306b\u5165\u308c\u3066\u3084\u308b\u3068<\/p>\n

\\begin{eqnarray}
\\boldsymbol{E}&=& \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\boldsymbol{r}’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’ \\\\
&=& \\frac{\\sigma}{4\\pi \\varepsilon_0} \\iiint\\frac{\\delta(x’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’
\\end{eqnarray}<\/p>\n

\u5404\u6210\u5206\u3054\u3068\u306b\u8a08\u7b97\u3059\u308b\u3068\uff0c\\(x\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
E_x&=&\\frac{\\sigma}{4\\pi \\varepsilon_0} \\iiint\\frac{\\delta(x’) (x -x’) }{\\left((x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right)^{3\/2}}dx’ dy’ dz’ \\\\
&=&\\frac{\\sigma x}{4\\pi \\varepsilon_0} \\int_{-\\infty}^{\\infty} dy’\\int_{-\\infty}^{\\infty}\\frac{1 }{\\left(x^2 + (y-y’)^2 + (z-z’)^2\\right)^{3\/2}} dz’ \\\\
\n&=& \\frac{\\sigma x}{4\\pi \\varepsilon_0} \\int_{-\\infty}^{\\infty} dy’ \\frac{2}{x^2 + (y-y’)^2}\u00a0 \\\\
&=& \\frac{\\sigma x}{2\\pi \\varepsilon_0} \\int_{-\\infty}^{\\infty} dy’ \\frac{1}{x^2 + (y-y’)^2}\u00a0 \\\\
\n&=& \\frac{\\sigma x}{2\\pi \\varepsilon_0} \\frac{\\pi}{|x|}\\\\
\n&=& \\frac{\\sigma}{2 \\varepsilon_0} \\frac{x}{|x|}
\\end{eqnarray}<\/p>\n

\\(y, z\\) \u6210\u5206\u304c\u30bc\u30ed\u3068\u306a\u308b\u3053\u3068\u306f\u7c21\u5358\u306b\u308f\u304b\u308b\u3060\u308d\u3046\u3002<\/p>\n

\u3053\u3053\u3067\u4f7f\u3063\u305f\u65b0\u305f\u306a\u7a4d\u5206\u306f<\/p>\n

$$\\int_{-\\infty}^{\\infty} \\frac{1}{x^2 + (y-y’)^2 }dy’=\u00a0\u00a0 \\frac{\\pi}{|x|}$$<\/p>\n

\u9762\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u96fb\u5834<\/h3>\n

$x=0$ \u306e $yz$ \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\u3053\u308c\u306b\u3088\u308b\u96fb\u5834\u306f<\/p>\n

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

$E_y$ \u306f<\/p>\n

\\begin{eqnarray}
\nE_y &=& \\frac{1}{4\\pi\\varepsilon_0}
\n\\iiint \\frac{\\rho(|x’|) \\left(y-y’\\right)}{|\\boldsymbol{r}-\\boldsymbol{r}’|^3} dV’ \\\\
\n&=& \\frac{1}{4\\pi\\varepsilon_0}
\n\\iint dx’\\,dz’ \\int_{-\\infty}^{\\infty} dy’ \\frac{y-y’}{\\left\\{(x-x’)^2+(y-y’)^2+(z-z’)^2 \\right\\}^{\\frac{3}{2}}}\\\\
\n&=& \\frac{1}{4\\pi\\varepsilon_0}
\n\\iint dx’\\,dz’ \\int_{-\\infty}^{\\infty} dY \\frac{Y}{\\left\\{(x-x’)^2+Y^2+(z-z’)^2 \\right\\}^{\\frac{3}{2}}}\\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\uff08\u88ab\u7a4d\u5206\u95a2\u6570\u304c $Y$ \u306e\u5947\u95a2\u6570\u3060\u304b\u3089 $\\int_{-\\infty}^{\\infty} dY$ \u3067\u30bc\u30ed\u3002\uff09<\/p>\n

$E_z$ \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u3067\uff0c\u88ab\u7a4d\u5206\u95a2\u6570\u304c $Z \\equiv z -z’$ \u306e\u5947\u95a2\u6570\u306b\u306a\u308b\u3053\u3068\u304b\u3089\u30bc\u30ed\u3002<\/p>\n

$E_x$ \u306f $x>0$ \u3092\u4eee\u5b9a\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nE_x &=& \\frac{1}{4\\pi\\varepsilon_0}
\n\\iiint \\frac{\\rho(|x’|) \\left(x-x’\\right)}{|\\boldsymbol{r}-\\boldsymbol{r}’|^3} dV’ \\\\
\n&=& \\frac{1}{4\\pi\\varepsilon_0} \\int dx’ \\rho(|x’|) \\left(x-x’\\right)
\n\\int dy’ \\int_{-\\infty}^{\\infty} dz’ \\frac{1}{\\left\\{(x-x’)^2+(y-y’)^2+(z-z’)^2 \\right\\}^{\\frac{3}{2}}} \\\\
\n&=& \\frac{1}{4\\pi\\varepsilon_0} \\int dx’ \\rho(|x’|) \\left(x-x’\\right)
\n\\int_{-\\infty}^{\\infty} dy’ \\frac{2}{(x-x’)^2+(y-y’)^2}\\\\
\n&=& \\frac{1}{2\\pi\\varepsilon_0} \\int dx’ \\rho(|x’|) \\left(x-x’\\right) \\frac{\\pi}{|x-x’|}\\\\
\n&=& \\frac{1}{2\\varepsilon_0} \\left\\{
\n\\int_{-\\infty}^{-x} dx’ + \\int^{x}_{-x} dx’ + \\int_{x}^{\\infty} dx’
\n\\right\\} \\rho(|x’|) \\frac{x-x’}{|x-x’|} \\\\
\n&=& \\frac{1}{2\\varepsilon_0} \\left\\{
\n\\int_{-\\infty}^{-x} dx’ \\rho(|x’|) + \\int^{x}_{-x} dx’ \\rho(|x’|) -\\int_{x}^{\\infty} dx’ \\rho(|x’|)
\n\\right\\} \\\\
\n&=& \\frac{1}{2\\varepsilon_0} \\left\\{
\n\\int^{\\infty}_{x} dt\\, \\rho(|t|) + Q_{|x|}-\\int_{x}^{\\infty} dx’ \\rho(|x’|)
\n\\right\\} \\quad(t \\equiv -x’)\\\\
\n&=& \\frac{Q_{|x|}}{2\\varepsilon_0}
\n\\end{eqnarray}<\/p>\n

$x<0$ \u306e\u5834\u5408\u306f\uff0c$Q_{|x|}$ \u306e\u90e8\u5206\u3060\u3051\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u308f\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\int^{x}_{-x} dx’ \\rho(|x’|) &=&\u00a0 \\int^{-|x|}_{|x|} dx’ \\rho(|x’|) \\\\
\n&=& -\\int_{-|x|}^{|x|} dx’ \\rho(|x’|) \\\\
\n&=& -Q_{|x|}
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c$x$ \u304c\u6b63\u8ca0\u3069\u3061\u3089\u3067\u3082\u6210\u308a\u7acb\u3064\u3088\u3046\u306b<\/p>\n

$$E_x(x) = \\frac{Q_{|x|}}{2\\varepsilon_0} \\frac{x}{|x|}$$<\/p>\n

\u7279\u306b \\(yz\\) \u5e73\u9762\u4e0a\u306e\u9762\u96fb\u8377\u5bc6\u5ea6\u306e\u5834\u5408\u306f \\(Q_{|x|} \\rightarrow \\sigma\\) \u3068\u3059\u308b\u3053\u3068\u3067\u7d50\u679c\u3092\u518d\u73fe\u3059\u308b\u3002<\/p>\n

 <\/p>\n

\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u96fb\u5834<\/h3>\n

\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$\\rho(\\boldsymbol{r}) = \\rho(\\sqrt{x^2 + y^2 + z^2}) = \\rho(r)$$<\/p>\n

\u3068\u66f8\u3051\u308b\u3002<\/p>\n

\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c$z$ \u8ef8\u4e0a\u306e\u96fb\u5834\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u3068\u3057\uff0c<\/p>\n

$$\\boldsymbol{r} = (x, y, z) = (0, 0, r)$$\u3068\u3059\u308b\u3002<\/p>\n

\u307e\u305f\uff0c\u7a4d\u5206\u5909\u6570\u3092\u6975\u5ea7\u6a19\u3067\u66f8\u304f\u3068<\/p>\n

\\begin{eqnarray}
\nx’ &=& r’ \\sin\\theta’ \\cos \\phi’ \\\\
\ny’ &=& r’ \\sin\\theta’ \\sin \\phi’\\\\
\nz’ &=& r’ \\cos\\theta’ \\\\
\ndV’ &=& (r’)^2 dr’\\,\\sin\\theta’ d\\theta’ \\,d\\phi’
\n\\end{eqnarray}<\/p>\n

\u300c3\u6b21\u5143\u6975\u5ea7\u6a19\u7cfb\u306e\u4f53\u7a4d\u8981\u7d20<\/a>\u300d\u3092\u53c2\u7167\u306e\u3053\u3068\u3002<\/p>\n

\u307e\u305f\uff0c<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{r} -\\boldsymbol{r}’ &=& (x-x’, y-y’, z-z’) \\\\
\n&=& (-r’ \\sin\\theta’ \\cos\\phi’, -r’ \\sin\\theta’ \\sin \\phi’, z-r’ \\cos\\theta’ ) \\\\
\n|\\boldsymbol{r} -\\boldsymbol{r}’| &=& \\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{1}{2}}
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n

$$\\boldsymbol{E} = \\frac{1}{4\\pi \\varepsilon_0} \\iiint \\frac{\\rho(r’) (\\boldsymbol{r} -\\boldsymbol{r}’)}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} dV’$$<\/p>\n

\u5404\u6210\u5206\u3054\u3068\u306b\u8a08\u7b97\u3059\u308b\u3068\uff0c\\(x\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
\nE_x &=& \\frac{1}{4\\pi \\varepsilon_0} \\int_0^{\\infty} (r’)^2 dr’ \\int_0^{\\pi} \\sin\\theta’ d\\theta’
\n{\\color{red}{\\int_0^{2\\pi} d\\phi’ }}
\n\\frac{\\rho(r’) (-r’ \\sin\\theta’ {\\color{red}{\\cos\\phi’}})}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\\(y\\) \u6210\u5206\u3082\u540c\u69d8\u306b\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nE_y &=& \\frac{1}{4\\pi \\varepsilon_0} \\int_0^{\\infty} (r’)^2 dr’ \\int_0^{\\pi} \\sin\\theta’ d\\theta’
\n{\\color{red}{\\int_0^{2\\pi} d\\phi’ }}
\n\\frac{\\rho(r’) (-r’ \\sin\\theta’ {\\color{red}{\\sin\\phi’}})}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\\(z\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
\nE_z &=& \\frac{1}{4\\pi \\varepsilon_0} \\int_0^{\\infty} (r’)^2 dr’ \\int_0^{\\pi} \\sin\\theta’ d\\theta’ \\frac{\\rho(r’) (r-r’ \\cos\\theta’ )}{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} \\int_0^{2\\pi} d\\phi’ \\\\
\n&=& \\frac{1}{4\\pi \\varepsilon_0} \\int_0^{\\infty} (r’)^2 dr’ \\frac{2\\pi \\rho(r’)}{r^2}
\n\\left( \\frac{r + r’}{|r + r’|} + \\frac{r -r’}{|r -r’|} \\right) \\\\
\n&=& \\frac{1}{4\\pi \\varepsilon_0} \\frac{1}{r^2} \\int_0^{r} 4\\pi \\rho(r’) (r’)^2 dr’ \\\\
\n&=& \\frac{Q_r}{4\\pi \\varepsilon_0} \\frac{1}{r^2}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067 \\(\\displaystyle Q_r \\equiv \\int_0^{r} 4\\pi \\rho(r’) (r’)^2 dr’ \\) \u306f\u534a\u5f84 $r$ \u306e\u7403\u5185\u306e\u5168\u96fb\u8377\u3067\u3042\u308b\u3002<\/p>\n

\u4ee5\u4e0a\u306e\u7d50\u679c\u3092 \\(\\boldsymbol{r} = (0, 0, r)\\) \u3068\u3057\u3066\u30d9\u30af\u30c8\u30eb\u5f0f\u3067\u8868\u3059\u3068<\/p>\n

$$\\boldsymbol{E} = \\frac{Q_r}{4\\pi \\varepsilon_0} \\frac{\\boldsymbol{r}}{r^3}$$<\/p>\n

\u3044\u3063\u305f\u3093\u3053\u306e\u3088\u3046\u306a\u30d9\u30af\u30c8\u30eb\u5f0f\u3067\u8868\u3055\u308c\u305f\u89e3\u306f\uff0c\u4e00\u822c\u306b \\(\\boldsymbol{r} = (x, y, z)\\) \u3068\u3057\u3066\u3082\u6210\u308a\u7acb\u3064\u3093\u3067\u3059\u3088\u3002<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u3053\u306e\u5834\u5408\u306e\u30af\u30fc\u30ed\u30f3\u306e\u6cd5\u5247\u306f
\n$$\\boldsymbol{F}(\\boldsymbol{r}) = \\frac{Q_r q}{4\\pi\\varepsilon_0} \\frac{\\boldsymbol{r}}{r^3}$$
\n\u3068\u306a\u308a\uff0c\u539f\u70b9\u306b\u70b9\u96fb\u8377 \\(Q\\) \u3092\u304a\u3044\u305f\u5834\u5408\u306e\u30af\u30fc\u30ed\u30f3\u306e\u6cd5\u5247\u3067 \\(Q\\) \u3092 \\(Q_r\\) \u306b\u7f6e\u304d\u63db\u3048\u305f\u5f62\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n

\u8a00\u3044\u63db\u3048\u308b\u3068\uff0c\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u304c\u3064\u304f\u308b\u96fb\u5834\u306f\uff0c\u534a\u5f84 $r$ \u306e\u5168\u96fb\u8377 $Q_r$ \u304c\u70b9\u96fb\u8377\u3068\u3057\u3066\u539f\u70b9\u306b\u3042\u308b\u5834\u5408\u306e\u96fb\u5834\u3068\u7b49\u4fa1\u3067\u3042\u308b\u3002\u3042\u308b\u3044\u306f\uff0c\u534a\u5f84 $r$ \u5185\u306b\uff08\u7403\u5bfe\u79f0\u3067\u3055\u3048\u3042\u308c\u3070\uff09\u3069\u306e\u3088\u3046\u306a\u5206\u5e03\u3092\u3057\u3066\u3044\u308b\u304b\u306b\u3088\u3089\u305a\uff0c\u534a\u5f84 $r$ \u306e\u5168\u96fb\u8377 $Q_r$ \u306e\u307f\u306b\u3088\u3063\u3066\u96fb\u5834\u304c\u304d\u307e\u308b\uff0c\u3068\u3044\u3063\u3066\u3082\u3088\u3044\u3067\u3042\u308d\u3046\u3002<\/p>\n

\u3053\u3053\u3067\u4f7f\u3063\u305f\u7a4d\u5206\u306f<\/p>\n

\\begin{eqnarray}
\n\\int_0^{\\pi} \\sin\\theta’ d\\theta’ \\frac{r-r’ \\cos\\theta’ }{\\left\\{ r^2 + (r’)^2 -2 r r’ \\cos\\theta’ \\right\\}^{\\frac{3}{2}}} &=& \\frac{1}{r^2}
\n\\left( \\frac{r + r’}{|r + r’|} + \\frac{r -r’}{|r -r’|} \\right) \\\\
\n&=&\\frac{1}{r^2}
\n\\left(1+ \\frac{r -r’}{|r -r’|}\\right) \\\\
\n&=& \\left\\{
\n\\begin{array}{ll}
\n{\\displaystyle \\frac{2}{r^2}} & (r’ \\leq r)\\\\
\n0 & (r < r’)
\n\\end{array}
\n\\right.
\n\\end{eqnarray}<\/p>\n

\u307e\u3068\u3081<\/h3>\n

\u96fb\u8377\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u96fb\u5834\u3092\u6c42\u3081\u308b\u5f0f<\/span><\/h4>\n

\\begin{eqnarray}
\\boldsymbol{E} &=&\u00a0 \\frac{1}{4\\pi \\varepsilon_0} \\iiint\\frac{\\rho(\\boldsymbol{r}’) (\\boldsymbol{r} -\\boldsymbol{r}’) }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3}dV’
\\end{eqnarray}<\/p>\n

\u3053\u306e\u4f53\u7a4d\u7a4d\u5206\u3092\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u308a\uff0c\u4ee5\u4e0b\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u305f\u3002<\/p>\n

\u70b9\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834<\/h4>\n

\u4f4d\u7f6e $\\boldsymbol{r}_i$ \u306b\u96fb\u8377 $q_i$ ($i = 1 \\dots N$) \u304c\u5206\u5e03\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$\\rho(\\boldsymbol{r}) =\\sum_{i=1}^N q_i \\, \\delta^3(\\boldsymbol{r} -\\boldsymbol{r}_i)$$<\/p>\n

\u96fb\u5834\u306f<\/p>\n

$$
\n\\boldsymbol{E} = \\frac{1}{4\\pi \\varepsilon_0}\u00a0 \\sum_{i=1}^N \\frac{q_i ( \\boldsymbol{r}\u00a0 -\\boldsymbol{r}_i)}{|\\boldsymbol{r}\u00a0 -\\boldsymbol{r}_i|^3}
\n$$<\/p>\n

\u96fb\u6c17\u53cc\u6975\u5b50\u306b\u3088\u308b\u96fb\u5834<\/span><\/h4>\n

\u96fb\u6c17\u53cc\u6975\u30e2\u30fc\u30e1\u30f3\u30c8 $\\boldsymbol{p} \\equiv q \\boldsymbol{d}$ \u304c\u3064\u304f\u308b\u96fb\u5834\u306f<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{E}
\n&=& \\frac{1}{4\\pi\\varepsilon_0}
\n\\left\\{ 3 \\frac{\\boldsymbol{r}\\cdot \\boldsymbol{p} }{r^5} \\boldsymbol{r} -\\frac{\\boldsymbol{p}}{r^3}\\right\\}
\n\\end{eqnarray}<\/p>\n

\u4e00\u69d8\u306a\u7dda\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834<\/span><\/h4>\n

$z$ \u8ef8\u4e0a\u306e\u4e00\u69d8\u306a\u7dda\u96fb\u8377\u5bc6\u5ea6 $\\lambda\\ (\\mbox{C}\/\\mbox{m})$ \u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6 $\\rho$ \u306f<\/p>\n

$$
\n\\rho(\\boldsymbol{r}) = \\lambda \\delta(x) \\delta(y)
\n$$<\/p>\n

\u96fb\u5834\u306f<\/p>\n

$$
\n\\boldsymbol{E} = \\frac{\\lambda}{2\\pi\\varepsilon_0} \\frac{\\boldsymbol{\\varrho}}{{\\varrho}^2}
\n$$<\/p>\n

\u3053\u3053\u3067\uff0c$z$ \u8ef8\u306b\u76f4\u4ea4\u3059\u308b\u4e00\u30d9\u30af\u30c8\u30eb\u3092 $\\boldsymbol{\\varrho} \\equiv (x, y, 0), \\ {\\varrho}^2 \\equiv \\boldsymbol{\\varrho} \\cdot\\boldsymbol{\\varrho}$ \u3068\u3057\u305f\u3002<\/p>\n

\u8ef8\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u96fb\u5834<\/span><\/h4>\n

$z$ \u8ef8\u306b\u3064\u3044\u3066\u8ef8\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$
\n\\rho(\\boldsymbol{r}) = \\rho(\\sqrt{x^2 + y^2})
\n$$<\/p>\n

\u96fb\u5834\u306f<\/p>\n

$$
\n\\boldsymbol{E} = \\frac{Q_{\\varrho}}{2\\pi \\varepsilon_0} \\frac{\\boldsymbol{\\varrho}}{{\\varrho}^2}, \\qquad \\boldsymbol{\\varrho} = (x, y, 0)
\n$$<\/p>\n

\u8ef8\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3067\u3042\u308c\u3070\uff0c\u305d\u306e\u5206\u5e03\u306e\u4ed5\u65b9\u306b\u3088\u3089\u305a\u306b\uff0c\u5186\u67f1\u5185\u306e\u5168\u96fb\u8377 $Q_{\\varrho}$ \u3067\u96fb\u5834\u304c\u6c7a\u307e\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\u4f53\u7a4d\u7a4d\u5206\u81ea\u4f53\u306f\u5168\u7a7a\u9593\u3067\u7a4d\u5206\u3057\u3066\u3044\u308b\u306b\u3082\u304b\u304b\u308f\u3089\u305a\uff0c$\\varrho$ \u306e\u5916\u5074\u306e\u96fb\u8377\u306f\u96fb\u5834\u306b\u5bc4\u4e0e\u3057\u306a\u3044\u3002 \u7279\u306b\u8ef8\u4e0a\u306e\u4e00\u69d8\u96fb\u8377\u5bc6\u5ea6\u306e\u5834\u5408\u306f $Q_{\\varrho} \\rightarrow \\lambda$ \u3068\u3059\u308c\u3070\u7d50\u679c\u3092\u518d\u73fe\u3059\u308b\u3002<\/p>\n

\u4e00\u69d8\u306a\u9762\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834<\/span><\/h4>\n

$x = 0$ \u306e $yz$ \u5e73\u9762\u4e0a\u306e\u4e00\u69d8\u306a\u9762\u96fb\u8377\u5bc6\u5ea6 $\\sigma\\ (\\mbox{C}\/\\mbox{m}^2)$ \u3092\u3042\u3089\u308f\u3059\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$
\n\\rho(\\boldsymbol{r}) = \\sigma \\delta(x)
\n$$<\/p>\n

\u96fb\u5834\u306f<\/p>\n

$$
\nE_x = \\frac{\\sigma}{2 \\varepsilon_0} \\frac{x}{|x|}, \\quad E_y = 0, \\quad E_z = 0
\n$$<\/p>\n

\u3042\u308b\u3044\u306f\uff0c\u5bfe\u79f0\u9762\u306b\u5782\u76f4\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3092 $\\boldsymbol{n}$ \u3068\u3059\u308b\u3068\uff0c\u30d9\u30af\u30c8\u30eb\u5f62\u3067<\/p>\n

$$\\boldsymbol{E} = \\frac{\\sigma}{2 \\varepsilon_0} \\boldsymbol{n}$$<\/p>\n

\u9762\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u96fb\u5834<\/h4>\n

$x=0$ \u306e $yz$ \u9762\u306b\u5bfe\u3057\u3066\u9762\u5bfe\u79f0\u306a\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$ \\rho(\\boldsymbol{r}) = \\rho(|x|)$$<\/p>\n

\u96fb\u5834\u306f<\/p>\n

$$
\nE_x = \\frac{Q_{|x|}}{2 \\varepsilon_0} \\frac{x}{|x|}, \\quad E_y = 0, \\quad E_z = 0
\n$$<\/p>\n

\u3042\u308b\u3044\u306f\uff0c\u5bfe\u79f0\u9762\u306b\u5782\u76f4\u306a\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3092 $\\boldsymbol{n}$ \u3068\u3059\u308b\u3068\uff0c\u30d9\u30af\u30c8\u30eb\u5f62\u3067<\/p>\n

$$\\boldsymbol{E} = \\frac{Q_{|x|}}{2 \\varepsilon_0} \\boldsymbol{n}$$<\/p>\n

\u9762\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3067\u3042\u308c\u3070\uff0c\u305d\u306e\u5206\u5e03\u306e\u4ed5\u65b9\u306b\u3088\u3089\u305a\u306b\uff0c\u9762\u304b\u3089\u306e\u8ddd\u96e2\u304c $|x|$ \u4ee5\u5185\u306e\u5168\u96fb\u8377 $Q_{|x|}$ \u3067\u96fb\u5834\u304c\u6c7a\u307e\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\u4f53\u7a4d\u7a4d\u5206\u81ea\u4f53\u306f\u5168\u7a7a\u9593\u3067\u7a4d\u5206\u3057\u3066\u3044\u308b\u306b\u3082\u304b\u304b\u308f\u3089\u305a\uff0c$|x|$ \u306e\u5916\u5074\u306e\u96fb\u8377\u306f\u96fb\u5834\u306b\u5bc4\u4e0e\u3057\u306a\u3044\u3002 \u7279\u306b\u9762\u4e0a\u306e\u4e00\u69d8\u96fb\u8377\u5bc6\u5ea6\u306e\u5834\u5408\u306f $Q_{|x|} \\rightarrow \\sigma$ \u3068\u3059\u308c\u3070\u7d50\u679c\u3092\u518d\u73fe\u3059\u308b\u3002<\/p>\n

\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u96fb\u5834<\/h4>\n

\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3092\u8868\u3059\u96fb\u8377\u5bc6\u5ea6\u306f<\/p>\n

$$
\n\\rho(\\boldsymbol{r}) = \\rho(\\sqrt{x^2 + y^2 + z^2}) = \\rho(r)
\n$$<\/p>\n

\u96fb\u5834\u306f<\/p>\n

$$
\n\\boldsymbol{E} = \\frac{Q_r}{4\\pi \\varepsilon_0} \\frac{\\boldsymbol{r}}{r^3}
\n$$<\/p>\n

\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u3067\u3042\u308c\u3070\uff0c\u305d\u306e\u5206\u5e03\u306e\u4ed5\u65b9\u306b\u3088\u3089\u305a\u306b\uff0c\u539f\u70b9\u304b\u3089\u306e\u8ddd\u96e2\u304c $r$ \u4ee5\u5185\u306e\u5168\u96fb\u8377 $Q_{r}$ \u3067\u96fb\u5834\u304c\u6c7a\u307e\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002\u4f53\u7a4d\u7a4d\u5206\u81ea\u4f53\u306f\u5168\u7a7a\u9593\u3067\u7a4d\u5206\u3057\u3066\u3044\u308b\u306b\u3082\u304b\u304b\u308f\u3089\u305a\uff0c$r$ \u306e\u5916\u5074\u306e\u96fb\u8377\u306f\u96fb\u5834\u306b\u5bc4\u4e0e\u3057\u306a\u3044\u3002 \u7279\u306b\u539f\u70b9\u306b\u3042\u308b\u70b9\u96fb\u8377\u306e\u5834\u5408\u306f $Q_{r} \\rightarrow q$ \u3068\u3059\u308c\u3070\u7d50\u679c\u3092\u518d\u73fe\u3059\u308b\u3002<\/p>\n

\u88dc\u8db3<\/h4>\n

\u4ee5\u4e0a\uff0c\u4f53\u7a4d\u7a4d\u5206\u3092\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\uff0c\u4e0e\u3048\u3089\u308c\u305f\u96fb\u8377\u5bc6\u5ea6 $\\rho(\\boldsymbol{r})$ \u304b\u3089\u76f4\u63a5\u96fb\u5834 $\\boldsymbol{E}$ \u3092\u6c42\u3081\u305f\u308f\u3051\u3060\u304c\uff0c\u9014\u4e2d\u306e\u7a4d\u5206\u306b\u306f\u306a\u304b\u306a\u304b\u624b\u3053\u305a\u3063\u305f\u3002\u4f7f\u3063\u305f\u7a4d\u5206\u3092 Maxima \u3067\u78ba\u8a8d\u3059\u308b\u306b\u306f\uff0c<\/p>\n