{"id":2687,"date":"2022-03-28T14:08:41","date_gmt":"2022-03-28T05:08:41","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2687"},"modified":"2026-01-19T09:36:26","modified_gmt":"2026-01-19T00:36:26","slug":"%e9%9d%99%e7%a3%81%e5%a0%b4%ef%bc%9a%e9%9b%bb%e6%b5%81%e5%af%86%e5%ba%a6%e3%81%8b%e3%82%89%e7%9b%b4%e6%8e%a5%e9%9d%99%e7%a3%81%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%e7%a3%81%e5%a0%b4%ef%bc%9a%e9%9b%bb%e6%b5%81%e5%af%86%e5%ba%a6%e3%81%8b%e3%82%89%e7%9b%b4%e6%8e%a5%e9%9d%99%e7%a3%81%e5%a0%b4%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/","title":{"rendered":"\u9759\u78c1\u5834\uff1a\u96fb\u6d41\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u78c1\u5834\u3092\u6c42\u3081\u308b"},"content":{"rendered":"

\u30a2\u30f3\u30da\u30fc\u30eb\u306e\u6cd5\u5247\u306a\u3069\u3092\u4f7f\u308f\u305a\u306b\uff0c\u30d3\u30aa-\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u3092\u4f7f\u3063\u3066\u96fb\u6d41\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u78c1\u5834\u3092\u6c42\u3081\u308b\u3002<\/p>\n

<\/p>\n

\u30d9\u30af\u30c8\u30eb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092\u4f7f\u3063\u305f\u9759\u78c1\u5834\u306e\u57fa\u672c\u65b9\u7a0b\u5f0f<\/h3>\n

\u30af\u30fc\u30ed\u30f3\u30b2\u30fc\u30b8\u6761\u4ef6\u3092\u8ab2\u3057\u305f\u30d9\u30af\u30c8\u30eb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb $\\boldsymbol{A}$ \u3092\u4f7f\u3046\u3068\uff0c\u9759\u78c1\u5834\u306e\u57fa\u672c\u65b9\u7a0b\u5f0f\u306f<\/p>\n

$$\u00a0 \\nabla^2 \\boldsymbol{A} = -\\frac{\\boldsymbol{J}}{\\varepsilon_0 c^2}, \\quad \\boldsymbol{B}\u00a0 = \\nabla\\times \\boldsymbol{A} $$
\n\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u306e\u3067\u3042\u3063\u305f\u3002\u7279\u306b\uff0c<\/p>\n

$$\u00a0 \\nabla^2 \\boldsymbol{A} = -\\frac{\\boldsymbol{J}}{\\varepsilon_0c^2}$$\u306f\u300c\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u300d\u3002\u300c\u65b9\u7a0b\u5f0f\u306e\u5f62\u304c\u540c\u3058\u306a\u3089\u89e3\u306e\u5f62\u3082\u540c\u3058<\/strong><\/span>\u300d\u3068\u3044\u3046\u539f\u5247\u306b\u3088\u308a\uff0c\u305f\u3060\u3061\u306b\u89e3\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\uff08\u300c\u9759\u96fb\u5834\uff1a\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306e\u89e3<\/a>\u300d\u3092\u53c2\u7167\u3002\uff09<\/p>\n

$$ \\boldsymbol{A} (\\boldsymbol{r})= \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{\\boldsymbol{J}(\\boldsymbol{r}’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|} \\,dV’$$<\/p>\n

\u30d3\u30aa – \u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247<\/h3>\n

\u96fb\u6d41\u5bc6\u5ea6 \\(\\boldsymbol{J}\\) \u304b\u3089\u76f4\u63a5\u78c1\u675f\u5bc6\u5ea6 \\(\\boldsymbol{B}\\) \u3092\u6c42\u3081\u308b\u5f0f\u3002<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{B} (\\boldsymbol{r}) = \\nabla\\times \\boldsymbol{A} (\\boldsymbol{r})&=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{\\boldsymbol{J}(\\boldsymbol{r}’)\\times (\\boldsymbol{r} -\\boldsymbol{r}’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} \\,dV’
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u304c\uff0c\u30d3\u30aa\u30fb\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247<\/strong><\/span>\u3067\u3042\u308b\u3002\u3053\u3053\u3067\u306f\u5ff5\u306e\u305f\u3081\uff0c<\/p>\n

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

\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304a\u304f\u3002<\/p>\n

\\(x\\) \u6210\u5206\u3092\u8a08\u7b97\u3059\u308b\u3068<\/p>\n

\\begin{eqnarray}
\n\\left(\\nabla\\times \\frac{\\boldsymbol{J}(\\boldsymbol{r}’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|} \\right)_x &=& \\frac{\\partial}{\\partial y} \\frac{J_z(\\boldsymbol{r}’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|} -\\frac{\\partial}{\\partial z} \\frac{J_y(\\boldsymbol{r}’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|} \\\\
\n&=& J_z(\\boldsymbol{r}’)\\frac{\\partial}{\\partial y}\\frac{1}{|\\boldsymbol{r} -\\boldsymbol{r}’|} -J_y(\\boldsymbol{r}’)\\frac{\\partial}{\\partial z}\\frac{1}{|\\boldsymbol{r} -\\boldsymbol{r}’|} \\\\
\n&=& J_z(\\boldsymbol{r}’) \\frac{- (y-y’)\\ }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} -J_y(\\boldsymbol{r}’) \\frac{- (z-z’)\\ }{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} \\\\
\n&=& J_y(\\boldsymbol{r}’) \\frac{(z-z’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} -J_z(\\boldsymbol{r}’) \\frac{(y-y’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} \\\\
\n&=& \\left(\\frac{\\boldsymbol{J}(\\boldsymbol{r}’)\\times (\\boldsymbol{r} -\\boldsymbol{r}’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} \\right)_x
\n\\end{eqnarray}<\/p>\n

\\(y, z\\) \u6210\u5206\u3082\u540c\u69d8\u3002<\/p>\n

\u4ee5\u4e0b\u3067\u306f\uff0c\u30d3\u30aa\u30fb\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u3092\u4f7f\u3063\u3066\uff0c\u3044\u304f\u3064\u304b\u306e\u7c21\u5358\u306a\u5834\u5408\u306b\u3064\u3044\u3066\u96fb\u6d41\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u78c1\u5834\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n

\u76f4\u7dda\u96fb\u6d41\u306b\u3088\u308b\u78c1\u5834<\/h3>\n

\\(z\\) \u8ef8\u4e0a\u306e\u96fb\u6d41 \\(\\boldsymbol{I} = (0, 0, I)\\) \u3092\u8868\u3059\u96fb\u6d41\u5bc6\u5ea6 \\(\\boldsymbol{J}\\) \u306f\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

$$\\boldsymbol{J} = (0, 0, J_z) = \\boldsymbol{I} \\delta(x) \\delta(y) = (0, 0, I \\delta(x) \\delta(y)) $$<\/p>\n

\u30d3\u30aa – \u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u306e \\(x\\) \u6210\u5206\u306f<\/p>\n

\\begin{eqnarray}
B_x &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_y(\\boldsymbol{r}’)\\cdot (z-z’) -J_z(\\boldsymbol{r}’)\\cdot (y-y’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} \\,dV’ \\\\
&=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint\\frac{ -I \\delta(x’) \\delta(y’) (y-y’)}{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}} \\,dx’ dy’ dz’\\\\
&=& \\frac{-I y}{4\\pi \\varepsilon_0 c^2} \\int_{-\\infty}^{\\infty} \\frac{1}{\\left\\{x^2 + y^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}}\\,dz’\\\\
&=& \\frac{-I y}{4\\pi \\varepsilon_0 c^2} \\frac{2}{x^2 + y^2} \\\\
&=& \\frac{1}{2\\pi \\varepsilon_0 c^2} \\frac{(\\boldsymbol{I}\\times\\boldsymbol{\\varrho})_x}{\\varrho^2}
\\end{eqnarray}
\u3053\u3053\u3067\uff0c\\(\\boldsymbol{I} = (0, 0, I), \\ \\boldsymbol{\\varrho} = (x, y, 0), \\ \\varrho^2 =\\boldsymbol{\\varrho}\\cdot\\boldsymbol{\\varrho} \\) \u3068\u3057\u305f\u3002\uff08$\\rho$ \u306f\u96fb\u8377\u5bc6\u5ea6\u3067\u4f7f\u3046\u305f\u3081\u3002\uff09\u307e\u305f\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d $\\boldsymbol{I}\\times\\boldsymbol{\\varrho}$ \u306f<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{I}\\times\\boldsymbol{\\varrho} &=&
\n\\left(I_y \\varrho_z -I_z \\varrho_y, I_z \\varrho_x -I_x \\varrho_z, I_x \\varrho_y -I_y \\varrho_x\\right) \\\\
\n&=& \\left( -I y, I x, 0\\right)
\n\\end{eqnarray}<\/p>\n

\u307e\u305f\uff0c\u30c7\u30eb\u30bf\u95a2\u6570\u3092\u542b\u3080\u7a4d\u5206\u3067\u306f
$$\\iint f(x’, y’, z’) \\delta(x’) \\delta(y’) \\,dx’ dy’ = f(0, 0, z’)$$
\n\u3068\u306a\u308b\u3053\u3068\u3092\u5229\u7528\u3057\u3066\u3044\u308b\u3002\u307e\u305f\uff0c\u4ee5\u4e0b\u306e\u7a4d\u5206\u7d50\u679c\u3082\u4f7f\u3063\u305f\u3002<\/p>\n

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

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

\\begin{eqnarray}
B_y &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_z(\\boldsymbol{r}’)\\cdot (x-x’) -J_x(\\boldsymbol{r}’)\\cdot (z-z’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} \\,dV’ \\\\
&=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint\\frac{\u00a0 I \\delta(x’) \\delta(y’) (x-x’)}{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}} \\,dx’ dy’ dz’\\\\
&=& \\frac{I x}{4\\pi \\varepsilon_0 c^2} \\int_{-\\infty}^{\\infty} \\frac{1}{\\left\\{x^2 + y^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}}\\,dz’\\\\
&=& \\frac{I x}{4\\pi \\varepsilon_0 c^2} \\frac{2}{x^2 + y^2} \\\\
&=& \\frac{1}{2\\pi \\varepsilon_0 c^2} \\frac{(\\boldsymbol{I}\\times\\boldsymbol{\\varrho})_y}{\\varrho^2}
\\end{eqnarray}<\/p>\n

 <\/p>\n

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

\\begin{eqnarray}
B_z &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_x(\\boldsymbol{r}’)\\cdot (y-y’) -J_y(\\boldsymbol{r}’)\\cdot (x-x’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} \\,dV’ \\\\
&=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{0\\cdot (y-y’) -0\\cdot (x-x’)}{|\\boldsymbol{r} -\\boldsymbol{r}’|^3} \\,dV’ \\\\
\n&=& 0
\\end{eqnarray}<\/p>\n

\u3061\u306a\u307f\u306b\uff0c\\( (\\boldsymbol{I}\\times\\boldsymbol{\\varrho})_z = 0\\)\u00a0 \u3060\u304b\u3089<\/p>\n

\\begin{eqnarray}
B_z &=& \\frac{1}{2\\pi \\varepsilon_0 c^2}\u00a0 \\frac{(\\boldsymbol{I}\\times\\boldsymbol{\\varrho})_z}{\\varrho^2}
\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\u3088\u3044\u3002\u307e\u3068\u3081\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\\boldsymbol{B} &=& \\frac{1}{2\\pi \\varepsilon_0 c^2}\u00a0 \\frac{\\boldsymbol{I}\\times\\boldsymbol{\\varrho}}{\\varrho^2}
\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c\\(\\boldsymbol{I} = (0, 0, I), \\ \\boldsymbol{\\varrho} = (x, y, 0)\\) \u3067\u3042\u3063\u305f\u3002<\/p>\n

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

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

\u5e73\u884c2\u76f4\u7dda\u96fb\u6d41\u304c\u3064\u304f\u308b\u78c1\u5834<\/h3>\n

\u91cd\u306d\u5408\u308f\u305b\u306e\u539f\u7406<\/strong><\/span>\u304b\u3089\uff0c<\/p>\n

$$\\boldsymbol{B} = \\frac{1}{2\\pi\\varepsilon_0 c^2} \\frac{\\boldsymbol{I}_1\\times \\left(\\boldsymbol{\\varrho} -\\boldsymbol{r}_1\\right)}{|\\boldsymbol{\\varrho} -\\boldsymbol{r}_1|^2}
\n+ \\frac{1}{2\\pi\\varepsilon_0 c^2} \\frac{\\boldsymbol{I}_2\\times \\left(\\boldsymbol{\\varrho} -\\boldsymbol{r}_2\\right)}{|\\boldsymbol{\\varrho} -\\boldsymbol{r}_2|^2}$$<\/p>\n

\u96fb\u6d41\u5bc6\u5ea6 $\\boldsymbol{J}$ \u304c2\u3064\u306e\u96fb\u6d41\u5bc6\u5ea6 $\\boldsymbol{J}_1, \\boldsymbol{J}_2$ \u306e\u91cd\u306d\u5408\u308f\u305b\uff08\u8db3\u3057\u7b97\uff09\u3067<\/p>\n

$$\\boldsymbol{J} = \\boldsymbol{J}_1 + \\boldsymbol{J}_2
\n= \\boldsymbol{I}_1 \\delta(x-x_1) \\delta(y-y_1)\u00a0 + \\boldsymbol{I}_2 \\delta(x-x_2) \\delta(y-y_2)$$<\/p>\n

\u3068\u66f8\u3051\u308b\u3068\u304d\u306b\uff0c$\\boldsymbol{J}$ \u306b\u3088\u3063\u3066\u3064\u304f\u3089\u308c\u308b\u78c1\u5834 $\\boldsymbol{B}$ \u306f\uff0c$\\boldsymbol{J}_1,\u00a0 \\boldsymbol{J}_2$ \u304c\u305d\u308c\u305e\u308c\u72ec\u7acb\u306b\u3064\u304f\u308b\u78c1\u5834 $\\boldsymbol{B}_1, \\boldsymbol{B}_2$ \u306e\u91cd\u306d\u5408\u308f\u305b\uff08\u8db3\u3057\u7b97\uff09\u3067<\/p>\n

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

\u3068\u66f8\u3051\u308b\u3053\u3068\u306f\u660e\u3089\u304b\u3067\u3042\u308d\u3046\u3002\u9759\u96fb\u5834\u306b\u304a\u3051\u308b\u91cd\u306d\u5408\u308f\u305b\u306e\u539f\u7406\u306e\u8aac\u660e\u306e\u3068\u3053\u308d<\/a>\u3092\u53c2\u7167\u3002<\/p>\n

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

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

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

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

\u5186\u96fb\u6d41\u306b\u3088\u308b\u78c1\u5834<\/h3>\n

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

\\(xy\\) \u5e73\u9762\u4e0a\u306e\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u534a\u5f84 \\(a\\) \u306e\u5186\u4e0a\u306b\u96fb\u6d41 \\(I\\) \u304c\u6d41\u308c\u3066\u3044\u308b\u3002\u3053\u308c\u3092\u8868\u3059\u96fb\u6d41\u5bc6\u5ea6 \\(\\boldsymbol{J}\\) \u306f\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

\\begin{eqnarray}
\n\\boldsymbol{J} &=& (J_x, J_y, 0) \\\\
\n&=& (-I \\frac{y}{\\varrho} \\,\\delta(\\varrho-a) \\,\\delta(z), I \\frac{x}{\\varrho} \\,\\delta(\\varrho-a) \\,\\delta(z), 0)\\\\
\n&=& (-I \\sin\\phi \\,\\delta(\\varrho-a) \\,\\delta(z), I \\cos\\phi \\,\\delta(\\varrho-a) \\,\\delta(z), 0)\\\\
\n&&\\ \\\\
\n\\varrho &=& \\sqrt{x^2 + y^2} \\\\
\nx &=& \\varrho \\cos\\phi\\\\
\ny &=& \\varrho \\sin\\phi
\n\\end{eqnarray}<\/p>\n

\u5ff5\u306e\u305f\u3081\uff0c\u88dc\u8db3\u8aac\u660e\u3002<\/p>\n

$xy$ \u5e73\u9762\u4e0a\u306e\u5186\u96fb\u6d41\u3092\u8868\u3059\u30d9\u30af\u30c8\u30eb $\\boldsymbol{I}$ \u306f<\/p>\n

$$\\boldsymbol{I} = (I_x, I_y, 0)$$<\/p>\n

\u3068\u66f8\u3051\u308b\u3002\u3053\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u304c\u96fb\u6d41 $I$ \u3067\u3042\u308a\uff0c\u307e\u305f\uff0c\u5186\u96fb\u6d41\u30d9\u30af\u30c8\u30eb\u306f\u539f\u70b9\u304b\u3089\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{\\varrho}=(\\varrho \\cos\\phi, \\varrho\\sin\\phi, 0)$ \u3068\u76f4\u4ea4\u3059\u308b\u304b\u3089<\/p>\n

\\begin{eqnarray}
\nI ^2&=& \\boldsymbol{I}\\cdot\\boldsymbol{I} = I_x^2 + I_y^2 \\tag{1}\\\\
\n\\boldsymbol{I}\\cdot\\boldsymbol{\\varrho} &=& I_x\\,\\varrho \\cos\\phi + I_y \\,\\varrho\\sin\\phi \\\\
\n&=& 0 \\\\
\n\\therefore\\ \\ I_x &=& -\\frac{\\sin\\phi}{\\cos\\phi} I_y \\tag{2}\\\\
\n\\therefore\\ \\ I^2 &=& I_y^2 \\left\\{\\frac{\\sin^2 \\phi}{\\cos^2 \\phi} + 1\\right\\}\u00a0 = \\frac{I_y^2}{\\cos^2\\phi}\\tag{3}\\\\
\n\\end{eqnarray}<\/p>\n

\u5186\u96fb\u6d41\u304c\u300c\u53cd\u6642\u8a08\u56de\u308a\u300d\u306b\u6d41\u308c\u308b\u3068\u3059\u308b\u3068\uff0c$\\phi = 0$ \u3067 $I_y > 0$ \u3067\u3042\u308b\u304b\u3089\uff0c$(3)$ \u5f0f\u3088\u308a<\/p>\n

$$I_y = I \\cos\\phi, \\quad\\therefore\\ \\ I_x = -I \\sin\\phi$$<\/p>\n

\u96fb\u6d41\u30d9\u30af\u30c8\u30eb\u3092\u4f7f\u3063\u3066\u96fb\u6d41\u5bc6\u5ea6\u30d9\u30af\u30c8\u30eb\u3092\u8868\u3059\u3068<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{J} &=& \\boldsymbol{I} \\,\\delta(\\varrho-a) \\,\\delta(z) \\\\
\n\\therefore\\ \\\u00a0 (J_x, J_y, 0)
\n&=& (I_x\\, \\delta(\\varrho-a) \\,\\delta(z), I_y\\, \\delta(\\varrho-a) \\,\\delta(z), 0) \\\\
\n&=&(-I \\sin\\phi \\,\\delta(\\varrho-a) \\,\\delta(z), I \\cos\\phi \\,\\delta(\\varrho-a) \\,\\delta(z), 0)
\n\\end{eqnarray}<\/p>\n

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

\u3053\u308c\u3092\u30d3\u30aa\u30fb\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u306b\u4ee3\u5165\u3059\u308c\u3070\u78c1\u5834\u304c\u6c42\u307e\u308b\u3002\u6210\u5206\u3054\u3068\u306b\u66f8\u304f\u3068\uff0c$x$ \u6210\u5206 $B_x$ \u306f<\/p>\n

\\begin{eqnarray}
\nB_x(\\boldsymbol{r}) &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_y(\\boldsymbol{r}’) \\cdot(z-z’) -J_z(\\boldsymbol{r}’)\\cdot (y-y’)}{\\left\\{ (x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right\\}^{\\frac{3}{2}}}\\,dV’ \\\\
\n&=&\\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{I \\frac{x’}{\\varrho’} \\,\\delta(\\varrho’-a) \\,\\delta(z’) (z-z’)}{\\left\\{ (x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ dz’\\\\
\n&=&\\frac{1}{4\\pi \\varepsilon_0 c^2} \\iint \\frac{I \\frac{x’}{\\varrho’} \\,\\delta(\\varrho’-a)\u00a0 z}{\\left\\{ (x-x’)^2 + (y-y’)^2 +z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’
\n\\end{eqnarray}<\/p>\n

$y$ \u6210\u5206 $B_y$ \u306f<\/p>\n

\\begin{eqnarray}
\nB_y(\\boldsymbol{r}) &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_z(\\boldsymbol{r}’)\\cdot (x-x’) -J_x(\\boldsymbol{r}’)\\cdot (z-z’)}{\\left\\{ (x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right\\}^{\\frac{3}{2}}}\\,dV’ \\\\
\n&=&\\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{I \\frac{y’}{\\varrho’} \\,\\delta(\\varrho’-a) \\,\\delta(z’) (z-z’)}{\\left\\{ (x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ dz’\\\\
\n&=&\\frac{1}{4\\pi \\varepsilon_0 c^2} \\iint \\frac{I \\frac{y’}{\\varrho’} \\,\\delta(\\varrho’-a)\u00a0 z}{\\left\\{ (x-x’)^2 + (y-y’)^2 + z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’
\n\\end{eqnarray}<\/p>\n

$z$ \u6210\u5206 $B_z$ \u306f<\/p>\n

\\begin{eqnarray}
\nB_z(\\boldsymbol{r}) &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_x(\\boldsymbol{r}’) \\cdot(y-y’) -J_y(\\boldsymbol{r}’) \\cdot(x-x’)}{\\left\\{ (x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right\\}^{\\frac{3}{2}}}\\,dV’ \\\\
\n&=&\\frac{-I }{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{\\left\\{\\frac{y’}{\\varrho’} (y-y’) + \\frac{x’}{\\varrho’} (x-x’)\\right\\} \\delta(\\varrho’-a) \\,\\delta(z’)}{\\left\\{ (x-x’)^2 + (y-y’)^2 + (z-z’)^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ dz’\\\\
\n&=&\\frac{-I }{4\\pi \\varepsilon_0 c^2} \\iint \\frac{\\left\\{\\frac{y’}{\\varrho’} (y-y’) + \\frac{x’}{\\varrho’} (x-x’)\\right\\} \\delta(\\varrho’-a) }{\\left\\{ (x-x’)^2 + (y-y’)^2 + z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’
\n\\end{eqnarray}<\/p>\n

$y = 0$ ($xz$ \u5e73\u9762) \u3067\u306e\u78c1\u5834<\/h4>\n

\u3053\u306e\u5186\u96fb\u6d41\u306f $z$ \u306e\u307e\u308f\u308a\u306b\u4efb\u610f\u306e\u56de\u8ee2\u3092\u884c\u3063\u3066\u3082\u5909\u308f\u3089\u306a\u3044\u3002\u3064\u307e\u308a\uff0c\u3053\u306e\u7cfb\u306f $z$ \u8ef8\u306b\u3064\u3044\u3066\u8ef8\u5bfe\u79f0\u306a\u78c1\u5834\u3092\u3064\u304f\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\uff0c$y=0$ \u306e $xz$ \u5e73\u9762\u4e0a\u3067\u8a08\u7b97\u3057\u3066\u3088\u3044\u3002\u3053\u306e\u3068\u304d\uff0c<\/p>\n

\\begin{eqnarray}
\nB_x(x,0,z) &=& \\frac{I}{4\\pi \\varepsilon_0 c^2}
\n\\iint \\frac{\\frac{x’}{\\varrho’} \\,\\delta(\\varrho’-a)\u00a0 z}{\\left\\{ (x-x’)^2 + (y’)^2 +z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ \\\\
\n&=& \\frac{I}{4\\pi \\varepsilon_0 c^2}
\n\\iint \\frac{z \\cos \\phi’ \\,\\delta(\\varrho’-a)}{\\left\\{ (x-\\varrho’ \\cos\\phi’)^2 + (\\varrho’ \\sin\\phi’ )^2 +z^2\\right\\}^{\\frac{3}{2}}}\\, \\varrho’\\,d\\varrho’\\,d\\phi’ \\\\
\n&=& \\frac{I}{4\\pi \\varepsilon_0 c^2}
\n\\int \\frac{a z \\cos \\phi’ }{\\left\\{ x^2 + z^2 + a^2 -2 a x \\cos\\phi’ \\right\\}^{\\frac{3}{2}}}\\,d\\phi’ \\\\
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\nB_y(x,0,z) &=& \\frac{I}{4\\pi \\varepsilon_0 c^2}
\n\\iint \\frac{ \\frac{y’}{\\varrho’} \\,\\delta(\\varrho’-a)\u00a0 z}{\\left\\{ (x-x’)^2 + (y’)^2 + z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ \\\\
\n&=& \\frac{I}{4\\pi \\varepsilon_0 c^2}
\n\\iint \\frac{\u00a0 z \\sin\\phi’\u00a0 \\,\\delta(\\varrho’-a)}{\\left\\{ (x-\\varrho’ \\cos\\phi’)^2 + (\\varrho’ \\sin\\phi’ )^2 +z^2\\right\\}^{\\frac{3}{2}}}\\, \\varrho’\\,d\\varrho’\\,d\\phi’ \\\\
\n&=& \\frac{I}{4\\pi \\varepsilon_0 c^2}
\n\\int \\frac{a z \\sin \\phi’ }{\\left\\{ x^2 + z^2 + a^2 -2 a x \\cos\\phi’ \\right\\}^{\\frac{3}{2}}}\\,d\\phi’ \\\\
\n&=& \\frac{I}{4\\pi \\varepsilon_0 c^2}
\n\\Bigl[-\\frac{1}{ax \\sqrt{x^2 + z^2 + a^2 -2 a x \\cos\\phi’}} \\Bigr]_0^{2\\pi} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\nB_z(x,0,z) &=& \\frac{-I }{4\\pi \\varepsilon_0 c^2}
\n\\iint \\frac{\\left\\{\\frac{y’}{\\varrho’} (-y’) + \\frac{x’}{\\varrho’} (x-x’) \\right\\} \\delta(\\varrho’-a) }{\\left\\{ (x-x’)^2 + (y’)^2 + z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ \\\\
\n&=& \\frac{-I }{4\\pi \\varepsilon_0 c^2}
\n\\iint \\frac{\\left\\{\\sin\\phi’\u00a0 (-\\varrho’ \\sin\\phi’ ) + \\cos\\phi’\u00a0 (x-\\varrho’ \\cos\\phi’)\\right\\} \\delta(\\varrho’-a) }{\\left\\{ (x-\\varrho’ \\cos\\phi’)^2 + (\\varrho \\sin\\phi’)^2 + z^2\\right\\}^{\\frac{3}{2}}}\\, \\varrho’\\,d\\varrho’\\,d\\phi’ \\\\
\n&=& \\frac{I }{4\\pi \\varepsilon_0 c^2}
\n\\int \\frac{a^2 -a x \\cos\\phi’}{\\left\\{x^2 + z^2 + a^2 -2 a x \\cos\\phi’ \\right\\}^{\\frac{3}{2}}}\\, d\\phi’
\n\\end{eqnarray}<\/p>\n

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

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

$z$ \u8ef8\u4e0a\u3067\u306e\u78c1\u5834<\/h4>\n

\u7279\u306b\u7c21\u5358\u306b\u8a08\u7b97\u3067\u304d\u308b $z$ \u8ef8\u4e0a $\\boldsymbol{r} = (0, 0, z)$ \u306e\u78c1\u5834\u3092\u6c42\u3081\u3066\u307f\u308b\u3068\uff0c$B_x$ \u306f<\/p>\n

\\begin{eqnarray}
\nB_x(0,0,z) &=&\\frac{I}{4\\pi \\varepsilon_0 c^2}
\n\\int \\frac{a z \\cos \\phi’ }{\\left\\{ z^2 + a^2 \\right\\}^{\\frac{3}{2}}}\\,d\\phi’ \\\\
\n&=& 0\\\\ \\ \\\\
\nB_y(0,0,z)
\n&=& 0 \\\\ \\ \\\\
\nB_z(0,0,z) &=& \\frac{I }{4\\pi \\varepsilon_0 c^2}
\n\\int \\frac{a^2}{\\left\\{z^2 + a^2\\right\\}^{\\frac{3}{2}}}\\, d\\phi’ \\\\
\n&=&\\frac{I }{4\\pi \\varepsilon_0 c^2}
\n\\frac{a^2}{\\left\\{z^2 + a^2\\right\\}^{\\frac{3}{2}}} \\times 2\\pi \\\\
\n&=& \\frac{I a^2}{2 \\varepsilon_0 c^2\\left\\{z^2 + a^2\\right\\}^{\\frac{3}{2}}}
\n\\end{eqnarray}<\/p>\n

 <\/p>\n

$z$ \u8ef8\u4e0a\u3067\u306f $B_x = 0, B_y = 0$ \u3067\u3042\u308a\uff0c$B_z$ \u6210\u5206\u306e\u307f\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u30d3\u30aa\u30fb\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u304b\u3089\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u305f\u3002<\/p>\n

\u5fae\u5c0f\u306a\u5186\u96fb\u6d41\u306b\u3088\u308b\u78c1\u5834\uff1a\u78c1\u6c17\u53cc\u6975\u5b50<\/h3>\n

\u5fae\u5c0f\u306a\u5186\u96fb\u6d41\u306e\u5834\u5408\u306f\uff0c\u5186\u96fb\u6d41\u306e\u534a\u5f84 $a$ \u304c\u975e\u5e38\u306b\u5c0f\u3055\u3044\u3068\u3057\u3066
\n$$a^2 \\ll r^2 = x^2 + y^2 + z^2$$
\n\u3092\u4eee\u5b9a\u3057\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8fd1\u4f3c\u3092\u884c\u3046\u3002\uff08$\\delta(\\varrho’-a)$ \u304c\u304b\u304b\u3063\u3066\u3044\u308b\u306e\u3067\uff0c$\\varrho’ = a$ \u3068\u3057\u3066\u3088\u3044\u304b\u3089\uff0c\u5b9f\u969b\u306b\u306f $(\\varrho’)^2 \\ll r^2$ \u3068\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u306a\u308b\u3002\uff09<\/p>\n

\\begin{eqnarray}
\n\\frac{1}{\\left\\{ (x-x’)^2 + (y-y’)^2 + z^2\\right\\}^{\\frac{3}{2}}}
\n&=& \\left\\{ r^2 -2x x’ -2y y’ + (\\varrho’)^2\\right\\}^{-\\frac{3}{2}} \\\\
\n&\\simeq& \\left\\{r^2\u00a0 -2x x’ -2y y’\\right\\}^{-\\frac{3}{2}} \\\\
\n&=& \\left\\{r^2 \\left(1 -2 \\frac{x \\varrho’ \\cos\\phi’ + y \\varrho’ \\sin\\phi’}{r^2} \\right)\\right\\}^{-\\frac{3}{2}} \\\\
\n&=& \\left(r^2\\right)^{-\\frac{3}{2}} \\left(1 -2 \\frac{x \\varrho’ \\cos\\phi’ + y \\varrho’ \\sin\\phi’}{r^2} \\right)^{-\\frac{3}{2}} \\\\
\n&\\simeq& \\frac{1}{r^3} \\left(1 +3 \\frac{x \\varrho’ \\cos\\phi’ + y \\varrho’ \\sin\\phi’}{r^2} \\right)\\\\
\n&=&\\frac{1}{r^3} +3 \\frac{x \\varrho’ \\cos\\phi’ + y \\varrho’ \\sin\\phi’}{r^5}
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u8fd1\u4f3c\u3092\u4f7f\u3046\u3068\uff0c\u5186\u96fb\u6d41\u304c\u3064\u304f\u308b\u78c1\u5834\u306f<\/p>\n

\\begin{eqnarray}
\nB_x(\\boldsymbol{r})
\n&=&\\frac{1}{4\\pi \\varepsilon_0 c^2} \\iint \\frac{I \\frac{x’}{\\varrho’} \\,\\delta(\\varrho’-a)\u00a0 z}{\\left\\{ (x-x’)^2 + (y-y’)^2 +z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ \\\\
\n&\\simeq& \\frac{I z}{4\\pi \\varepsilon_0 c^2} \\iint \\cos\\phi’ \\delta(\\varrho’ -a) \\left\\{ \\frac{1}{r^3} +3 \\frac{x \\varrho’ \\cos\\phi’ + y \\varrho’ \\sin\\phi’}{r^5}\\right\\} \\,\\varrho’ \\,d\\varrho’ \\, d\\phi’\\\\
\n&=& \\frac{I z}{4\\pi \\varepsilon_0 c^2} \\int_0^{2\\pi}\u00a0 \\cos\\phi’ \\left\\{ \\frac{1}{r^3} +3 \\frac{x a \\cos\\phi’ + y a \\sin\\phi’}{r^5}\\right\\} \\,a\\, d\\phi’\\\\
\n&=& \\frac{I z}{4\\pi \\varepsilon_0 c^2} \\int_0^{2\\pi} 3 \\frac{x a^2 \\cos^2\\phi’}{r^5} \\, d\\phi’\\\\
\n&=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2} 3 \\frac{z x}{r^5}
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\nB_y(\\boldsymbol{r})
\n&=&\\frac{1}{4\\pi \\varepsilon_0 c^2} \\iint \\frac{I \\frac{y’}{\\varrho’} \\,\\delta(\\varrho’-a)\u00a0 z}{\\left\\{ (x-x’)^2 + (y-y’)^2 +z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ \\\\
\n&\\simeq& \\frac{I z}{4\\pi \\varepsilon_0 c^2} \\iint \\sin\\phi’ \\delta(\\varrho’ -a) \\left\\{ \\frac{1}{r^3} +3 \\frac{x \\varrho’ \\cos\\phi’ + y \\varrho’ \\sin\\phi’}{r^5}\\right\\} \\,\\varrho’ \\,d\\varrho’ \\, d\\phi’\\\\
\n&=& \\frac{I z}{4\\pi \\varepsilon_0 c^2} \\int_0^{2\\pi}\u00a0 \\sin\\phi’ \\left\\{ \\frac{1}{r^3} +3 \\frac{x a \\cos\\phi’ + y a \\sin\\phi’}{r^5}\\right\\} \\,a\\, d\\phi’\\\\
\n&=& \\frac{I z}{4\\pi \\varepsilon_0 c^2} \\int_0^{2\\pi} 3 \\frac{y a^2 \\sin^2\\phi’}{r^5} \\, d\\phi’\\\\
\n&=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2} 3 \\frac{z y}{r^5}
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\nB_z(\\boldsymbol{r}) &=&\\frac{-I }{4\\pi \\varepsilon_0 c^2} \\iint \\frac{\\left\\{\\frac{y’}{\\varrho’} (y-y’) + \\frac{x’}{\\varrho’} (x-x’)\\right\\} \\delta(\\varrho’-a) }{\\left\\{ (x-x’)^2 + (y-y’)^2 + z^2\\right\\}^{\\frac{3}{2}}}\\, dx’ dy’ \\\\
\n&\\simeq& \\frac{I a}{4\\pi \\varepsilon_0 c^2}\\int_0^{2\\pi} \\,d\\phi’ \\left(a-x\\cos\\phi’ -y\\sin\\phi’ \\right)\\left\\{ \\frac{1}{r^3} +3 \\frac{x \\varrho’ \\cos\\phi’ + y \\varrho’ \\sin\\phi’}{r^5}\\right\\} \\\\
\n&=& \\frac{I a^2}{4\\pi \\varepsilon_0 c^2}\\int_0^{2\\pi}\\,d\\phi’ \\left\\{\\frac{1}{r^3} -3\\frac{x^2}{r^5} \\cos^2\\phi’\u00a0 -3\\frac{y^2}{r^5} \\sin^2\\phi’\u00a0 \\right\\} \\\\
\n&=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2}\\left\\{ \\frac{2}{r^3} -3\\frac{x^2}{r^5} -3\\frac{y^2}{r^5} \\right\\} \\\\
\n&=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2} \\frac{2(x^2 + y^2 + z^2) -3 x^2 -3 y^2}{r^5}\\\\
\n&=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2} \\frac{3 z^2 -(x^2 + y^2 + z^2) }{r^5}\\\\
\n&=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2} \\left\\{3 \\frac{z^2}{r^5} -\\frac{1}{r^3}\\right\\}
\n\\end{eqnarray}<\/p>\n

\u7d50\u679c\u3092\u307e\u3068\u3081\u308b\u3068<\/p>\n

\\begin{eqnarray}
\nB_x &=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2}\u00a0 \\frac{3}{r^5} z x\\\\
\nB_y &=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2}\u00a0 \\frac{3 }{r^5}z y \\\\
\nB_z &=& \\frac{I \\pi a^2}{4\\pi \\varepsilon_0 c^2} \\left\\{3 \\frac{z^2}{r^5} -\\frac{1}{r^3}\\right\\}
\n\\end{eqnarray}<\/p>\n

\u3069\u3063\u304b\u3067\u898b\u305f\u3053\u3068\u304c\u3042\u308b\u5f62\u3067\u3059\u306d\u3002\u305d\u3046\u3067\u3059\uff0c\u96fb\u6c17\u53cc\u6975\u5b50\u304c\u3064\u304f\u308b\u9060\u65b9\u306e\u96fb\u5834<\/a><\/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\\}\\\\ \\ \\\\
\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

\u3068\u305d\u3063\u304f\u308a\u3067\u3059\uff01<\/p>\n

\"\"<\/p>\n

\u96fb\u6c17\u53cc\u6975\u5b50\u30e2\u30fc\u30e1\u30f3\u30c8\u306b\u306a\u3089\u3063\u3066\uff0c\u78c1\u6c17\u53cc\u6975\u5b50\u30e2\u30fc\u30e1\u30f3\u30c8 $\\boldsymbol{\\mu}$ \u3092<\/p>\n

$$\\boldsymbol{\\mu} \\equiv (0, 0, \\mu) \\equiv (0, 0, I \\pi a^2)$$<\/p>\n

\u3068\u5b9a\u7fa9\u3059\u308c\u3070\uff0c\u5fae\u5c0f\u306a\u5186\u96fb\u6d41\uff08\u9762\u7a4d $S = \\pi a^2$ \u306e\u5186\u5468\u4e0a\u3092\u6d41\u308c\u308b\u96fb\u6d41 $I$\uff09\u304c\u3064\u304f\u308b\u9060\u65b9\u306e\u78c1\u5834\u306f\uff0c\u78c1\u6c17\u53cc\u6975\u5b50\u30e2\u30fc\u30e1\u30f3\u30c8 $\\boldsymbol{\\mu}, \\\u00a0 \\mu = |\\boldsymbol{\\mu}| = I S$ \u304c\u3064\u304f\u308b\u53cc\u6975\u5b50\u78c1\u5834<\/p>\n

$$\\boldsymbol{B} = \\frac{1}{4\\pi\\varepsilon_0 c^2} \\left\\{3 \\frac{\\boldsymbol{r}\\cdot\\boldsymbol{\\mu} }{r^5} \\boldsymbol{r} -\\frac{\\boldsymbol{\\mu}}{r^3} \\right\\}$$
\n\u3067\u3042\u308b\uff01\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

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

\u5730\u7403\u78c1\u5834\u3082\uff0c\u5927\u5c40\u7684\u306b\u898b\u308c\u3070\u5317\u6975\u3092 $S$ \u6975\uff0c\u5357\u6975\u3092 $N$ \u6975\u3068\u3059\u308b\u53cc\u6975\u5b50\u78c1\u5834\u3067\u3042\u308b\u304c\uff0c\u5730\u7403\u5185\u90e8\u306b\u4f55\u304b\u68d2\u78c1\u77f3\u304c\u3042\u308b\u306e\u3060\u3068\u8003\u3048\u308b\u3088\u308a\u306f\uff0c\u5730\u7403\u5185\u90e8\u306b\u5186\u96fb\u6d41\u304c\u6d41\u308c\u3066\u3044\u3066\uff0c\u305d\u308c\u306b\u3088\u3063\u3066\u78c1\u5834\u304c\u3064\u304f\u3089\u308c\u3066\u3044\u308b\u306e\u3060\u3068\u7406\u89e3\u3059\u308b\u3002<\/p>\n

\u30bd\u30ec\u30ce\u30a4\u30c9\u3092\u6d41\u308c\u308b\u96fb\u6d41\u306b\u3088\u308b\u78c1\u5834<\/h3>\n

\u30bd\u30ec\u30ce\u30a4\u30c9\u306f1\u672c\u306e\u96fb\u7dda\u3092\u87ba\u65cb\u72b6\u306b\u5bc6\u306b\u5dfb\u3044\u305f\u3082\u306e\u3067\u3042\u308b\u3002<\/p>\n

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

$z$ \u8ef8\u3092\u4e2d\u5fc3\u3068\u3057\uff0c\u534a\u5f84 $a$ \u3067\u5358\u4f4d\u9577\u3055\u3042\u305f\u308a $n$ \u56de\u5dfb\u304d\u306e\u300c\u5341\u5206\u306b\u9577\u3044\u300d\u30bd\u30ec\u30ce\u30a4\u30c9\u3092\u6d41\u308c\u308b\u96fb\u6d41 $I$ \u306b\u3088\u308b\u78c1\u5834\u3092\u6c42\u3081\u308b\u3002\u7e70\u308a\u8fd4\u3059\u304c\u30bd\u30ec\u30ce\u30a4\u30c9\u306f1\u672c\u306e\u96fb\u7dda\u3092\u87ba\u65cb\u72b6\u306b\u5bc6\u306b\u5dfb\u3044\u305f\u3082\u306e\u3067\u3042\u308b\u306e\u3060\u304c\uff0c\u7c21\u5358\u306e\u305f\u3081\u306b\u4ee5\u4e0b\u3067\u306f\u5186\u96fb\u6d41\u56de\u8def\u306e\u91cd\u306d\u5408\u308f\u305b\u3068\u3057\u3066\u6271\u3046\u3002<\/p>\n

\u307e\u305a\uff0c1\u56de\u5dfb\u304d\u306e\u5186\u96fb\u6d41\u3092\u3042\u3089\u308f\u3059\u96fb\u6d41\u5bc6\u5ea6 $\\boldsymbol{J}$ \u306f<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{J} &=& (J_x, J_y, 0) \\\\
\n&=& (-I\\,\\delta(z) \\sin\\phi \\,\\delta(\\varrho-a), I\\,\\delta(z) \\cos\\phi \\,\\delta(\\varrho-a), 0)
\n\\end{eqnarray}<\/p>\n

\u3067\u3042\u3063\u305f\u304b\u3089\uff0c\u5358\u4f4d\u9577\u3055\u3042\u305f\u308a $n$ \u56de\u5dfb\u304d\u306e\u5341\u5206\u306b\u9577\u3044\u30bd\u30ec\u30ce\u30a4\u30c9\u306e\u5834\u5408\u306f\uff0c\u4e0a\u8a18\u306e\u8868\u5f0f\u3067<\/p>\n

$$I\\,\\delta(z)\\ \\ \\rightarrow \\ \\ n I$$<\/p>\n

\u3068\u7f6e\u304d\u63db\u3048\u308c\u3070\u3088\u304f\uff0c\u30bd\u30ec\u30ce\u30a4\u30c9\u3092\u6d41\u308c\u308b\u96fb\u6d41\u5bc6\u5ea6 $\\boldsymbol{J}$ \u306f<\/p>\n

\\begin{eqnarray}
\n\\boldsymbol{J} &=& (J_x, J_y, 0) \\\\
\n&=& (-n I \\sin\\phi \\,\\delta(\\varrho-a), n I \\cos\\phi \\,\\delta(\\varrho-a), 0)
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u3002\u3053\u308c\u3092\u30d3\u30aa\u30fb\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u306b\u4ee3\u5165\u3057\u3066\uff0c\u5404\u6210\u5206\u3054\u3068\u306b\u78c1\u675f\u5bc6\u5ea6\u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n

\\begin{eqnarray}
\nB_x &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_y(\\boldsymbol{r}’)\\cdot(z-z’) -J_z(\\boldsymbol{r}’)\\cdot(y-y’)}{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}}\\, dV’ \\\\
\n&=& \\frac{n I}{4\\pi \\varepsilon_0 c^2} \\iint\\,dx’ dy’ \\int_{-\\infty}^{\\infty}\\,dz’\u00a0 \\frac{\\cos\\phi’ \\,\\delta(\\varrho’-a)\\cdot(z-z’)}{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}} \\\\
\n&=& \\frac{n I}{4\\pi \\varepsilon_0 c^2} \\!\\!\\iint\\!\\! dx’ dy’ \\cos\\phi’ \\delta(\\varrho’-a)\\!\\!\\int_{-\\infty}^{\\infty}\\!\\!\u00a0 \\frac{(z-z’) \\,dz’}{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\u540c\u69d8\u306b\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nB_y &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_z(\\boldsymbol{r}’)\\cdot(x-x’) -J_x(\\boldsymbol{r}’)\\cdot(z-z’)}{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}}\\, dV’ \\\\
\n&=& \\frac{-n I}{4\\pi \\varepsilon_0 c^2} \\iint\\,dx’\\, dy’ \\int_{-\\infty}^{\\infty}\\,dz’\u00a0 \\frac{\\sin\\phi’ \\,\\delta(\\varrho’-a)\\cdot(z-z’)}{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}} \\\\
\n&=& \\frac{-n I}{4\\pi \\varepsilon_0 c^2} \\iint\\!\\! dx’ dy’ \\sin\\phi’ \\delta(\\varrho’-a)\\!\\!\\int_{-\\infty}^{\\infty} \\!\\! \\frac{(z-z’)\\,dz’ }{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}} \\\\
\n&=& 0
\n\\end{eqnarray}<\/p>\n

\u30bd\u30ec\u30ce\u30a4\u30c9\u306e\u5185\u90e8\u30fb\u5916\u90e8\u306b\u3088\u3089\u305a\uff0c$B_x = 0, \\ B_y = 0$ \u3067\u3042\u308b\u3053\u3068\u304c\u30d3\u30aa\u30fb\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u304b\u3089\u76f4\u63a5\u8a08\u7b97\u3067\u304d\u305f\u3002<\/p>\n

\u6700\u5f8c\u306b\uff0c$B_z$ \u306f<\/p>\n

\\begin{eqnarray}
\nB_z &=& \\frac{1}{4\\pi \\varepsilon_0 c^2} \\iiint \\frac{J_x(\\boldsymbol{r}’)\\cdot(y-y’) -J_y(\\boldsymbol{r}’)\\cdot(x-x’)}{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}}\\, dV’ \\\\
\n&=& \\frac{n I}{4\\pi \\varepsilon_0 c^2}
\n\\iint\\,dx’\\, dy’ \\int_{-\\infty}^{\\infty}\\,dz’\u00a0 \\frac{ \\left\\{\\frac{y’}{\\varrho’}(y’-y) + \\frac{x’}{\\varrho’} (x’ -x) \\right\\} \\,\\delta(\\varrho’-a) }{\\left\\{(x-x’)^2 + (y-y’)^2 + (z-z’)^2 \\right\\}^{\\frac{3}{2}}} \\\\
\n&=& \\frac{n I}{2\\pi \\varepsilon_0 c^2}
\n\\iint\\,dx’\\, dy’ \\frac{ \\left\\{\\frac{y’}{\\varrho’}(y’-y) + \\frac{x’}{\\varrho’} (x’ -x) \\right\\} \\,\\delta(\\varrho’-a) }{(x-x’)^2 + (y-y’)^2 } \\\\
\n&=&\\frac{n I}{2\\pi \\varepsilon_0 c^2}
\n\\iint \\varrho’ \\,d\\varrho’ \\,d\\phi’\u00a0 \\frac{ \\left\\{\\varrho’ -y\\sin\\phi’ -x\\cos\\phi’ \\right\\} \\,\\delta(\\varrho’-a) }{(x-\\varrho’ \\cos\\phi’ )^2 + (y-\\varrho’ \\sin\\phi’)^2 } \\\\
\n&=&\\frac{n I}{2\\pi \\varepsilon_0 c^2}
\n\\int\u00a0 \\,d\\phi’\u00a0 \\frac{ a^2\u00a0 -a y\\sin\\phi’ -a x\\cos\\phi’\u00a0 }{(x-a \\cos\\phi’ )^2 + (y-a \\sin\\phi’)^2 } \\\\
\n&=&\\frac{n I}{2\\pi \\varepsilon_0 c^2}
\n\\int_0^{2\\pi}\u00a0 \\,d\\phi’\u00a0 \\frac{ a^2\u00a0 -a\u00a0 y\\sin\\phi’ -a x\\cos\\phi’\u00a0 }{x^2 + y^2 + a^2 -2 a x \\cos\\phi’ -2 a y \\sin\\phi’ } \\\\ \\ \\\\
\n&=& \\left\\{
\n\\begin{array}{ll}
\n\\frac{n I}{\\varepsilon_0 c^2}\u00a0 & ( \\sqrt{x^2 + y^2} < a)\\\\ \\ \\\\
\n0 & (\\sqrt{x^2 + y^2} > a)
\n\\end{array}
\n\\right.
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c\u4ee5\u4e0b\u306e\u7a4d\u5206\u7d50\u679c\u3092\u4f7f\u3063\u305f\u3002<\/p>\n

\\begin{eqnarray}
\n\\int_0^{2\\pi}\u00a0 \\,d\\phi’\u00a0 \\frac{ a^2\u00a0 -a\u00a0 y\\sin\\phi’ -a x\\cos\\phi’\u00a0 }{x^2 + y^2 + a^2 -2 a x \\cos\\phi’ -2 a y \\sin\\phi’ } &=& \\left\\{
\n\\begin{array}{ll}
\n2\\pi\u00a0 & ( \\sqrt{x^2 + y^2} < a)\\\\ \\ \\\\
\n0 & (a < \\sqrt{x^2 + y^2} )
\n\\end{array}
\n\\right.
\n\\end{eqnarray}<\/p>\n

\u3064\u307e\u308a\uff0c$z$ \u8ef8\u304b\u3089\u306e\u8ddd\u96e2 $\\sqrt{x^2 + y^2}$ \u304c\u30bd\u30ec\u30ce\u30a4\u30c9\u306e\u534a\u5f84 $a$ \u3088\u308a\u3082\u5c0f\u3055\u3044\u30bd\u30ec\u30ce\u30a4\u30c9\u306e\u5185\u90e8 ($\\sqrt{x^2 + y^2} < a$) \u3067\u306f $\\displaystyle B_z = \\frac{n I}{\\varepsilon_0 c^2} $ \u3068\u3044\u3046\u4e00\u5b9a\u306e\u78c1\u5834\u304c\uff0c$z$ \u8ef8\u304b\u3089\u306e\u8ddd\u96e2 $\\sqrt{x^2 + y^2}$ \u304c\u30bd\u30ec\u30ce\u30a4\u30c9\u306e\u534a\u5f84 $a$ \u3088\u308a\u3082\u5927\u304d\u3044\u30bd\u30ec\u30ce\u30a4\u30c9\u306e\u5916\u90e8 ($\\sqrt{x^2 + y^2} > a$) \u3067\u306f $B_z = 0$ \u3068\u306a\u308b\u3053\u3068\u304c\uff0c\u30d3\u30aa\u30fb\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u304b\u3089\u76f4\u63a5\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u305f\u3002<\/p>\n

\"\"<\/a>\"\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

\u30a2\u30f3\u30da\u30fc\u30eb\u306e\u6cd5\u5247\u306a\u3069\u3092\u4f7f\u308f\u305a\u306b\uff0c\u30d3\u30aa-\u30b5\u30d0\u30fc\u30eb\u306e\u6cd5\u5247\u3092\u4f7f\u3063\u3066\u96fb\u6d41\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u9759\u78c1\u5834\u3092\u6c42\u3081\u308b\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2561,"menu_order":26,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2687","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2687","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"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=2687"}],"version-history":[{"count":90,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2687\/revisions"}],"predecessor-version":[{"id":10651,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2687\/revisions\/10651"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2561"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2687"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}