<\/p>\n
\\( \\nabla\\cdot\\boldsymbol{B} = 0\\) \u306f\u5909\u308f\u3089\u306a\u3044\u306e\u3067\uff0c\u9759\u78c1\u5834\u306e\u3068\u304d\u3068\u540c\u69d8\u306b\u30d9\u30af\u30c8\u30eb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092 \\(\\boldsymbol{B} \\equiv \\nabla\\times\\boldsymbol{A}\\) \u306e\u3088\u3046\u306b\u5c0e\u5165\u3067\u304d\u308b\u3002<\/p>\n
\u3057\u304b\u3057\uff0c<\/p>\n
$$\\nabla\\times\\boldsymbol{E} + \\frac{\\partial \\boldsymbol{B}}{\\partial t} = \\boldsymbol{0}$$
\n\u3064\u307e\u308a\uff0c\\(\\nabla\\times\\boldsymbol{E} \\neq \\boldsymbol{0}\\) \u3068\u306a\u3063\u3066\u3057\u307e\u3046\u305f\u3081\u306b\uff0c\u9759\u96fb\u5834\u306e\u3068\u304d\u3068\u5168\u304f\u540c\u3058\u3088\u3046\u306b\u3057\u3066\u30b9\u30ab\u30e9\u30fc\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092\u5c0e\u5165\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u3002\u3069\u3046\u3057\u3088\u3046\uff1f<\/p>\n
\u3053\u3093\u306a\u3068\u304d\u3082\uff0c\u614c\u3066\u305a\u9a12\u304c\u305a\uff0c\\(\\boldsymbol{B} = \\nabla\\times\\boldsymbol{A}\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\uff0c
\n$$\\nabla\\times\\boldsymbol{E} + \\frac{\\partial}{\\partial t}\\left(\\nabla\\times\\boldsymbol{A}\\right) = \\boldsymbol{0}$$<\/p>\n
<\/p>\n
\u504f\u5fae\u5206\u306f\u4ea4\u63db\u53ef\u80fd \\(\\displaystyle \\frac{\\partial}{\\partial t}\\nabla = \\nabla \\frac{\\partial}{\\partial t}\\) \u3067\u3042\u308b\u304b\u3089
\n\\begin{eqnarray}
\n\\nabla\\times\\boldsymbol{E} + \\nabla\\times\\frac{\\partial\\boldsymbol{A}}{\\partial t} &= & \\boldsymbol{0}\\\\
\n\\nabla\\times \\left(\\boldsymbol{E} + \\frac{\\partial\\boldsymbol{A}}{\\partial t} \\right) &= & \\boldsymbol{0}\\\\
\n\\therefore\\ \\ \\boldsymbol{E} + \\frac{\\partial\\boldsymbol{A}}{\\partial t} &\\equiv& -\\nabla\\phi \\\\
\n\\therefore\\ \\ \\boldsymbol{E} &=& -\\nabla\\phi -\\frac{\\partial\\boldsymbol{A}}{\\partial t}
\n\\end{eqnarray}<\/p>\n
\u6642\u9593\u5909\u52d5\u304c\u306a\u3044\u5834\u5408\u306f\uff0c\u30af\u30fc\u30ed\u30f3\u30b2\u30fc\u30b8\u6761\u4ef6<\/strong><\/span> \\(\\nabla\\cdot\\boldsymbol{A} = 0\\) \u3092\u8ab2\u3059\u3053\u3068\u3067\uff0c\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u304c\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f<\/strong><\/span>\u306e\u5f62\u306b\u7d71\u4e00\u7684\u306b\u66f8\u304b\u308c\uff0c\u305f\u3060\u3061\u306b\u5b8c\u5168\u306a\u89e3\u304c\u3082\u3068\u307e\u3063\u305f\u304c\uff0c\u6642\u9593\u5909\u52d5\u304c\u3042\u308b\u5834\u5408\u306f\uff0c\u30af\u30fc\u30ed\u30f3\u30b2\u30fc\u30b8\u6761\u4ef6<\/strong><\/span>\u3067\u306f\u306a\u304f\uff0c\u5225\u306e\u30b2\u30fc\u30b8\u6761\u4ef6\u304c\u4fbf\u5229\u3067\u3042\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n \u305d\u308c\u306f\u30ed\u30fc\u30ec\u30f3\u30c4\u30b2\u30fc\u30b8\u6761\u4ef6<\/strong><\/span>\u3068\u547c\u3070\u308c\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u308b\u3002 \n \u306a\u305c\u3053\u306e\u30b2\u30fc\u30b8\u6761\u4ef6\u304c\u4fbf\u5229\u3067\u3042\u308b\u304b\u306f\uff0c\u3053\u306e\u5f8c…<\/p>\n <\/p>\n \u6642\u9593\u5909\u52d5\u3092\u3059\u308b\u96fb\u78c1\u5834\u306e\u5834\u5408\u306f\uff0c\u96fb\u5834\uff0c\u78c1\u5834\u304c\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u308b\u3002\u8d64\u8272\u90e8\u5206\u304c\u5909\u66f4\u7b87\u6240\u3002<\/p>\n $$\\boldsymbol{E} = -\\nabla\\phi {\\color{red}{-\\frac{\\partial\\boldsymbol{A}}{\\partial t}}}, \\quad\\boldsymbol{B} = \\nabla\\times\\boldsymbol{A}$$<\/p>\n \u307e\u305f\uff0c\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306b\u5bfe\u3059\u308b\u30b2\u30fc\u30b8\u6761\u4ef6\u3082\uff0c\u30af\u30fc\u30ed\u30f3\u30b2\u30fc\u30b8\u6761\u4ef6\u3067\u306f\u306a\u304f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30ed\u30fc\u30ec\u30f3\u30c4\u30b2\u30fc\u30b8\u6761\u4ef6\u3092\u8ab2\u3059\u3068\uff0c\u3044\u308d\u3044\u308d\u3068\u4fbf\u5229\u306b\u306a\u308b\u3002\u8d64\u8272\u90e8\u5206\u304c\u5909\u66f4\u7b87\u6240\u3002 \\begin{eqnarray} \u3092\u4f7f\u3063\u3066\uff0c\u30de\u30af\u30b9\u30a6\u30a7\u30eb\u65b9\u7a0b\u5f0f\u3092$\\boldsymbol{E}$ \u3068 $\\boldsymbol{B}$ \u3067\u8868\u3059\u3068<\/p>\n \\begin{eqnarray} $(1)$ \u5f0f\u306e $\\boldsymbol{E}$ \u306b\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u8868\u8a18\u3092\u4ee3\u5165\u3057\u3066\u30ed\u30fc\u30ec\u30f3\u30c4\u30b2\u30fc\u30b8\u6761\u4ef6\u3092\u4f7f\u3046\u3068\uff0c<\/p>\n \\begin{eqnarray} $(4)$ \u5f0f\u3082\u540c\u69d8\u306b\u3057\u3066<\/p>\n \\begin{eqnarray} \u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50<\/strong><\/span>\uff08Laplace \u306e\u5f62\u5bb9\u8a5e\u5f62\u3067 Laplacian \u30e9\u30d7\u30e9\u30b7\u30a2\u30f3<\/strong><\/span>\u3068\u3082\u3044\u3046\uff09 $\\nabla^2 $ \u306f $$\\square \\equiv \\triangle -\\frac{1}{c^2} \\frac{\\partial^2}{\\partial t^2} = \\nabla^2 -\\frac{1}{c^2} \\frac{\\partial^2}{\\partial t^2} $$<\/p>\n \u30ed\u30fc\u30ec\u30f3\u30c4\u30b2\u30fc\u30b8\u6761\u4ef6\u3092\u8ab2\u3057\u305f\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306f\uff0c\u4ee5\u4e0b\u306e\u5f0f\u306b\u5f93\u3046\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n \\begin{eqnarray} \u3053\u306e\u5f0f\u3082\u3061\u3083\u3093\u3068\u89e3\u3051\u308b\u306e\u3067\u3042\u308b\u304c\uff0c\u305d\u306e\u8a71\u306f\u307e\u305f\u5225\u306e\u6a5f\u4f1a\u306b…<\/p>\n \u3061\u306a\u307f\u306b\uff0c\u4e0a\u306e\u5f0f\u3067\u53f3\u8fba\u304c\u30bc\u30ed\uff08\u30bc\u30ed\u30d9\u30af\u30c8\u30eb\uff09\u3068\u306a\u308b\u5f0f\u306f\uff08\u901f\u3055\u304c $c$ \u3067\u3042\u308b\u6ce2\u3092\u3042\u3089\u308f\u3059\uff09\u300c\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u300d\u3068\u547c\u3070\u308c\uff0c\u3042\u3068\u3067\u51fa\u3066\u304d\u307e\u3059\u3002<\/p>\n Laplace \u306f\u300c\u30e9\u30d7\u30e9\u30b9\u300d\uff0c\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\u306e\u610f\u5473\u306e Laplacian \u306f\u300c\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u300d\u3002<\/p>\n d’Alembert<\/strong><\/span> \u306f\u300c\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb<\/strong><\/span>\u300d\uff08\u6700\u5f8c\u306e t \u306f\u767a\u97f3\u3057\u306a\u3044\uff09\u3002\u3067\u306f\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u6f14\u7b97\u5b50\u306e\u610f\u5473\u306e d’Alembertian<\/strong><\/span> \u306f\u300c\u30c0\u30e9\u30f3\u30d9\u30fc\u30ea\u30a2\u30f3\u300d\u304b\u3068\u3044\u3046\u3068\u305d\u3046\u3067\u306f\u306a\u304f\uff0cWikipedia \u306b\u3088\u308c\u3070\u300c\u30c0\u30e9\u30f3\u30d9\u30eb\u30b7\u30a2\u30f3<\/strong><\/span>\u300d\u3002<\/p>\n \u53c2\u8003\uff1a<\/p>\n
\n$$\\frac{1}{c^2} \\frac{\\partial \\phi}{\\partial t} + \\nabla\\cdot\\boldsymbol{A} = 0$$<\/p>\n\u3053\u3053\u307e\u3067\u306e\u307e\u3068\u3081<\/h3>\n
\n$${\\color{red}{\\frac{1}{c^2} \\frac{\\partial \\phi}{\\partial t}}} + \\nabla\\cdot\\boldsymbol{A} = 0$$<\/p>\n$\\boldsymbol{E}$ \u3068 $\\boldsymbol{B}$ \u3067\u8868\u3057\u305f\u30de\u30af\u30b9\u30a6\u30a7\u30eb\u65b9\u7a0b\u5f0f<\/h3>\n
\n\\boldsymbol{D} &=& \\varepsilon_0 \\boldsymbol{E}\\\\
\n\\boldsymbol{H} &=& \\frac{1}{\\mu_0} \\boldsymbol{B} = \\varepsilon_0 c^2 \\boldsymbol{B}
\n\\end{eqnarray}<\/p>\n
\\nabla\\cdot \\boldsymbol{E} &=& \\frac{\\rho}{\\varepsilon_0}\u00a0 \\tag{1}\\\\
\\nabla\\cdot\\boldsymbol{B} &=& 0\u00a0 \\tag{2}\\\\
\\nabla\\times\\boldsymbol{E} + \\frac{\\partial \\boldsymbol{B}}{\\partial t} &=& \\boldsymbol{0}\u00a0 \\tag{3}\\\\
\\nabla\\times\\boldsymbol{B} -\\frac{1}{c^2} \\frac{\\partial \\boldsymbol{E}}{\\partial t} &=& \\frac{\\boldsymbol{J}}{\\varepsilon_0 c^2} \\tag{4}
\\end{eqnarray}<\/p>\n
\n\\nabla\\cdot\\boldsymbol{E} &=& \\nabla\\cdot\\left\\{ -\\nabla\\phi -\\frac{\\partial\\boldsymbol{A}}{\\partial t}\\right\\} \\\\
\n&=& -\\nabla^2 \\phi -\\frac{\\partial}{\\partial t} \\nabla\\cdot\\boldsymbol{A} \\\\
\n&=& -\\nabla^2 \\phi +\\frac{1}{c^2} \\frac{\\partial^2 \\phi}{\\partial t^2}\u00a0 \\\\
\n&=& \\frac{\\rho}{\\varepsilon_0} \\\\ \\ \\\\
\n\\therefore\\ \\ \\left(\\nabla^2 -\\frac{1}{c^2} \\frac{\\partial^2}{\\partial t^2}\\right) \\phi &=& -\\frac{\\rho}{\\varepsilon_0}
\n\\end{eqnarray}<\/p>\n
\n\\nabla\\times\\boldsymbol{B} -\\frac{1}{c^2} \\frac{\\partial \\boldsymbol{E}}{\\partial t}
\n&=& \\nabla\\times \\left\\{\\nabla\\times \\boldsymbol{A}\\right\\} + \\frac{1}{c^2} \\left\\{ \\nabla\\frac{\\partial \\phi}{\\partial t} + \\frac{\\partial^2 \\boldsymbol{A}}{\\partial t^2} \\right\\}\\\\
\n&=& \\nabla (\\nabla\\cdot\\boldsymbol{A}) -\\nabla^2 \\boldsymbol{A} + \\frac{1}{c^2} \\left\\{ \\nabla\\frac{\\partial \\phi}{\\partial t} + \\frac{\\partial^2 \\boldsymbol{A}}{\\partial t^2} \\right\\}\\\\
\n&=& -\\nabla^2 \\boldsymbol{A} + \\frac{1}{c^2} \\frac{\\partial^2 \\boldsymbol{A}}{\\partial t^2}
\n+ \\nabla \\left(\\frac{1}{c^2} \\frac{\\partial \\phi}{\\partial t}\u00a0 + \\nabla\\cdot\\boldsymbol{A}\\right) \\\\
\n&=& -\\nabla^2 \\boldsymbol{A} + \\frac{1}{c^2} \\frac{\\partial^2 \\boldsymbol{A}}{\\partial t^2} \\\\
\n&=& \\frac{\\boldsymbol{J}}{\\varepsilon_0 c^2} \\\\ \\ \\\\
\n\\therefore\\ \\ \\left(\\nabla^2 -\\frac{1}{c^2} \\frac{\\partial^2}{\\partial t^2}\\right) \\boldsymbol{A} &=& -\\frac{\\boldsymbol{J}}{\\varepsilon_0 c^2}
\n\\end{eqnarray}<\/p>\n\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u6f14\u7b97\u5b50\u3067\u8868\u3057\u305f\u96fb\u78c1\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306e\u5f0f<\/h3>\n
\n$$\\nabla^2 = \\nabla\\cdot\\nabla \\equiv \\triangle$$
\n\u3068\u3082\u66f8\u304f\u306e\u3067\uff08\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\u3092\u30ae\u30ea\u30b7\u30a2\u6587\u5b57\u306e\u30c7\u30eb\u30bf\u306e\u5927\u6587\u5b57 $\\Delta$\u00a0 \u3067\u66f8\u304f\u3068\u3059\u308b\u8a18\u8ff0\u3082\u6563\u898b\u3055\u308c\u308b\u304c\uff0c\u305d\u306e\u610f\u898b\u306b\u306f\u4e0e\u3057\u306a\u3044\u3002\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\u306f\u30b9\u30ab\u30e9\u30fc\u6f14\u7b97\u5b50\u3060\u304b\u3089 $\\triangle$ \u3068\u66f8\u304f\u3079\u304d\u304b\u3068\u601d\u3046\uff09\uff0c\u3053\u308c\u306b\u30a4\u30f3\u30b9\u30d1\u30a4\u30a2\u3055\u308c\u3066\u30c0\u30e9\u30f3\u30d9\u30fc\u30eb\u6f14\u7b97\u5b50<\/strong><\/span> $\\square$ \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\u3068\uff0c<\/p>\n
\n\\square\\,\u00a0 \\phi &=& -\\frac{\\rho}{\\varepsilon_0}\\\\
\n\\square\\,\u00a0 \\boldsymbol{A} &=& -\\frac{\\boldsymbol{J}}{\\varepsilon_0 c^2}
\n\\end{eqnarray}<\/p>\nd’Alembertian (d’Alembert operator) \u306e\u8aad\u307f\u65b9<\/h3>\n