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電磁気まとめ

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ローレンツゲージにおけるマクスウェルの方程式

証明はここ

    \boldsymbol{A} = \left(\frac{\phi}{c}, A_x, A_y, A_z \right)        \tag{電磁ポテンシャル}
    \boldsymbol{i} = \left(\rho c, i_x, i_y, i_z \right)        \tag{4元電流密度}
    \Box \boldsymbol{A} = - \mu_0 \boldsymbol{i}        \tag{ローレンツゲージにおけるマクスウェルの方程式}
    \mathbf{div} \boldsymbol{A} + \frac{1}{c^2} \frac{∂ \phi}{∂t} = 0       \tag{ローレンツ条件}

相対論的なマクスウェルの方程式

証明はここ

    \boldsymbol{A} = \left(\frac{\phi}{c}, A_x, A_y, A_z \right)        \tag{電磁ポテンシャル}
    \boldsymbol{i} = \left(\rho c, i_x, i_y, i_z \right)        \tag{4元電流密度}
    \Box A^\mu - ∂^\mu \left(∂_\nu A^\nu \right)
    = - \mu_0 i^\nu        \tag{相対論的なマクスウェルの方程式}
    ∂_\nu i^\nu = 0        \tag{相対論的な電荷の保存則}
    ∂_\nu A^\nu = 0        \tag{相対論的なローレンツ条件}

ゲージ変換

    \boldsymbol{A}^{’} = \boldsymbol{A} + \mathbf{grad} \chi
    \phi^{’} = \phi - \frac{∂ \chi}{∂t}

電磁気でよく使用するベクトル公式

    \mathbf{rot} (\mathbf{rot} \boldsymbol{X}) = \mathbf{grad} (\mathbf{div} \boldsymbol{X}) - \triangle \boldsymbol{X}
    \mathbf{rot} \cdot\mathbf{grad} \phi = \boldsymbol{0}
        \mathbf{div} \cdot\mathbf{rot} \boldsymbol{X} = 0
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