Electrodynamics/Magnetic Stress Tensor

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Differential Version[edit]

\mathbf{F} = \frac{q}{c}\mathbf{v} \times \mathbf{B}

Volume Integral Version[edit]

\mathbf{F} = \int_V(\mathbf{j} \times \mathbf{B})dV

Magnetic Stress Tensor[edit]

\mathbb{T}_M = \frac{1}{4\pi} \begin{bmatrix}
 B_x^2 - \frac{B^2}{2} & B_x B_y & B_x B_z \\
 B_x B_y & B_y^2 - \frac{B^2}{2} & B_y B_z \\
 B_x B_z & B_y B_z & B_z^2 - \frac{B^2}{2}
\end{bmatrix}

Surface Integral Version[edit]

\mathbf{F} = \int_S \mathbb{T}_M \mathbf{n} dA

Electromagnetic Stress Tensor[edit]

If we add our two stress tensors together, piece-wise, we will get a combined electromagnetic stress tensor:

\mathbb{T}_{EM} = \mathbb{T}_E + \mathbb{T}_M