Electrodynamics/Magnetic Stress Tensor

Differential Version

$𝐅=\frac{q}{c}𝐯\times 𝐁$

Volume Integral Version

$𝐅=\underset{V}{\int }(𝐣\times 𝐁)dV$

Magnetic Stress Tensor

${𝕋}_M=\frac{1}{4\pi }\left[\begin{array}{ccc}{B}_x^2-\frac{B^2}{2}& B_x{B}_y& B_x{B}_z\\ B_x{B}_y& B_y^2-\frac{B^2}{2}& B_y{B}_z\\ B_x{B}_z& B_y{B}_z& B_z^2-\frac{B^2}{2}\end{array}\right]$

Surface Integral Version

$𝐅=\underset{S}{\int }{𝕋}_M𝐧dA$

Electromagnetic Stress Tensor

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

${𝕋}_{EM}={𝕋}_E+{𝕋}_M$