Electrodynamics/Magnetic Stress Tensor

Differential Version

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

Volume Integral Version

$\mathbf {F} =\int _{V}(\mathbf {j} \times \mathbf {B} )dV$

Magnetic Stress Tensor

$\mathbb {T} _{M}={\frac {1}{4\pi }}{\begin{bmatrix}B_{x}^{2}-{\frac {B^{2}}{2}}&B_{x}B_{y}&B_{x}B_{z}\\B_{x}B_{y}&B_{y}^{2}-{\frac {B^{2}}{2}}&B_{y}B_{z}\\B_{x}B_{z}&B_{y}B_{z}&B_{z}^{2}-{\frac {B^{2}}{2}}\end{bmatrix}}$

Surface Integral Version

$\mathbf {F} =\int _{S}\mathbb {T} _{M}\mathbf {n} dA$

Electromagnetic Stress Tensor

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

$\mathbb {T} _{EM}=\mathbb {T} _{E}+\mathbb {T} _{M}$