Electrodynamics/Magnetic Stress Tensor

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

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

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

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

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

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