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:

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