Force on a Charge

When we want to discuss the force on a charge due to a charge distribution, there are two options. The first is a more traditional method: an integral over a volume containing the charge distribution. The second method is less traditional but is easier to do: a surface integral over a special stress tensor.

Volume Integral Version

${\displaystyle \mathbf {F} =\int _{V}\rho \mathbf {E} dV}$

Electrostatic Stress Tensor

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

Surface Integral Version

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

The Maxwell Stress Tensor

T_ij is called the Maxwell Stress Tensor, it has two indices and is not a vector so is given a double arrow.

${\displaystyle \mathbb {T} _{ij}=\epsilon _{0}\left(E_{i}E_{j}-{\frac {1}{2}}\delta _{ij}E^{2}\right)+{\frac {1}{\mu _{0}}}\left(B_{i}B_{j}-{\frac {1}{2}}\delta _{ij}B^{2}\right)}$