Electrodynamics/Electrostatic Stress Tensor

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

$𝐅=\underset{V}{\int }\rho 𝐄dV$

Electrostatic Stress Tensor

${𝕋}_E=\frac{1}{4\pi }\left[\begin{array}{ccc}{E}_x^2-\frac{E^2}{2}& E_x{E}_y& E_x{E}_z\\ E_x{E}_y& E_y^2-\frac{E^2}{2}& E_y{E}_z\\ E_x{E}_z& E_y{E}_z& E_z^2-\frac{E^2}{2}\end{array}\right]$

Surface Integral Version

$𝐅=\underset{S}{\int }𝕋𝐧dA$

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.

${𝕋}_{ij}={ϵ}_0(E_i{E}_j-\frac{1}{2}{\delta }_{ij}{E}^2)+\frac{1}{{\mu }_0}(B_i{B}_j-\frac{1}{2}{\delta }_{ij}{B}^2)$