Maxwell's Stress Tensor

Measures pressure of electric and magnetic fields

$T_{ij}\equiv {\epsilon }_0\left(E_i{E}_j-\frac{1}{2}{\delta }_{ij}{E}^2\right)+\frac{1}{{\mu }_0}\left(B_i{B}_j-\frac{1}{2}{\delta }_{ij}{B}^2\right)$

• $i$ and $j$ are refer to coordinates $x,y,z$

$T_{xx}=\frac{1}{2}{\epsilon }_0(E_x^2-E_y^2-E_z^2)+\frac{1}{2{\mu }_0}(B_x^2-B_y^2-B_z^2)$

• Example tensor component

$f=\nabla \cdot \stackrel{\Rightarrow }{T}-{\epsilon }_0{\mu }_0{\partial }_{t}S$

${\text{d}}_t{\vec{p}}_{\text{mech}}=-{\epsilon }_0{\mu }_0{\text{d}}_t{\int }_{\mathcal{V}}\vec{S}\mathrm{d}\tau +{\oint }_{\mathcal{S}}\stackrel{\Rightarrow }{T}\cdot \mathrm{d}\vec{a}$

• Conservation of momentum