Ethan's Notes

    Maxwell's Stress Tensor

    #Physics
    Measures pressure of electric and magnetic fields

    $\displaystyle T_{ij}\equiv {\varepsilon}{0}\left( E{i}E_{j}-\frac{1}{2}\delta_{ij}E^{2} \right)+\frac{1}{{\mu}{0}}\left( B{i}B_{j}-\frac{1}{2}\delta_{ij}B^{2} \right)$

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

    $\displaystyle T_{xx}=\frac{1}{2}{\varepsilon}{0}(E{x}^{2}-E_{y}^{2}-E_{z}^{2})+\frac{1}{2{\mu}{0}}(B{x}^{2}-B_{y}^{2}-B_{z}^{2})$

    • Example tensor component

    $\displaystyle f=\nabla \cdot\stackrel{\Rightarrow}{T}-\varepsilon_{0}{\mu}{0}\partial{t}S$

    $\displaystyle \text{d}{t}\vec{p}{\text{mech}}=-{\varepsilon}{0}{\mu}{0}\text{d}_{t}\int {\mathcal{V}}\vec{S} , \mathrm{d}\tau+\oint{\mathcal{S}}\stackrel{\Rightarrow}{T}\cdot\mathrm{d}\vec{a}$