Chapter 4: Differentation

### Maxwell's Equations II

It is straightforward to translate Maxwell's equations into the language of differential forms. We obtain \begin{align} {*}d{*}E &= 4\pi\rho \\ {*}d{*}B &= 0 \\ {*}dE + \dot B &= 0 \\ {*}dB - \dot E &= 4\pi J \end{align} but it is customary to take the Hodge dual of both sides, resulting in \begin{align} d{*}E &= 4\pi\,{*}\rho \\ d{*}B &= 0 \\ dE + {*}\dot B &= 0 \\ dB - {*}\dot E &= 4\pi \,{*}J \end{align} Differentiating the last equation, and using the first, brings the continuity equation to the form $${*}d{*}J + \dot\rho = 0$$ and making the Ansatz \begin{align} B &= {*}dA \\ E &= -d\Phi - \dot A \end{align} automatically solves the middle two (source-free) equations.