Maxwell's Equation in Empty Space

Question

Consider Maxwell's equation $$ \begin{cases} \frac{\partial B}{\partial t} = - \text{c} \cdot \text{curl} (E), \\ \frac{\partial E}{\partial t} = \text{c} \cdot \text{curl} (B), \\ \text{div} (E) = 0, \\ \text{div} (B) = 0 \end{cases} $$ for empty space. How do the second two equations play a role when solving this system of PDEs? Specifically, when using finite differences, why do the first two not supply enough information? It appears to me that, when doing finite differences, each step forward in time is completely determined by the first two equations.

Answer 1

If the pair $(E,B)$ satisfy the first two equations then $(E',B') = (E + \nabla\phi(x), B + \nabla\psi(x))$ also satisfy the same equations: $$\frac{dE'}{dt} = \frac{dE}{dt} = c\nabla\times B = c\nabla\times B'$$ $$\frac{dB'}{dt} = \frac{dB}{dt} = -c\nabla\times E = -c\nabla\times E'$$ since $\nabla\times \nabla f \equiv 0$. Thus the first two equations only determine $(E,B)$ up to a time-independent curl-free part. To get a closed system we therefore need the last two equations.

Taking the divergence of your first two equations we get $$\frac{d(\nabla\cdot E)}{dt} = 0 = \frac{d(\nabla\cdot B)}{dt}$$ so the divergence of the fields are conserved. Thus for a numerical evaluation the last two equations does not enter in the evolution (in a perfect world where the numerical method does not introduce a divergence). The way it enters is in the initial conditions.

The zero divergence conditions can serve as a good check that the numerical method is working as intended (evaluate the divergence of $E$ and $B$ as you integrate and check that it stays close to zero).