Maxwell equations and electric and magnetic fields produced repeatedly

Question

Consider the two Maxwell equations (in the case of no conduction currents): $$\mathrm{rot}\bf{E}=-\frac{\partial \bf{B}}{\partial t}\tag{1}$$ $$\mathrm{rot}\bf{B}=\mu_0 \epsilon_0\frac{\partial \bf{E}}{\partial t}\tag{2}$$

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Suppose for example that I have a time dependent $\bf{B}$ in a capacitor: I can use $\bf{(1)}$ in order to find the "generated" field $\bf{E}$. But then this field $\bf{E}$ is time dependent itself, so I can use $\bf{(2)}$ and find the magnetic field $\bf{B}$ generated by $\bf{E}$ calculated before and I would find a magnetic field different from the one I started with.

I'm quite confused about this: can I go on forever? That is, can I find a new $\bf{E}$, then a new $\bf{B}$ and so on (with each field different form the previous one)?

And if this makes sense, what does it mean? Is this about electromagnetic waves propagation or would it just be a process of "correcting" the "original" field $\bf{B}$ (and the $\bf{E}$ found at first) (a correction that becomes neglectable from a certain point on)?

Answer 1

What you're describing is an 'iterative numerical method': you make an approximation, calculate a correction, then update your approximation... continue until your corrections are smaller than the accuracy you want. In some situations this can work, but it requires that the method is convergent---i.e. that the corrections tend to get smaller, and the approximation converges towards a unique solution. This is not necessarily the case.

In regards to wave propagation: I think you could make a good philosophical analogy, but physically I think these are entirely separate things. The iterative process applies to a certain fixed time (or set of times), and applies to static or dynamic situations (i.e. with or without waves); while the wave propagation is a relationship from one time to the next. You could of course use numerical methods (including iterative ones) to solve for time-series wave propagation with the right initial conditions.

Answer 2

Yes, indeed this process can go on forever in free space. The Maxwell equations in free space (i.e. as you have them here together with $\nabla\cdot\mathbf{E}=0$ and $\nabla\cdot\mathbf{B}=0$), describe electromagnetic waves. To see this, take the curl of both sides of each equation. Then, by using the "curl of a curl" identity

\begin{align} \nabla \times\nabla\times\mathbf{E}&=\nabla(\nabla\cdot\mathbf{E})-\nabla^2\mathbf{E}\\ &=-\nabla^2\mathbf{E}\\ &=-\frac{\partial}{\partial t}\nabla\times\mathbf{B}\\ &=-\mu_0\epsilon_0\frac{\partial^2}{\partial t^2}\mathbf{E} \end{align}

where we have used that $\nabla\cdot\mathbf{E}=0$ in free space, and $\nabla\cdot\mathbf{B}=0$ always. It follows that

\begin{align} \nabla^2\mathbf{E}=\mu_0\epsilon_0\frac{\partial^2}{\partial t^2}\mathbf{E}. \end{align}

That is, the electric field obeys the wave equation. In an almost$-$identical fashion, it can be shown that $\mathbf{B}$, too, satisfies the wave equation. Nice observation.