# A question about induced fields

This is something that I just realized bothers me, after thinking about it too much. Faradays law tells us a changing magnetic field creates a circulating electric field. This electric field has a time dependence and therefore by Amperes law creates a circulating magnetic field. This is usually where flux/emf problems stop. But won't the time dependence of the induced circulating B field create a new second E field which will then create a third B field and so on ad infinitum? This is how EM waves propagate, so why don't we consider this chain in flux problems?

We do! (Or at least, we should, to be complete.) Imagine the basic arrangement of a bar magnet, north up, moving upward toward a solenoid. Faraday's Law has a very important negative sign for this question, which dictates that the new current generated by this increasing flux is going to be clockwise from the top (use the right-hand rule, and flip it for the negative sign).

$\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$

Now go to Ampère's circuit law, and you'll see that the current generated will produce a magnetic field with flux lines pointing DOWN in the middle.

$\nabla \times \vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial \vec{E}}{\partial t}$

This will act to oppose the bar magnet going up. Thus we not only take it into account, but it is essential for energy to be conserved. It doesn't result in a cascading chain, because it's simply opposing some of the initial flux change inserted into the system by the motion of the bar magnet. The world would be a very messy place without that negative sign in Faraday's Law.

Ampere's law doesn't say that in this example you would have a time varying magnetic field (which is what you would need in order to generate another electric field), but rather just some sort of magnetic field which behaves according to the equation:

$$\nabla \times \textbf{B} = \mu_0 \textbf{J} + \mu_0 \epsilon_0 \frac{\partial \textbf{E}}{\partial t}$$

Nothing here requires the magnetic field to vary with time, only have a particular variation in space. Therefore, the magnetic field would not need to generate an additional electric field.