Induction in closed loops

Question

I'm trying to understand the law of current induction in closed loops.

I just saw this example, of a non-moving circuit in a constant magnetic field $B$. On it, a sliding (conducting metal bar) which moves with a velocity $v$.

In this case, I saw that the EMF generated is $BLv$, since this is the flux change in either the right or the left loops.

So first of all, I'm wondering why we can deduce the EMF from each loop separately? Don't they affect one another? What if the flux change wasn't equal in both of the loops? I've drawn such a case here:

There's a non moving black half-closed loop, above it the moving bar and above it another non-moving half-closed red loop.

So in this case, the flux change in the black loop is different than the flux change in the red loop. What would then be the EMF?

Summing up, I'm trying to understand why in the first example only one loop was considered when calculating the EMF, and how would that logic apply in a case such as the second case, where there is no symmetry in the flux of the two loops.

Thanks!

Answer 1

Each loop (even a loop consisting of multiple smaller loops) has an EMF along it given by Faraday's law of induction, also called the flux rule: $$\mathcal{E}=-\frac{d}{dt}\iint\limits_S\vec{B}·d\vec{A}\ .$$ It is okay for two adjacent loops to have different EMFs.

The only way nearby loops can affect the EMF along one another if currents are induced in them. Then, because of the mutual inductance of the loops, the current along one affects the magnetic flux through the other. But a current induced in a loop affects the magnetic flux even through itself, due to its self inductance, so this phenomenon doesn't only occur when you have multiple loops.

But if the magnetic fields due to induced currents are small (e.g. wires have high resistance), this is not a problem, and you can calculate the EMF along each loop independently.