ehrenfest said:
I never learned how to use Lenz's law to calculate the Faraday induced by when a uniform B-field that fills all of space is reduced. I can use Lenz's law to calculate the emf induced in a loop! This is the heart of my problem here!
Ah, that's the problem!
First ask yourself the reverse question: if I have some current loop with a current I, it produces a B field. In what direction is this B field? This you should know from the right-hand rule. Suppose we have a current loop in the page, running counterclockwise. Which direction is B, in the interior of the loop? Into the page, or out of the page?
Now, let's go back to Faraday's Law. Suppose we have a loop of wire, in the page again, and a B field pointing out of the page. We know that if the B field is
changing in time, then a current will be induced. We also know that the amount of current induced will be proportional to the total flux of B field through the interior of the loop. But which way will the current go? Here's a way to think about it:
In order to induce a current, you need to expend
energy, to get the charge to flow against the resistance of the wire in question. You can't get energy from nothing; energy is always conserved.
Also, once we have a current flowing through the loop, we know (via Ampere's Law), that the current will produce its
own B field! So now we have two possibilities. Either:
A. The B field produced by the loop will
agree with dB_ext/dt, or
B. The B field produced by the loop will
oppose with dB_ext/dt.
Note that the time derivative of B_ext must be used, because the current is proportional to this time derivative (and therefore the sign of the current must depend on the sign of the time derivative).
Now, suppose that possibility A happens. That is, if B_ext increases, then B_wire points in the same direction as B_ext, and vice versa. What happens to the energy of the system? What happens to the current? Note that the current induced in the wire is due to the TOTAL B; it reacts to its own B as well as the external B. This is called self-inductance. (Hint: you get runaway solutions in this case.)
Now, suppose that possibility B happens. That is, if B_ext increases, then B_wire
opposes B_ext, etc. What happens to the energy? What happens to the current? In this case, you should have stable solutions.
The end result is that the B field set up by the wire loop MUST oppose the
change in the external B field, or else the mathematics explodes. Knowing that, then, which way must the current go in order to set up this B field?