This was an old qual problem I was just looking at. Let me describe how to solve it, and then ask another question I had that my solution doesn't answer.
According to Lenz's Law, (or really to Maxwell's equation \vec{\nabla} \text{x} \vec{E} = -\frac{\partial{\vec{B}}}{\partial{t}}), the induced electric field due to a change in magnetic flux \Phi = \vec{B} \cdot \vec{A} (where \vec{A} is the area vector normal to the surface through which the magnetic field is "fluxing") is given by
(1) \oint{\vec{E} \cdot d\vec{l}}=-\frac{d\Phi}{dt}
In this case (a ring of radius a), \Phi = \pi a^2 \vec{B}, and due to symmetry \vec{E} is the same magnitude along the ring, and due to Lenz's Law \vec{E} will point in the direction such that it opposes the change in flux through the ring, or in the \hat{\phi}-direction (parallel to d\vec{l}). Using this information, (1) becomes
(2) 2\pi a E= \pi a^2 \frac{dB}{dt}
which simplifies to
(2') E= \frac{a}{2}\frac{dB}{dt}
So it might seem that we can't calculate this time derivative, but we don't need to! Recall Newton's second law in rotational form:
(3) \vec{\tau}= \frac{d\vec{L}}{dt}
where \vec{\tau} is the total torque on the object, and \vec{L} is its angular momentum. Now, the torque on the ring is due to the induced electric field, and the infinitesimal torque d\tau on an infinitesimal length of the ring dl is
(4) d\tau=a d F=a \lambda E dl
where \lambda is the linear charge density; i.e., \lambda=\frac{q}{2\pi a}. I'm dropping vector signs here because I'm lazy, but we know that the force is in the direction of the electric field, and the torque is in the \hat{z}-direction. Now the total torque is obtained by integrating these infinitesimal torques around the ring, giving
(5) \tau=2\pi a^2 \lambda E
Finally, we substitute in (3) and (2'), leaving us with
(6) \frac{dL}{dt}=\lambda \pi a^3 \frac{dB}{dt}
Now we integrate over time, and we get the final angular momentum of the ring,
(7) L=\lambda \pi a^3 B
So that's fine, but the question I have is, is there a way to get this result using only the initial momentum and final momentum of the electromagnetic fields alone? The problem I have is, this angular momentum acquired by the ring comes from the magnetic field. But I can't figure out what the initial angular momentum of the field is using Poynting's vector, because there's no electric field inside the ring (right? I tried using a Gaussian surface to see this), and therefore the cross-product of the E and B-fields inside the ring is 0. But that can't be right... the static magnetic field must have some angular momentum, or else there wouldn't be any way for the ring to acquire angular momentum. What am I missing here? Any ideas are appreciated.
