Suppose you have two superconducting loops, concentric, in a plane. Also suppose that their radii, $R_1$ (outer) and $R_2$ (inner) have the same order of magnitude (so you can't assume $B$ through the inner loop is constant everywhere over the inner loop's surface.)
If a constant current $I_1$ is ramped up from $0$ in the outer loop, what will the induced current in the inner loop be?
I begin by attempting to evaluate the mutual inductance of the loops: $$L = \frac{\phi}{I} = \mu_0 \int_{0}^{R_2} r \int_{0}^{\pi} \frac{(R_1 - rcos(\theta))R_1 d\theta}{(r^2 + R_1^2 - 2rR_1cos(\theta))^{3/2}} dr$$ where the inner integral comes from the expression for $B$ at an arbitrary point in the plane of a current loop, and the outer integral is just the integration over the inner radius to compute the flux. Aside from the profound ugliness of that expression, I don't really have a problem with it -- it's just that with $R = 0$, knowing that $\varepsilon = -L \frac{dI}{dt}$ doesn't seem super helpful.
