I've recently been reading the Griffiths Electrodynamics book, and came across this problem (7.17, 4th edition) and its solution:
I don't have any objections for the solution to part a). I have, however, a hard time justifying the solution to part b). Shouldn't the magnetic field (and hence the magnetic flux) by the solenoid at a distance far away from the loop be 0? And therefore shouldn't the change in flux be $\Delta\Phi = \pi{a^2}\mu_0nI - 0 = \pi{a^2}\mu_0nI$ instead of $\Delta\Phi = \pi{a^2}\mu_0nI - (-\pi{a^2}\mu_0nI) = 2\pi{a^2}\mu_0nI?$