Alternatively, in numbers. I'm ignoring constant factors and only doing proportional calculations. Where you see an = sign you should think "proportional to".
Suppose we have a magnet at the origin with magnetic moment 1 in the z direction: m = [0,0,1]. The magnet is moving in vertically (in the z direction). Further we have the pipe vertically with radius 1. The problem now is to determine the current through the pipe. Since the problem is symmetrical around the z-axis we can consider just the current though the points r = [1,0,z].
The magnetic field due to the dipole is B(r) = 3r(m dot r)/r^5 - m/r^3. The flux through the pipe at r = [1,0,z] is just the x component of the magnetic field, since the pipe is vertical. Plugging in and simplifying we get Phi = 3z/sqrt(1+z^2)^5. Since the magnet is moving in the z direction, lets say z = t, to compute the change in Phi we need to compute dPhi/dz. If we put that into Wolfram Alpha we get a picture like this:
As you can see you get a strong current in one direction around the pipe at the middle of the magnets, and two weaker currents in the opposite direction above and below the magnet.
This doesn't yet say anything about the case of the slit, but it does at least show that the currents are indeed moving like I sketched for the no slit case in the picture of my other response.
Okay I see how the currents move with the slit - thank you for the visuals! I just found a video using a solid vs. slit pipe that has a good example of the timing difference - not nearly as dramatic as my memory of the demonstration, but still informative!
Suppose we have a magnet at the origin with magnetic moment 1 in the z direction: m = [0,0,1]. The magnet is moving in vertically (in the z direction). Further we have the pipe vertically with radius 1. The problem now is to determine the current through the pipe. Since the problem is symmetrical around the z-axis we can consider just the current though the points r = [1,0,z].
The magnetic field due to the dipole is B(r) = 3r(m dot r)/r^5 - m/r^3. The flux through the pipe at r = [1,0,z] is just the x component of the magnetic field, since the pipe is vertical. Plugging in and simplifying we get Phi = 3z/sqrt(1+z^2)^5. Since the magnet is moving in the z direction, lets say z = t, to compute the change in Phi we need to compute dPhi/dz. If we put that into Wolfram Alpha we get a picture like this:
http://www.wolframalpha.com/input/?i=d%2Fdx+x%2Fsqrt%281%2Bx...
As you can see you get a strong current in one direction around the pipe at the middle of the magnets, and two weaker currents in the opposite direction above and below the magnet.
This doesn't yet say anything about the case of the slit, but it does at least show that the currents are indeed moving like I sketched for the no slit case in the picture of my other response.