Magnetic field on the surface of a solenoid

Question

Given an infinite dielectric cylinder of radius $r$, with uniform surface charge density $\sigma$, rotating about its axis with a constant angular velocity $\omega$, find the magnitude of magnetic field on its surface.

I understand that this rotating cylinder behaves like a solenoid, and we can thus compute that the external magnetic field is $0$, and that the magnetic field inside it is a constant value (which we can find using Ampere's Law). But what is the magnetic field at the surface?

Do we sort of assume that an Amperian loop slices all the charges in half? Because the answer to a follow-up question asking for the pressure on the surface due to magnetic forces seemed to suggest a value that was half the field inside the cylinder.

Answer 1

You are right: in surface modeling, the field is discontinuous and one cannot speak of the field “on the surface”.

For the following, the factor ½ comes from the fact that we distinguish the field created by an elementary surface $dS$ on which we seek the force and the field created by the rest of the solenoid: we can show that in the solenoid, the two are equal (and they are opposite on the outside of the solenoid)