Question #42673

A long cylindrical conducting shell of inner radius $R_1$ and outer radius $R_2$ is placed in a uniform magnetic field $\vec{B}$ pointing in the $+z$ direction. The magnetic field exists everywhere in space and its magnitude varies with time as $B(t) = \alpha t$, where $\alpha$ is a positive constant. A point charge $q$ is moved slowly by an external agent from point $P$ to point $Q$ through the conducting material. In cylindrical coordinates $(r, \phi, z)$ with the z-axis along the cylinder's axis, the coordinates of $P$ and $Q$ are $(R_2, 0, 0)$ and $(R_2, \pi, 0)$ respectively.

Consider two paths for moving the charge: Path A: A semi-circular arc along the outer surface of the shell ($r = R_2$). Path B: A path consisting of three segments: radially inward from $P$ to $(R_1, 0, 0)$, then a semi-circular arc along the inner surface ($r = R_1$) to $(R_1, \pi, 0)$, and finally radially outward to $Q$.

Assuming the charge is moved slowly enough that radiation and self-magnetic field effects are negligible, what is the difference in the work done by the external agent along Path A and Path B, i.e., $\Delta W = W_A - W_B$?