“Hidden” Momentum in a Charged, Rotating Cylinder

The cylinder (of radius a) and the plates of the capacitor (in the planes y = ±b) can move independently and without friction in the x-direction, where the z-axis is that of the cylinder. That axis can be electrically charged with the opposite sign to that of the cylinder such that the cylinder as a whole has zero charge. The (linear, isotropic) medium between the capacitor plates, including the interior of the cylinder, has relative permittivity and relative permeability μ. Ignore gravity. This problem is a variant of that considered by Shockley [1], which is itself a variant of the Feynman disk paradox [2], which can be traced back to a discussion by Poincaré (1896) [3, 4], who built on a prescient argument of Darboux (1878) [5]. The geometry of this problem is also related to that of the Aharonov-Bohm effect [6].