Electromagnetic Momentum of a Capacitor in a Uniform Magnetic Field

Analytic calculations of the electric field of a parallel-plate capacitor are notoriously difficult [1, 2, 3, 4, 5, 6, 7, 8]. The usual goal of such efforts is a calculation of the correction to the hard-edge approximation, C = A/4πd (in Gaussian units), of the capacitance for plates of area A separated by distance d. Calculate instead the electromagnetic momentum of the parallel-plate capacitor if it resides in a uniform magnetic field that is parallel to the capacitor plates. Consider also the case of a capacitor whose electrodes are caps of polar angle θ0 < π/2 on a sphere of radius a. In both cases, the remaining space is vacuum.