Parallel Plate Capacitor with a Spherical Boss
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A large parallel plate capacitor is made up of two plane conducting sheets with seperation D, oneo f which has a small hemispherical boss of radius a on its inner surface (D >> a). The conductor with the boss is kept at zero potential, and the other conductor is at a potential such that far from the boss the electric field between the plates is Eo.
(a) Calculate the surface-charge densities at an arbitrary point on the plane and on the boss, and sketch their behavior as a function of distance (or angle).
(b) Show that the total charge on the boss has the magnitude 3*pi*e0*E0*a^2.
(c) If, instead of the other conducting sheet at a different potential, a point charge q is placed directly above the hemispherical boss at a distance d from its center, show that the charge induced on the boss is:
q' = -q[1 - (d^2-a^2)/(d*sqrt*d^2+a^2)].
The solution shows in detail how to find the charge density on a spherical boss inside a parallel plate capacitor, and how to use the method of images to find the charge density when one plate is removed.