Two thin conducting hemispherical shells share the same centre and the same…
PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Conductors, capacitance and boundary-value electrostatics
Electricity, Magnetism and Electromagnetism, Ch Conductors, screening and capacitance
Problem
Two thin conducting hemispherical shells share the same centre and the same equatorial plane. The inner one has radius a and carries charge +Q spread over it; the outer one has radius b>a and carries −Q. The gap between them is evacuated, and the two shells are held apart by insulating spacers of negligible size. Idealise the arrangement by assuming that the field in the gap is purely radial, has the same magnitude at all directions within the half-space occupied by the shells, and is zero everywhere else, so that fringing at the rim is neglected.
((a)) Using Gauss's law on a hemispherical surface of radius r (a<r<b) closed by the flat equatorial disc of the same radius, and then a radial line integral, derive the capacitance C of this pair in terms of ε0, a and b.
((b)) Confirm your result independently by computing the field energy 21ε0∫E2dV in the gap and comparing it with Q2/2C.
((c)) Evaluate C for a=4.0cm and b=5.0cm.