Question

We want to design a spherical vacuum capacitor, formed by an inner spherical conductor with radius b
and an outer spherical shell of given radius a.

We want to design a spherical vacuum capacitor, formed by an inner spherical conductor with radius b and an outer spherical shell of given radius a. We want the capacitor to be able to store the greatest amount of electrical energy subject to the constraint that the electric field strength at the surface of the inner sphere may not exceed Eo- What radius b should be chosen for the inner spherical conductor and how much energy can be stored? For convenience, choose b=ka, with k-constant

Answer 1

Assume there is a charge Q on the inner shell and a charge - Q on the outer shell, then the magnitude of Electric field between the shell is E = Q/r^2 r^ The potential difference is V = integral_a^b Q/r^2 dr = Q (a - b/ab) and capacitance c = Q/V = ab/a - b The energy stored by this capacitor is U = 1/2C Q^2 = 1/2 (a - b/ab) Q^2 The energy in the capacitor will depend on how much charge is an it. If we were allowed to put arbitrary amounts on, the energy would have no maximum. However For a radius b, the maximum field near the inner sphere gives Us the maximum allowed charge. This gives us the maximum stored energy for a given capacitor. E_0 = Q_max/b^2: U_max = 1/2 (a - b/ab) E_o^2 b^4 rightarrow Q_max = E_0 b^2 U_max = 1/2 ab^3 - b^4/a E_0^2 (partial differential U_max/db) _b max = 0 rightarrow partial differential/partial