Potential of a Charged Isolated Conductor — Question 3

Question 3

Equal surface voltages give larger charge density on the smaller sphere

Two separate distant conducting spheres of radii a and b have the same positive potential V relative to infinity. There are no nearby external sources.

1. Find each charge and surface density.

2. Find their charging energies and the ratios of charge, density and energy for b=3a.

Question 3 - Solution

Equal surface voltages give larger charge density on the smaller sphere

(a) Method and calculation

$$Q_a=Va/k,\quad Q_b=Vb/k.$$$$\sigma_a=\frac{V}{4\pi ka},\quad\sigma_b=\frac{V}{4\pi kb}.$$

(b) Interpretation and further calculation

$$U_i=\int_0^{Q_i}\frac{kq\prime}{R_i}dq\prime=\frac{V^2R_i}{2k}.$$$$b=3a:\quad Q_b/Q_a=3,\quad\sigma_b/\sigma_a=1/3,\quad U_b/U_a=3.$$

Physical check

At fixed voltage, larger radius stores more charge and energy while reducing surface density and field.