Conductors and Insulators — Question 10

Question 10

A surface density model distinguishes redistribution from net charging

A spherical conductor of radius $R$ has a supplied equilibrium surface-charge model $\sigma(\theta)=\sigma_0+\sigma_1\cos\theta$, produced by a net charge and an external influence. The angle $\theta$ is measured from the north axis. Use $dA=2\pi R^2\sin\theta\,d\theta$, $\sigma_0=2.0$ nC/m$^2$, $\sigma_1=6.0$ nC/m$^2$, and $R=.50$ m.

1. Integrate to find the total surface charge and show why the $\cos\theta$ term contributes no net charge.

2. Find the northern and southern hemisphere charges. Determine whether opposite signs on the two hemispheres are compatible with a positive total charge.

Question 10 - Solution

A surface density model distinguishes redistribution from net charging

(a) Method and calculation

$$Q=2\pi R^2\int_0^\pi(\sigma_0+\sigma_1\cos\theta)\sin\theta\,d\theta=4\pi R^2\sigma_0.$$$$\int_0^\pi\cos\theta\sin\theta\,d\theta=0,\quad Q=2\pi\,\mathrm{nC}.$$

(b) Interpretation and further calculation

$$Q_N=2\pi R^2(\sigma_0+\sigma_1/2)=2.5\pi\,\mathrm{nC}.$$$$Q_S=2\pi R^2(\sigma_0-\sigma_1/2)=-.5\pi\,\mathrm{nC},\quad Q_N+Q_S=2\pi\,\mathrm{nC}.$$The antisymmetric term moves charge between hemispheres without changing the total.

Physical check

The prescribed density requires an external influence; an isolated spherical conductor without one has a uniform equilibrium density.