Answer to Question #138826 in Electricity and Magnetism for Rohan

Question

A spherical surface of radius R has a surface charge density σ = σ0 sinθ in standard notation of spherical coordinates with the origin at the center of the surface. σ0 is a constant. The electric potential at far away points will vary with the distance from the center as

(a) 1/r

(b) 1/r2

(c) 1/r3

(d) 1/r4

(Note:- This question have one or more than one correct choice(s) out of the four given choices. Any number of options may be correct.)

Answer 1

In standard notation $\theta$ is the zenith angle, so it changes from 0 to $\pi$ . Let us first determine the total charge of the sphere. If the total charge is not equal to zero, the monopole term of the potential will not be zeroth, so it will determine the main part of the potential.

$Q_{\text{total}} = \int\limits_{0}^{2\pi}\int\limits_0^{\pi} (\sigma_0 \sin\theta)\cdot R^2\sin\theta\, d\theta d\phi = 2\pi \sigma_0 R^2 \int\limits_0^{\pi} \sin^2\theta \,d\theta= 2\pi \sigma_0 R^2 \cdot \dfrac{\pi}{2} = \pi^2 \sigma_0 R^2.$

We see that the total charge is not equal to 0, so in the multipole expansion the monopole term will not be zeroth. Therefore, the monopole term is proportional to $\dfrac{Q_{\text{total}}}{r}$ and the total potential will be approximately proportional to 1/r