Answer to Question #134274 in Electricity and Magnetism for Rohan

Question

Charge is distributed on the surface of a sphere with a surface charge density σ=σ₀ cos θ in spherical coordinates. This distribution is

(a) Spherically Symmetric

(b) cylindrically symmetric

(c) Planar Symmetric

(d) none of these

(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

We can represent spherical coordinates in cartesian as following

(picture from Wikipedia). We can see here that $cos \theta$ gives us projection on z-axis. So all values of $σ_0 cos θ$ are equal for the same value of theta. Cosine is an even function, $cos\theta = cos(\pi-\theta)$

So, values are the same for +z and -z. This gives us planar symmetry relative to the plane $\theta =0$ or X0Y in cartesian coordinates.

Answer: (c) Planar Symmetric