Two uniformly charged spheres are superposed with slight displacement. What's the surface density?

Question

*Note: This is from the Second Volume of Feynman's Lectures on Physics : Mainly Electromagnetism and Matter

And this is the excerpt from the book:

If the relative displacement of the two spheres is small, the net charge is equivalent to a surface charge (on a spherical surface), and the surface charge density will be proportional to the cosine of the polar angle.

i.e. $\sigma = \sigma_{0}cos\theta $

What is that theta?? What is polar angle?? This came out of nowhere.

Please derive this quantitatively if possible.

Answer 1

Please see the following diagram:

This is the picture you need to draw - two spheres, slightly displaced. You can now see how the angle $\theta$ is defined (the dashed lines are supposed to go through the center of the system, midway between $C_1$ and $C_2$. It doesn't quite look like that...). When the distance between the centers is very small, you can compute the net surface charge. Do this by considering the distance of the center of each sphere to the sphere centered on their common center - you will find that for small distance, things cancel out and you end up with the cosine relationship (and an $1/r^3$, incidentally - this is a dipole).

See if this is enough to get you going. Remember homework questions are not intended to be answered with "complete" answers...