Question

Q1. SEPARATION OF VARIABLES - SPHERICAL SIGMA The surface charge density on a sphere (radius R) is a constant, σ0 (As usual, assume V(r = ∞) = 0, and there is no charge anywhere inside or outside, it's ALL on the surface!)

i) Using the methods of section 3.3.2 (i.e. explicitly using separation of variables in spherical coordinates), find the electrical potential inside and outside this sphere.

ii) Discuss your answer, explain how you might have just "written it down" without doing all that work! (Be explicit - what about all the specific coefficients you got in i?) Also - can you think of a fairly simple (realistic) physical/experimental setup that might yield a situation like this?

iii) Now, suppose the surface charge density is +σ0 on the entire northern hemisphere, but -σ0 on the entire southern hemisphere. Again, find voltage inside and outside. (This time, you will in principle need an infinite sum of terms - but for this problem, just work out explicitly what the first two nonzero terms are. (In both cases, for V(r < R), and V(r >R)) Note: some terms you might have expected to be present will vanish. Explain physically or mathematically why the first "zero" term really *should* be zero.

Any help is greatly appreciated!

Answer 1

To solve this problem the details calculations are given below

Poteaia at the ceste O is 4 > SunRce charge den 6, 0 215 1-020 o a 2 fo aes emph