Download Physics 3310 Homework 7: Electrical Potential and Multipoles - Prof. Edward R. Kinney and more Assignments Physics in PDF only on Docsity! Phys 3310, HW #7, Due in class Wed Mar 5 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. Griffiths solves a generic example problem, for which part i above is a simple special case (and for that matter, so is part iii). But, please work through the details on your own - you're welcome to use Griffiths to guide you if/whenever you need it, but in the end, solve the whole problem yourself and show your work! Q2. SEPARATION OF VARIABLES - CONCENTRIC SPHERES Two concentric spherical surfaces have radii of a and b. If the potential on the inner surface, at r=a, is just a nonzero constant (call it V in ) and the potential on the outer surface is given by V (b,!) =V out P 1 (cos!) (i.e. =V out cos!) , find the potential in the region between the two surfaces (a < r <b). Q3. SEPARATION OF VARIABLES - DISK A disk of radius R has a uniform surface charge density σ0. Way back on Set #2 you found the E-field along the axis of the disk (and on the midterm, you again solved a very similar (but harder) version of this where σ was not uniform). You can check for yourself by direct integration, (but don't have to): I claim that along the z axis, (i.e. θ=0), V (r,! = 0) = " 0 2# 0 r 2 + R 2 $ r( ) i) Find the potential away from the axis (i.e nonzero θ) , for distances r > R, by using the result above and fiddling with the Legendre formula, Griffiths' 3.72 on page 140. You will in principle need an infinite sum of terms here - but for this problem, just work out explicitly what the first two *non-zero* terms are. (It might help to remember that Pl(1) is always equal to 1, and you will have to think mathematically about how the formula above behaves for r>>R) ii) Griffiths Chapter 3.4 talks about the "multipole expansion". Look at your answer to part i, and compare it to what Griffiths says it should look like (generically) on page 148. Discuss - does your answer make some physical sense? Note that there is a "missing term" - why is that? Can you think of some physical situation that might look a little like this problem?
