128 Week 3: Potential Energy and Potential
Problem 4.
Find by direct integration the potential on the axis of a thin disk of charge with surface charge
densityσand radiusR. Then expand the result to leading order in the two limitsR≫zandz≫R
and interpret the potentials in both of these cases.
Problem 5.
How much work is required to assemble a uniform ball of charge with total (final) chargeQand
radiusR? Hint: This is the same as the potential energy of the sphere, so usedU =V dqand
imagine “building” the sphere a layer of thicknessdrat a time. Alternatively, compute the work
directly by bringing a chargedqin from infinity against the electric field of the charge already there
(and distributed as a sphere of radiusr).
Problem 6.
Compute the potential difference ∆V between: a) Two conducting spherical shells of radiusaand
bwith a charge +Qon the inner one and charge−Qon the outer one. b) Two (infinitely long)
conducting cylindrical shells of radiusaandbwith a charge per unit length +λon the inner one
and charge per unit length−λon the outer one. c) Two (infinite) conducting sheets of charge, one
with charge +σon thexyplane and with with charge−σparallel to the first one but atz=d.
Great! Now you’ve donealmost all the workrequired to understandCapacitance!
Problem 7.
Three thin conducting spherical shells have radiia < b < crespectively. Initially the shell with
radiusahas a charge +Qand the shell with radiusbhas a charge−Q. You connect the shells with
radiiaandcusing a thin wire that passes through a tiny (insulated!) hole throughthe middle shell
and wait for the charge to come to a new equilibrium. What is: a) The charge on all three shells?
b) The potential at all points in space (this is quite a bit of work, but when you’re done you’ll really
have the hang of this down)?
Problem 8.
Two rings of chargeQand radiusR(uniformly distributed) are located atz=±Rand have the
same (z) axis. A small bead of massmwith chargeqis threaded on a frictionless string along the
zaxis. If the bead is displaced a small distance +z 0 ≪Rfrom the origin, describe the subsequent
motion of the bead in detail. (Hint: That means findz(t) and the approximate periodTor angular
frequencyωof harmonic oscillation for the bead, in case that wasn’t clear.)