0.1. POTENTIAL ENERGY OF A MAGNETIC DIPOLE 225
Problem 7.
R, M, QωztA disk of radiusRand thicknesst, with uniform charge densityρqand uniform mass densityρm
is rotating at angular velocity~ω=ωˆz.
Consider a tiny differential chunk of the disk’s volumedV =dA tlocated atr, θin cylindrical
polar coordinates. Note that this chunk is orbiting thez-axis at angular frequencyωin a circular
path.
a) Find the magnetic momentdmzof this chunk in terms ofρq,ω,dVand its coordinates.
b) Find the angular momentumdLzof this chunk in terms ofρm,ω,dVand its coordinates.
c) Doing the two (simple) integrals, express them in terms of the total charge and total mass of
the disk, respectively, and show that the magnetic moment of the disk is given by~m=μB~L, where
μB= 2 QM.
d) What do you expect the magneticfieldof this disk to look like on thezaxis forz≫R?
(Answer in terms of~mis fine.)
Advanced Problem 8.
Using the insight gained from the previous two problems, consider any of the symmetric distributions
of charge and mass, where the mass distribution is the same as the charge distribution and where
both are “balanced” rotationally. Find a relationship betweendI(the moment of inertia of a small
chunk of massdmat a radiusr) anddmz(the magnetic moment of thesamesmall chunk of chargedq
at the radiusr) to show that for all distributions with sufficient (balanced) symmetry thatLz=Iω,
mz= 2 QMLz. This result therefore holds for spheres, cylinders, disks, rods (in a plane), spherical or
cylindrical shells, etc.