254 Week 7: Sources of the Magnetic Field
Advanced Problem 9.
I R/2
R
R
I R/2
zxy(A)A pair of Helmholtz coils is made up of two loops of wire withNturns and radiusRcarrying
a currentIper turn. They both are concentric with thezaxis with centers atz=±R/2. Show
that atz= 0: dBdzz = 0 andd
(^2) Bz
dz^2 = 0. This means that the magnetic field is quite “flat” in the
middle of a Helmholtz coil.
Advanced Problem 10.
Find the magnetic field on the axis of a uniform disk of charge with radiusR, massM, and
chargeQ. This should be fairly easy toset upat this point and everybody ought to be able to do
it. The resulting integral overr, however, will require integration by parts to solve, in particular
∫
r^3 dr
(r^2 +z^2 )^3 /^2
If you makeu=r^2 andv= (r^2 +z^2 ) this is pretty easy, but it is still a bit difficult for non-majors.
A good challenge problem, though, for non-majors who want to improve their math and physics
skills!
Express the answer in terms of the magnetic moment of the disk (computed previously) and
show that its limiting form asz≫Ris that of a dipole. Note also that this is the spinning disk
that you demonstrated wouldprecessin the previous chapter when placed in a strong field! At this
point you should understand spinning disks of charge (as dipoles) pretty well!