252 Week 7: Sources of the Magnetic Field
Problem 4.
JinR/2
RA cylindrical conductor of radiusRaligned with thezdirection has a cylindricalholeof radius
R/2 centered atx=R/2 also aligned with thezdirection. The conductor carries acurrent density
J~=Jzˆ(and obviouslyJ~= 0 in the hole). Find the magnetic field at all points inside the hole.
Problem 5.
Using the Biot-Savart law:
a) Find theB~-field on thezaxis of a circular current loop of radiusaandNturns carrying a
currentIin thex−yplane (centered on the origin).b)Set upthe integral to be done to find thevB-field on thezaxis of adiskin thex−yplane of
uniform charge densityσand radiusathat is rotating with angular frequenceωaround thez
axis.(A)Do this integral (requires integration by parts a couple of times).Problem 6.
Based on theanalogybetween electric and magnetic dipoles, deduce the probable form ofthe mag-
netic field of a spherical ball of chargeQ, massM, and radiusRthat is rotating at angular velocity
ωon a) its axis of rotation; b) at a point in the plane that passes through the ball perpendicular
to the axis of rotation; in both casesfarfrom the ball of charge, that is, forz≫Randx≫Rfor
a ball spinning around thezaxis. Note that it is quite a bit of work to actually derive this result
(though it can be done). This is part of the point of multipolar expansions – once one knows the
form of the field for any given multipolar moment, one merely has to compute that moment for a
give charge-current density to discover the (far) field “for free”.