Description
Problems
1. Magnetic dipole moment:
A current distribution produces the vector potential
A(r, θ, ϕ) = ˆϕ
µ0
4π
A0 sin θ
r
exp(−λr).
(a) (50 pts) Find the magnetic (dipole) moment of this current distribution. Hint:
Find the current from the vector potential and then follow the definition of the
magnetic moment. Avoid directly performing integrals, if possible.
2. A current loop:
A filamentary current loop traverses eight edges of a cube with side length 2b as shown
in the figure. The origin is placed at the center of the cube.
KS P2: SFK Trim: 246mm × 189mm Top: 10.544mm Gutter: 18.98mm
4-11 CUUK1954/Zangwill 978 0 521 89697 9 August 9, 2012 1Problems 363
11.10 Biot-Savart at the Origin Show that the first non-zero term in an interior Cartesian multipole expansion
of the vector potential can be written in the form A(r) = (µ0/4π)G × r where G is a constant vector. Show
that the associated magnetic field is a Biot-Savart field.
11.11 Purcell’s Loop A filamentary current loop traverses eight edges of a cube with side length 2b as shown
below.
(a) Find the magnetic dipole moment m of this structure.
(b) Do you expect a negligible or a non-negligible magnetic quadrupole moment? Place the origin of
coordinates at the center of the cube as shown.
I x
y
z
2b
11.12 Dipole field from Monopole Field If such a thing existed, the magnetic field of a point particle with
magnetic charge g at rest at the origin would be Bmono(r) = (µ0gr/(4πr3). Show that the magnetic field of
a point magnetic dipole m is B = −(m · ∇)Bmono/g at points away from the dipole.
11.13 The Spherical Magnetic Dipole Moment Show that the formula for the magnetic dipole moment derived
in Example 11.1,
m = 3
2µ0
!
sphere
d3
r B(r),
is consistent with the spherical multipole expansion of the vector potential derived in Section 11.4,
A(r) = µ0
4π
“∞
ℓ=1
“ℓ
m=−ℓ
4π
2ℓ + 1
1
iℓ
MℓmL
Yℓm(#)
r ℓ+1 .
11.14 No Magnetic Dipole Moment Show that a current density with vector potential A(r) = f (r)r has zero
magnetic dipole moment.
11.15 A Spherical Superconductor A superconductor has the property that its interior has B = 0 under all
conditions. Let a sphere (radius R) of this kind sit in a uniform magnetic field B0.
(a) Place a fictitious point magnetic dipole m at the center of the sphere. Find m from the matching condition
on the normal component of B.
(b) In reality, the dipole field in part (a) is created by a current density K which appears on the surface of
the sphere. Find K from the matching condition on the tangential component of B.
(c) Confirm your answer in part (a) by computing the magnetic dipole moment associated with K from
(a) (50 pts) Find the magnetic dipole moment m of this structure.
