Show that the formula for the magnetic dipole moment derived in Example 11.1, is consistent with the
Question:
Show that the formula for the magnetic dipole moment derived in Example 11.1,
is consistent with the spherical multipole expansion of the vector potential derived in Section 11.4,
Transcribed Image Text:
Example 11.1 Let B(r) be the magnetic field produced by a current density j(r) that lies entirely
inside a spherical volume V of radius R. Show that the magnetic moment of j(r) is
3
2μ0
m =
Therefore,
This problem is the magnetic analog of Example 4.1, which expressed the electric dipole moment
of a charge distribution using a spherical average of the electric field produced by the charge.
Solution: Assume first that j(r) does not lie entirely inside V. If we place the origin of coordinates
at the center of V, the Biot-Savart law (10.15) gives
d³r B(r)= Ho
fa'r
==
E(r')=
4π V
[a³r B
= [ @³rs(³) × [d³r}
The r integral is exactly the "electric" field E(r) due to a uniform charge density p(r) = 4.7 €0.
From Gauss' law or otherwise, this is
pr 4π
3€0
=
pV
4 €0
d³r B(r).
=
= [d²rB®) = 1 [d²³²°*° x1(r^)=;
fd³rr -
3V
r'
Example 11.1 Let B(r) be the magnetic field produced by a current density j(r) that lies entirely
inside a spherical volume V of radius R. Show that the magnetic moment of j(r) is
3
2μ0
m =
Therefore,
This problem is the magnetic analog of Example 4.1, which expressed the electric dipole moment
of a charge distribution using a spherical average of the electric field produced by the charge.
Solution: Assume first that j(r) does not lie entirely inside V. If we place the origin of coordinates
at the center of V, the Biot-Savart law (10.15) gives
d³r B(r)= Ho
fa'r
==
E(r')=
4π V
[a³r B
= [ @³rs(³) × [d³r}
The r integral is exactly the "electric" field E(r) due to a uniform charge density p(r) = 4.7 €0.
From Gauss' law or otherwise, this is
pr 4π
3€0
=
pV
4 €0
d³r B(r).
=
= [d²rB®) = 1 [d²³²°*° x1(r^)=;
fd³rr -
3V
r' R.
plo
4л
Ja
r'>R
d³rj(r) x r
r3
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