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13 Electromagnetism I
Fig. 13.31 Magnetic moment
due to a rotating charged
sphere
di =
ω
2π
dq =
3qω
4π
r 2 sin θ dθ dr
R 3
(3)
where we have used (2) and (1).
The magnetic moment due to di is
dμ = (di)(dA) =
3q ωr 2 sin θ dθ dr
4π R 3
(π r
2 sin
2
θ)
=
3
4
ω qr 4 dr sin
3
θ dθ
R 3
(4)
where dA is the area enclosed by the circle of radius r sin θ .
Total magnetic moment
μ =
dμ =
3
4
ω q
R 3
R
0
r
4 dr
π
0
sin
3
θ dθ =
3
4
ω q
R 3
R 5
5
4
3
=
ω q R 2
5
(5)
The angular momentum of a sphere about a diameter is
L = I ω =
2
5
m R
2
ω
(6)
∴
μ
L
=
q
2m
(7)
13 Electromagnetism I
Fig. 13.31 Magnetic moment
due to a rotating charged
sphere
di =
ω
2π
dq =
3qω
4π
r 2 sin θ dθ dr
R 3
(3)
where we have used (2) and (1).
The magnetic moment due to di is
dμ = (di)(dA) =
3q ωr 2 sin θ dθ dr
4π R 3
(π r
2 sin
2
θ)
=
3
4
ω qr 4 dr sin
3
θ dθ
R 3
(4)
where dA is the area enclosed by the circle of radius r sin θ .
Total magnetic moment
μ =
dμ =
3
4
ω q
R 3
R
0
r
4 dr
π
0
sin
3
θ dθ =
3
4
ω q
R 3
R 5
5
4
3
=
ω q R 2
5
(5)
The angular momentum of a sphere about a diameter is
L = I ω =
2
5
m R
2
ω
(6)
∴
μ
L
=
q
2m
(7)