13.3 Solutions
623
μ =
dμ =
R
0
ω qr 3 dr
R 2 =
ω q R 2
4
13.61 Magnetic moment
μ = NiA = (1)(i)(πr
2
)
∴ i =
μ
πr 2 =
6.4 × 10 21
π (6.4 × 10 6 ) 2 = 4.98 × 10
7 A
13.62 Magnetic energy density
u B =
1
2μ 0
B
2
But
B =
μ 0 i
2r
∴ u B =
1
2μ 0
μ 0 i
2r
2
=
1
8
μ 0 i 2
r 2 =
4π × 10 −7 × (100) 2
8 × (0.1) 2
= 0.157 J/m
3
13.63 u B =
1
2μ 0
B 2 =
(13.5) 2
2 × 4π × 10 −7 = 7.25 × 10 7 J/m 3
13.64 τ max = iAB = (i) (πr 2 ) B
But
l = 2πr → r =
l
2π
∴
τ max = πi
l
2π
2
B =
l 2 i B
4π
13.65 Let the sphere of radius R rotate about the z-axis, Fig. 13.31. Consider a
spherical shell of radius r (r < R) concentric with the sphere. Consider a
volume element symmetrical about the z-axis.
dV = 2πr
2 sin θ d θ dr
(1)
where θ is the polar angle and r is the distance of the volume element from
the centre. The charge dq residing in the volume element is
dq =
3 q
4 π R 3 dV
(2)
The current due to rotation of charge is
623
μ =
dμ =
R
0
ω qr 3 dr
R 2 =
ω q R 2
4
13.61 Magnetic moment
μ = NiA = (1)(i)(πr
2
)
∴ i =
μ
πr 2 =
6.4 × 10 21
π (6.4 × 10 6 ) 2 = 4.98 × 10
7 A
13.62 Magnetic energy density
u B =
1
2μ 0
B
2
But
B =
μ 0 i
2r
∴ u B =
1
2μ 0
μ 0 i
2r
2
=
1
8
μ 0 i 2
r 2 =
4π × 10 −7 × (100) 2
8 × (0.1) 2
= 0.157 J/m
3
13.63 u B =
1
2μ 0
B 2 =
(13.5) 2
2 × 4π × 10 −7 = 7.25 × 10 7 J/m 3
13.64 τ max = iAB = (i) (πr 2 ) B
But
l = 2πr → r =
l
2π
∴
τ max = πi
l
2π
2
B =
l 2 i B
4π
13.65 Let the sphere of radius R rotate about the z-axis, Fig. 13.31. Consider a
spherical shell of radius r (r < R) concentric with the sphere. Consider a
volume element symmetrical about the z-axis.
dV = 2πr
2 sin θ d θ dr
(1)
where θ is the polar angle and r is the distance of the volume element from
the centre. The charge dq residing in the volume element is
dq =
3 q
4 π R 3 dV
(2)
The current due to rotation of charge is