79.
(a) The energy density at any point is given by u
B
= B
2
/2µ
0
, where B is the magnitude of the magnetic
field. The magnitude of the field inside a toroid, a distance r from the center, is given by Eq. 30-26:
B = µ
0
iN/2πr, where N is the number of turns and i is the current. Thus
u
B
=
1
2µ
0
µ
0
iN
2πr
2
=
µ
0
i
2
N
2
8π
2
r
2
.
(b) We evaluate the integral U
B
=
u
B
d
V over the volume of the toroid. A circular strip with radius
r, height h, and thickness dr has volume d
V = 2πrh dr, so
U
B
=
µ
0
i
2
N
2
8π
2
2πh
b
a
dr
r
=
µ
0
i
2
N
2
h
4π
ln
b
a
.
Substituting in the given values, we find
U
B
=
(4π
× 10
−7
T
·m/A)(0.500A)
2
(1250)
2
(13
× 10
−3
m)
4π
ln
95 mm
52 mm
=
3.06
× 10
−4
J .
(c) The inductance is given in Sample Problem 31-11:
L =
µ
0
N
2
h
2π
ln
b
a
so the energy is given by
U
B
=
1
2
Li
2
=
µ
0
N
2
i
2
h
4π
ln
b
a
.
This the exactly the same as the expression found in part (b) and yields the same numerical result.