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.