Magnetic Field Sources - Magnetocaloric Energy Conversion: From Theory to Applications - page 60

By the integration of the vector:

Z

2 p

h i d u

l 0 Ia
4 p a 2

B ¼

z cos u i þ z sin u j þ a k ð 3 : 76 Þ

3
2

ð þ z 2 Þ

0

One can now obtain the solution for the magnetic fl ux density in each of the

directions. However, the magnetic fl ux density in directions x and y equals zero.

B x ¼ B y ¼ 0 ð 3 : 77 Þ

Therefore, the magnetic fl ux density vector acts in the direction z with a mag-

nitude of:

Z

2 p

l 0 Ia 2
4 p a 2

l 0 Ia 2
2 a 2

B z ¼ B ð z Þ¼ d u ¼ ð 3 : 78 Þ

2 2

ð þ z 2 Þ ð þ z 2 Þ

0

In the centre of the ring, when z = 0, we obtain:

l 0 I
2 a

B ¼ ð 3 : 79 Þ

Let us focus now on the number of single N turns of an electric wire (Fig. 3.19 ).
We will consider, in contrast to the case in Eqs. ( 3.7 )

( 3.9 ), an example of a nite

-

solenoid.
By considering the previous example in Fig. 3.18 , we will rst de ne the number

of turns per unit length as:

N
L

n ¼ ð 3 : 80 Þ

Fig. 3.19 A nite circular
coil (solenoid) with N turns of
wire