Magnetic Field Sources - Magnetocaloric Energy Conversion: From Theory to Applications - page 59

Fig. 3.18 Magnetic fl ux density as a result of a circular loop that carries a steady current

The vector product between d l and

~ r equals:

h i

d l k

ad u sin u i þ cos u j cos u i sin u j ~ r ¼ z a a ð 3 : 71 Þ

where:

d u¼

d u

d a

d u

~

d l

sin u i cos u j ¼ a þ a ð 3 : 72 Þ

It follows from Eq. ( 3.71 ):

h i d u

d l cos u i sin u j k r ¼ az þ z þ a ð 3 : 73 Þ

The magnetic fl ux density can now be de ned as:

h i d u

l 0 l 0

4 p r 3 Ia z

r ! ¼

4 p r 3 Id l

k

dB cos u i sin u j ¼ þ z þ a ð 3 : 74 Þ

h i d u

l 0 Ia
4 p a 2

k

dB cos u i sin u j ¼ z þ z þ a ð 3 : 75 Þ

3
2

z 2

ð þ Þ