Magnetic Field Sources - Magnetocaloric Energy Conversion: From Theory to Applications - page 38

3.1 Introduction

Before starting, we will provide some basic de nitions, which are necessary for an

understanding of the magnetism.

3.1.1 Magnetic Field and Magnetic Induction

From the Biot - Savart law [ 6 ] we consider the generation of a magnetic eld H due

to the steady current fl ow I (constant over time) in a long conductor. Then the

eld d ! at some radial distance r from the elemental length of conductor
d ! can be de

magnetic

ned as: the vector product, which is perpendicular to the plane given
by d ! and r ! , and where r ! ¼ ! , and therefore ! represents a unit vector in

r

the radial direction:

d ! ¼

1
4 p r 3 I d

1
4 p r 2 I d

r ! ¼ !

~ s ~ s ð 3 : 1 Þ

In Fig. 3.1 , a long electrical conductor carries the electric current. If we measure
the magnetic eld at the point P for the element dL, this can be, according to

Eq. ( 3.1 ), de ned as:

1
4 p r 2 I dsu

I cos a

4 p a

90 a

dH ¼ sin ð Þ ¼ d a ð 3 : 2 Þ

a d a
cos

where s = a ¼ tg a ) ds ¼

2 a and a ¼ r cos a .

For the in nitely long electrical conductor at a distance of a, the magnetic eld

strength can be derived as:

2

Z

I cos a d a

4 p a ¼

I
2 p a

Ha ðÞ¼ ð 3 : 3 Þ

2

Fig. 3.1 The magnetic eld induced around a long conductive wire