## B_{θ }= −

µ_{0}M 4πr^{3 }

sin θ.

(43)

## The field strength B = |B| thus is

B=

B 2 r + B 2 θ =

µ_{0}M 4πr^{3 }

4 − 3 sin^{2 }θ,

(44)

going to zero as 1/r^{3 }when r increases toward infinity. The fundamental reason why this dipole field is important is simply that it is the term in complete multipole expansion (39) that decay slowest with increasing distance. Sufficiently far away from any source, the dipole field will dominate over the other terms in the multipole expansion (39).

4.2

# Field lines

A field line is a curve which everywhere is parallel to some given field. Denoting the field by B, we

thus have

dr = constant · B

(45)

where dr is a line element along the field line. In Cartesian coordinates we have dr = dx xˆ +dy yˆ + dz zˆ, and hence the field line equation is

dx B_{x }

=

dy B_{y }

=

dz B_{z }

.

(46)

ˆˆ In spherical coordinates, dr = dr rˆ + r dθ θ + r sin θ dφ φ, and thus

dr B_{r }

=

r dθ B_{θ }

=

r sin θ dφ B_{φ }

(47)

is the equation of the field lines. For the dipole field (41), there is no B_{φ}, so the last term in (47) disappears. The remaining

equation for the field lines of the dipole field is dr

2 cos θ

=

r dθ

sin θ

.

(48)

This is a spearable ordinary differential equation, which may be written as

dr

cos θ

r

=2

sin θ

dθ

(49)

=⇒ ln r = ln sin^{2 }θ + C =⇒ r = r_{0 }sin^{2 }θ. This is then the equations of the dipole field lines, some of which are plotted in Figure 3.

(50) (51)

14