# 4 Magnetic fields

4.1

## The dipole field

In the static case (∂/∂t = 0), it follows from Maxwell’s equations (3) – (6) that the magnetic field

## B is governed by the two equations

∇ · B = 0,

(35)

sometimes known as Gauss’ law for the magnetic field or the condition of no magnetic monopoles,

and

∇×B =

µ

_{0}j

,

(36)

called Amp`ere’s law. In perfect vacuum, there are no free charges that can carry a current, so here (36) reduces to ∇ × B = 0. From vector analysis, we know that this is a sufficient condition for the existence of a

scalar magnetic potential Ψ such that

B = −∇Ψ. Combining with (35), we get a Laplace equation for the magnetic potential:

(37)

∇^{2}Ψ = −∇ · (∇Ψ) = −∇ · B = 0.

(38)

If all sources of the magnetic field (all currents) are contained inside a sphere of radius r = a, then B can be described by Ψ outside this sphere. From the course in ”Mathematical methods of physics”, we recall that in spherical coordinates, the general solution of the Laplace equation (38) is

the multipole expansion

Ψ=

∞

l

A r lm l+1

Y_{lm}(θ, φ)

(39)

l=0 m=−l

where we have neglected terms that grow with distance r, as all fields should decrease with distance outside the source sphere r = a. The functions Y_{lm }are known as spherical harmonics. This expansion is very nice as it splits the potential into terms with different r-dependence: the higher the l-value, the faster the potential, and thus the field, decays with increasing r. For the case of the magnetic potential the term l = 0 will be missing, since this term does not fulfill the requirement that ∇ · B = 0 everywhere, as required by (35). Therefore, the first non-vanishing terms in the expansion (39) are the l = 1 terms, which are known as the dipole terms. With a suitable choice of coordinate axis (putting the symmetry axis along the dipole moment M of the sources r < a, i.e. M = Mzˆ, only the m = 0 term need to be used, so that the first term in the expansion (39) is

cos θ Ψ = A_{10 }r^{2 }

.

(40)

The coefficient is related to the dipole moment M by A_{10 }find that the dipole magnetic field is

= −µ_{0}M/4π, so taking the gradient, we

B = −∇Ψ = −

µ_{0}M 4πr^{3 }

ˆ 2 cos θ rˆ + sin θ θ

(41)

or

B_{r }= −

µ_{0}M 2πr^{3 }

cos θ

(42)

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