In classical electrodynamics, a * magnetic dipole moment* is associated with a current loop (see diagram below), and represents the magnitude and orientation of a magnetic dipole. It is a pseudo-vector, whose direction is perpendicular to the plane of the loop and given by the right hand thumb rule.

The magnitude of a magnetic dipole moment of a current loop is defined as

where is current, and is the area of the loop.

Rewriting eq60 in terms of (where is charge, is time and is the tangential velocity of the charged particle) and using , we have , where is the mass of the charged particle. Substitute eq59 in , we have the relation between magnetic dipole moment and angular momentum:

where is the ** classical gyromagnetic ratio**.

When placed in an external magnetic field , the magnetic dipole moment experiences a ** torque**, whose energy is

###### Question

How is eq62 derived?

###### Answer

In order to rotate a current loop, we must do work against the torque due to the magnetic field . For a rotating system (see diagram I below), , where according to convention, the negative sign is added so that work on the system by the field is positive.

For small , . Since and , we have

Diagram II below shows a square current loop with side 1 parallel to side 3 and side 2 parallel to side 4 (not shown in diagram). The length of each side is .

Since and (),

where is the area of the loop.

Substitute eq60 in the above equation,

Comparing eq62a and eq62b, torque can also be expressed as

The torque exerted by the magnetic field on the magnetic dipole tends to rotate the dipole towards a lower energy state. So, we let with and eq63 becomes , or simply:

Other than a torque, the external magnetic field may exert another force on the current loop. For a magnetic field pointing in the -direction, , where is a scalar function. We substitute eq65 in to give

where , and are unit vectors, and is the gradient of the external magnetic field along the -direction.

For a uniform field, (i.e. a constant) and ; while for an inhomogenous field, , where is a scalar representing the change in along the -axis, and .

Substituting eq61 in eq62,

For an inhomogenous magnetic field pointing in the -direction, the above equation becomes

Eq68 is used as a starting point in analysing results from the ** Stern-Gerlach experiment**.

###### Question

Does the magnetic field violate ** Maxwell’s 2^{nd} equation** of where ?

###### Answer

The field should be , which satisfies . However, the precession of the spin magnetic moment of a silver atom around in the Stern-Gerlach experiment is so fast that the -component of the spin moment averages to zero, resulting in an effective field of interacting with the atom.

###### Question

Why is ?

###### Answer

A simple way of showing is to consider a two-dimensional diagrammatic representation of the vector function below, where each point in the two-dimensional space is associated with a vector.

When , the successive points in the negative direction and the positive direction are associated with the vectors and respectively. Similarly, when , the successive points in the negative direction and the positive direction are associated with the vectors and respectively. Clearly, .