Magnetic dipole moment: torque, energy and field equations

[pervect]

here seems to be some interest in magnetic dipoles, such as spinning electrons, and current loops. So I thought I would start a thread and present some of the relevant equations that describe the forces and fields generated by magnetic dipoles. These equations are very similar to those for electric dipoles, BTW.

A current loop with an area A and carrying a current i has a
magnetic dipole moment of $$\mu = i A$$. The dipole moment is sometimes expressed as a vector $$\vec{\mu}$$ in which case the vector is perpendicular to the area A.

Some useful properties of the diople moment are given below

Torque generated by an external field $$\vec{\mu} \times \vec{B}$$

Energy in an external field $$-\vec{\mu} \cdot \vec{B}$$

Field from dipole at distant points along axis |B| = $$\frac {\mu_0}{2 \pi} \frac {\mu}{r^3}$$

Field from dipole at distant points along bisector |B| = $$\frac {\mu_0}{4 \pi} \frac {\mu}{r^3}$$

Field from dipole, vector form $$\vec{B} = \frac {\mu_0 \mu}{4 \pi r^3} (2 cos(\theta) \vec{r} + sin(\theta) \vec{\theta})$$

Net force on dipole from a constant magnetic field zero

Net force on a dipole from a varying magnetic field $$\nabla (\vec{\mu} \cdot \vec{B})$$

Note that the force between two dipoles will drop off with the 4th power of the distance - as the field generated by a dipole is proportional to 1/r^3, the gradient of the field is proportional to 1/r^4, and the force will be the dipole moment multiplied by the field gradient.

 

[krab]

Net force on dipole from a constant magnetic field zero

Net force on a dipole from a varying magnetic field $$\nabla (\vec{\mu} \cdot \vec{B})$$

"Constant" or "varying" usually means w.r.t. time. I think here you mean w.r.t. space, so common usage is "uniform" or "non-uniform".

 

[pervect]

"Constant" or "varying" usually means w.r.t. time. I think here you mean w.r.t. space, so common usage is "uniform" or "non-uniform".

Yes, that's what I mean. To develop a net force, one needs the field to be different at the two ends of the dipole, which means that the field must be varying in space.