What is the force between two magnetic dipoles?

Question

1. What is the force between two magnetic dipoles?

2. If I have two current loops parallel to each other with currents $I_1$ and $I_2$ and radii $R_1$ and $R_2$ a distance $z$ from each other, what is the force between them?

3. What would change if they were two solenoids instead of current loops?

4. Would the same hold if it was two magnets?

5. Or a magnet and a solenoid?

Answer 1

The field of a magnetic dipole is, at long distances (in some naturalized unit convention), equal to

$$ B_i = \partial_i ({\mu \cdot r \over 4\pi r^2})$$

where $\mu$ is the magnetic moment vector (the current times area of the loop, in the direction perpendicular to the area of the loop, times N for a solenoid with N windings, or just a given value for a fixed magnet). This form is universal for all magnetic dipoles at long distances, so it's the same for small loops, for small magnets and for small solenoids.

The form is easy to understand, because it's the form of the field for an electric dipole

The response of a second rigid magnetic dipole to this field is a torque which tends to align the magnetic moment along the direction of the field, plus a force proportional to the field gradient. These are also determined by the magnetic dipole of the magnet.

$$ T = \mu \times B $$

$$ F_j = - \mu^i \partial_i B_j $$

The first tends to align the magnetic dipole with the field, and this tends to happen quickly, the second moves the aligned dipole to the region of stronger field, and this happens more slowly.

These equations give a complete determination of the forces and torques acting between two dipoles, but if you substitute in the dipole field, the expressions become complicated and unilluminating. The picture is that there is a $1/r^3$ magnetic field which will align magnets so that their magnetic moment points along it (in the presence of dissipation), and then will lead to an attractive force as the aligned magnet drifts to a region of stronger field.

Answer 2

The equation for the force depends on how far you are from the current source. If you are far away, then the dipole-dipole interaction formulas can be used. In that case, all of the configurations will give the same force if they have the same dipole moments. If you are closer, then more complicated specific equations must be used for each configuration.