# Magnetic dipole

1. Jun 24, 2011

### rbwang1225

When I study the mechanism of gamma-ray emission from pulsars, I got a statement saying that the quantity $\sin^2\theta/r$ is conserved along any dipolar magnetic field line.

Does any body know how to derive this claim?

2. Jun 24, 2011

### Bill_K

The field of a magnetic dipole is B = (m/r3)(3r(r·z) - z) where r, z are unit vectors.

Let c = sin2Θ/r = (r2 - z2)/r3. Take the gradient of c and show that it is perpendicular to B. This shows that the lines c = const are the magnetic lines of force.

3. Jun 24, 2011

### rbwang1225

Sorry, I do not understand why the gradient of c is perpendicular to B means that the lines c = const are the magnetic lines of force. And is c a constant?

Regards

4. Jun 24, 2011

### Polyrhythmic

If the gradient, i.e. the "change" of some quantity is perpendicular to some other field, it means that the quantity has no change in the direction of the field. It is therefore constant along the lines of that field.

5. Jun 24, 2011

### rbwang1225

Wow! I got it, but could you tell me what kind of text have this kind of derivations? calculus? or vector analysis?

Thanks!

6. Jun 24, 2011

### Polyrhythmic

It should be covered in any introduction to vector analysis. Texts on electromagnetism might also help.