Deriving the Claim: \sin^2\theta/r Conservation in Pulsar Gamma-Ray Emission

[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 anybody know how to derive this claim?

Thanks in advance.

 

[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.

 

[rbwang1225]

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.

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

 

[Polyrhythmic]

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

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.

 

[rbwang1225]

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

Thanks!

 

[Polyrhythmic]

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

Thanks!

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