The discussion revolves around the conservation of the quantity \(\sin^2\theta/r\) along dipolar magnetic field lines in the context of gamma-ray emission from pulsars. Participants explore the derivation of this claim, examining the mathematical relationships and physical implications involved.
Participants do not reach a consensus on the derivation process or the implications of the gradient being perpendicular to the magnetic field. Confusion remains regarding the constancy of \(c\) and the interpretation of the mathematical relationships.
There are unresolved questions about the assumptions underlying the derivation and the definitions of the quantities involved. The discussion reflects varying levels of understanding of the mathematical concepts presented.
This discussion may be of interest to those studying astrophysics, particularly in the areas of pulsar emissions, magnetic fields, and vector calculus.
Bill_K said: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 said: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
rbwang1225 said:Wow! I got it, but could you tell me what kind of text have this kind of derivations? calculus? or vector analysis?
Thanks!