Gyromagnetic ratio

Hi,

The gyromagnetic ratio is the ratio of the magnetic dipole moment to the angular momentum.

I really don't get how the fact that the gyromagnetic ratio of a rotating circular loop of mass M and charge Q is g = Q/2M implies that the gyromagnetic ratio of a uniform rotating spere is also g = Q/2M!

 

No.

This is exactly what's throwing me off! In principle, since the sphere could be considered as an infinity of circular loops of different sizes, its dipole moment is the sum of all those. But since the sphere is uniform, the ratio of its mass to its charge is the same at every point of it. Hence, the dipole moment of each loops is the same at Q/2M, making the total dipole of the sphere infinite!

 

Griffiths says:

The result of part (a) was of course that g = Q/2M for a ring.

So he seems to be saying that just by reasoning, we can determine that it is the same. I exposed my reasoning to you; what is flawed in it?

 

I did it by integration.

Does anyone know what is the argument that allows one to conclude that the gyromagnetic ratio of any solid of revolution is Q/2M ?

 

Sure but I'm convinced that nevertheless, their ratio is the same:

[tex]\frac{dQ}{dM} = \frac{dQ/dV}{dM/dV} = \frac{\rho_q}{\rho_m} = \frac{Q}{M}[/tex]

 

For the sphere ok. But that seemed almost like magic you know. There are all these constant that come in from integration and they end up being 1/2 at the end. How do I know it wasn't a big coincidence and that for another random revolution figure, it will be different?

Edit: Ok, now I see it from the integral, by generalizing to an arbitrary volume of revolution, that it will always give Q/2M. Thanks for the hint Tide.