Is Vm a reliable measure of magnetic fields in the presence of current?

[sachi]

I'm confused about the status of Vm. We are asked under what circumstances we can write B = - Mu0*grad(Vm).
I think the idea is that when there is no current density (displacement or conduction) we can write curl(B) = 0. Therefore we can write B as a grad of a scalar potential, as curl grad = 0 always. The only problem is that even if curl B isn't zero, all this means is that B is not a conservative field, so that Vm isn't path independent i.e it is not well defined. this just means that if we find Vm using different paths we get different values of Vm which differ by a constant, therefore Vm includes an arbitrary constant. the only problem is that this constant disappears on differentiation anyway, so surely we can still write B = -Mu0*grad (Vm) even when there is a current density? thanks for your help.

 

[Physics Monkey]

The point is that if $$V_m$$ is a well behaved function then $$\nabla \times (\nabla V_m) = 0$$ so that the magnetic scalar potential cannot describe a magnetic field in regions where current is nonzero.