Zero curl of magnetic field and Ampére's law

Question

Is it true that if I a have zero curl of the magnetic field, I will get a zero current density using Ampére's law? I know that on the surface of the conductor it doesn't have to be true (the current density is going to be infinite here), but is it true that using Ampére's law here is useless?

Answer 1

Note Maxwell-Ampere's law

$$ \frac{1}{\mu_0}\left(\vec\nabla \times \mathbf{B}\right) = \left(\mathbf{J}_{\rm enc} + \epsilon_0 \frac{\partial\mathbf{E}}{\partial t}\right). $$

• $\mathbf{J}_{\rm enc}$ is the current density enclosed by a path.
• $\epsilon_0 \frac{\partial\mathbf{E}}{\partial t}$ is the displacement current density, due to a time-varying, nonconservative electric field.

If ${\rm curl\,}\mathbf{B}$ is zero then the entire right hand side is zero also --- which means there is no net current density.

The question then begs whether the possibility that you have described can really exist. Because we know by Gauss' law applied to magnetism that

$$ \vec \nabla \cdot \mathbf{B} = 0,$$

we know all magnetic fields must curl. North poles must connect to south poles. Sure, magnetic fields can be locally uniform, but they must curl at some point. The only way that $\nabla \times \mathbf{B} = 0$ is if there is no magnetic field at all, if we are talking about the field at all locations in space.

Answer 2

If you have a magnetic field which has curl 0 around some closed loop (not necessarily everywhere), i.e. $\nabla\times\vec{B}=0$ around some closed loop $C$, then it means that any surface $S$ for which $C$ is the boundary must have 0 net current passing through it (including displacement currents). This is due to Ampere's law and it is completely general. If you have a magnetic field which has curl 0 everywhere, then since the divergence of the magnetic field also has to be 0 everywhere, it implies that you have no magnetic field anywhere. This is due to the Helmholtz decomposition theorem - that any (sufficiently smooth and rapidly decaying) vector field can be decomposed into a curl-free (pure divergence) and a divergence-free (pure curl) component. No magnetic field anywhere implies no current (including displacement current) anywhere.

As the curl of the magnetic field is itself a field (i.e. has a value for all points in space), it is not clear what precisely you mean by "have zero curl of the magnetic field". Whether you mean you have 0 curl around some closed loop, or everywhere, or simply at a point.