Curl free magnetic field

1. Jun 5, 2013

ShayanJ

In the wikipedia page for guiding center,the following lines are written about curvature drift of charged particles.
My problem is the part that tells curl of B is zero in a vacuum.
Although I know Maxwell equations permit such a situation(with $\vec{B}=\vec{B}(\mbox{only space variables}) \ and \ \vec{E}=0$),I don't understand how that can happen.
Obviously a current has caused the existence of the magnetic field and so there should be a changing electric field.But there is no electric field and that seems strange to me.
I'll appreciate any ideas.
Thanks

2. Jun 5, 2013

marcusl

Maxwell's equation relates curl of B to density and to the time derivative of E. If you are in a region that as no currents (i.e., away from wires) and the fields are static (no time variation), then curl B = 0.

3. Jun 5, 2013

ShayanJ

Well,I know that.
The problem is, how can the electric field be static when there is a current?(I know that we're assuming no current at the point we're calculating the curl of B at.But a current elsewhere produces a changing electric field everywhere.)

Last edited: Jun 5, 2013
4. Jun 5, 2013

marcusl

The current can be static--a DC current from a battery, for instance. Or there may be no current at all, as when the magnetic field in your source-free region arises from a permanent magnet. Either way, J=0 in the vacuum and $\frac{\partial \mathbf E}{\partial t}=0$.

5. Jun 5, 2013

vanhees71

Somewhere should be sources of the magnetic field. In the static case you have
$$\vec{\nabla} \times \vec{H}=\vec{j}+\frac{1}{c} \vec{\nabla} \times \vec{M},$$
where $\vec{M}$ is the magnetization of your permanent magnet. Of course, everywhere else, the curl vanishes, and there you can have locally a potential. Other than the elecric field a magnetic field can never be globally a gradient field!