Why is the curl of the magnetic field zero in a vacuum?

[ShayanJ]

In the wikipedia page for guiding center,the following lines are written about curvature drift of charged particles.

The expression for the grad-B drift above can be rewritten for the case when $\nabla B$ is due to the curvature. This is most easily done by realizing that in a vacuum, Ampere's Law is $\nabla\times\vec{B} = 0$. In cylindrical coordinates chosen such that the azimuthal direction is parallel to the magnetic field and the radial direction is parallel to the gradient of the field, this becomes
$\nabla\times\vec{B} = \frac{1}{r} \frac{\partial}{\partial r} \left( r B_\theta \right) \hat{z} = 0$
Since$r B_\theta$is a constant, this implies that
$\nabla B = - B \frac{\vec{R}_c}{R_c^2}$
and the grad-B drift velocity can be written
$\vec{v}_{\nabla B} = -\frac{\epsilon_\perp}{q} \frac{\vec{B}\times \vec{R}_c}{R_c^2 B^2}$

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

 

[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.

 

[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.)

 

[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 $\frac{\partial \mathbf E}{\partial t}=0$.

 

[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!