How does ∇ × J = 0 relate to B = 0 in Maxwell's equations?

[It's me]

Homework Statement

Prove that a current density J(r, t) such that ∇ × J = 0 implies the magnetic field B = 0.

Homework Equations

Maxwell's equations, vector calculus

The Attempt at a Solution

I've played around with Maxwell's equations and with the properties of vector calculus but I can't reach the necessary conclusion. Any hints would be greatly appreciated.

 

[andrewkirk]

I think the problem may have been mis-stated because, as stated, it looks false.
Consider a point outside a long, straight wire carrying a steady, direct current. ∇ × J=0 at that point because J=0 in an open neighbourhood of that point. But B is not zero. It is a stable, nonzero field that runs around the wire.
The proposition fails inside the wire too: see these calcs.

 

[It's me]

Thank you very much, I had the feeling something was wrong when the math just didn't agree with the statement.
Do you know of any property that is similar to the one I was trying to prove? I mean, if the problem is mis-stated, any ideas as to what the correct statement is?

 

[andrewkirk]

Given those conditions, if we also have $\frac{\partial\mathbf{E}}{dt}=0$, which will for instance be correct if the current is steady, then we can deduce that $\nabla^2\mathbf{B}=0$. Perhaps they meant that.

 

[It's me]

Ok thanks, I really appreciate your help.