# Prove ∇ × J = 0 means B=0

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1. Mar 27, 2016

### It's me

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Maxwell's equations, vector calculus

3. 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.

2. Mar 27, 2016

### 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.

3. Mar 27, 2016

### 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?

4. Mar 27, 2016

### 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.

5. Mar 27, 2016

### It's me

Ok thanks, I really appreciate your help.