# Magnetic field created by a long wire

1. Aug 8, 2015

### Caio Graco

The magnetic field created by an electric current in a long straight wire is conservative or not conservative?

A field is conservative when its circulation closed path is zero.

For amperiana curve that surrounds the wire circulation is non-zero, but to a curve which does not involve the wire circulation is zero. And now?

2. Aug 8, 2015

### Hesch

Well, the magnetic field created by an electric current in a long straight wire will always surround/include the wire.

3. Aug 8, 2015

### Caio Graco

But I can choose amperiana curve so as not to move the wire and thus the movement is not null unlike the curve surrounding the wire. Then the magnetic field is conservative or not?

4. Aug 8, 2015

### Staff: Mentor

The path integral of a conservative vector field is independent of the path. It just depends on the end points. So if it is zero for some closed paths and not for others then it is not conservative.

5. Aug 13, 2015

### stedwards

The line integral is not over a simply connected space. The z-axis is not included. Things get a little tricky. This can get into some very abstract things involving cohomology groups that I would like to know more of besides the buzz-word.

Last edited: Aug 13, 2015
6. Aug 13, 2015

### stedwards

The scalar field about the wire is $f= \frac{\mu_0 I}{2 \pi} [ log(r) + c ]$.

$B = df = \frac{\mu_0 I}{2 \pi r}d\phi$. B first appears to be a conservative field.

$B = B_\phi d\phi$

But $\oint B_\phi d\phi = \frac{\mu_0 I}{2 \pi r} 2 \pi n$, where n is the winding number.

$B$ is locally conservative everywhere but at $r=0$, though not globally conservative.

Last edited: Aug 13, 2015