# Cylindrical symmetric magnetic field

1. Jun 10, 2016

### Maike

1. The problem statement, all variables and given/known data
Suppose the magnetic field line pattern is cylindrical symmetric. Explain with Stokes theorem that the field decreases like 1/r (with r the distance from the axis of the cylinder).

2. Relevant equations
Stokes theorem

3. The attempt at a solution
I was thinking of a circular loop around the axis. The line integral around this loop is B*2*pi*r. But I don't really know what I can say about the curl of B if you only know about the cylindrical symmetry.

Thanks in advance for helping me out!!

2. Jun 10, 2016

Integrating the Maxwell equation for $\nabla \times B$: $\$
$\int \nabla \times B \cdot dA=\int \mu_oJ \cdot dA=\mu_o I$. By Stokes theorem $\int \nabla \times B \cdot dA=\oint B \cdot dl$. One question to ask at this point-why is it that $\oint B \cdot dl =B 2 \pi r$? i.e. How do we know that $B$ is constant along the path of the integral?

3. Jun 11, 2016

### Maike

Yes, that is correct. $B_{\phi}$ is constant along the circular path. $J$ often points along the z-direction in problems with cylindrical symmetry, but it isn't a strict requirement.