Cylindrical symmetric magnetic field

[Maike]

Homework Statement

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).

Homework Equations

Stokes theorem

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!

 

[Charles Link]

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?

 

[Maike]

Thanks very much for your answer!
I think B is constant along the path because of the cylindrical symmetry. Or is that a wrong conclusion?
Are you sure by the way that ∫J⋅da = I in this case? How can you know that J and da point in the same direction? Or doesn't it matter?

 

[Charles Link]

Thanks very much for your answer!
I think B is constant along the path because of the cylindrical symmetry. Or is that a wrong conclusion?
Are you sure by the way that ∫J⋅da = I in this case? How can you know that J and da point in the same direction? Or doesn't it matter?

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.