# Cylindrical symmetric magnetic field

## 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).

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!!

Homework Helper
Gold Member
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?

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?

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