# Electric field angular momentum

1. Jan 10, 2015

### skrat

1. The problem statement, all variables and given/known data
A straight line with charge density $\lambda$ is in the middle of a large isolation cylinder, that can rotate around it's axis (line). Moment of inertia for that cylinder per unit of length is $l$ and electric charge density applied on the cylinder is $\frac{\lambda }{2\pi a}$ where $a$ is the radius of the cylinder. At first we have a magnetic field $B$, which is than slowly turned off. Calculate the angular velocity $\omega$ after the magnetic filed is turned off. [Feynman paradox]

2. Relevant equations

3. The attempt at a solution
Angluar momentum before $$\Gamma =\varepsilon _0 \int EB dV=\frac{\lambda l B a^2}{2}$$ And after...? Yup, i have i guesstion here. I know that angular momentum should have two terms, one $J\omega$ and the other one that comes from the magnetic filed induced due to the rotation of the charged cylinder. I do have a couple of question concerning that second term.

To get that second term, I have to calculate the magnetic filed $B_2$ due to the rotation of the cylinder with angular velocity $\omega$.
I have two options but I can't decide which one is right or which one is not. Here they are:
1)
Simply $$B=\frac{\mu _0 NI}{l}=\frac{\mu _0}{l}\frac{\lambda l \omega}{2\pi}$$ where I used $N=1$ and $I=\frac{e\omega }{2\pi}$. I don't really understand this option - but it is the one we wrote at university. My solution differs a bit:
2) Let's ignore that $B$ and $j$ are vectors for a moment $$\nabla \times B=\mu _0 j$$ $$\nabla \times B=\mu _0 (\sigma 2\pi l a)(r\omega)$$ if $l$ is the length of the cylinder. Integrating the last equation and using stokes law on the LHS leaves me with $$\int B ds =\mu _0 \sigma 2\pi la\omega \int r 2\pi r dr$$ which sadly leads to a different result than option 1). because $$B=\frac{\mu _0\lambda l \omega a^2}{3}$$

I seriously doubt we got it all wrong at university, so my question here is: Why is my solution wrong? And could somebody explain the option 1.) - especially where this $I=\frac{e\omega }{2\pi}$ comes from.

2. Jan 10, 2015

### skrat

Blaaaa, ok, nevermind. :D I found my mistake now.

The charge on the cylinder is only on the surface and not throughout the entire cylinder. This changes $$\nabla \times B=\mu _0 (\sigma 2\pi l a)(r\omega)$$ to $$\nabla \times B=\mu _0 (\sigma 2\pi l a)(a\omega)$$ and leads to the same result as in option 1.).

thanks, anyway.

3. Jan 11, 2015

### skrat

Ok, I take that back. Something is wroooong :(

Question: Does one calculate the magnetic field $B$ inside a rotating cylinder with charge density $\sigma$ ?