Missing a Factor of 2 in a Poynting Vector Verification

[the-brammo]

The question reads:

A fat wire, radius a, carries a constant current I, uniformly distributed over its cross section. A narrow gap in the wire, of width w << a, forms a parallel-plate capacitor.

I have drawn a red box at the bottom of the page where the Poynting theorem is supposedly verified - however it seems to be a factor of 2 out. I am happy that the derivation for u_em is correct, it must be something to do with the very last line. Could someone please point me in the right direction, excuse the pun.

 

[Haborix]

You should be careful about taking the divergence in cylindrical coordinates.

 

[the-brammo]

Is this a hint? I know there are some steps are missing, does a factor of two comes out when the divergence of s is taken in cylindrical coordinates?

 

[Haborix]

Is this a hint? I know there are some steps are missing, does a factor of two comes out when the divergence of s is taken?

Yes, it was a hint. The divergence of a vector field with only a radial component is given by $$\nabla \cdot (A_s\hat{s})=\frac{1}{s}\frac{\partial}{\partial s}(sA_s).$$

So in your example $A_s =s$ and the divergence becomes

$$\nabla \cdot (s \hat{s})=\frac{1}{s}\frac{\partial}{\partial s}(s^2)=2.$$

 

[the-brammo]

Thanks so much, this answers my question perfectly.