Calculating Power Dissipation on Cylinder Surface

[likephysics]

Homework Statement

I am trying to calculate power dissipated over a cylindrical surface using poynting vector -
$$\oint$$ ExH ds

I know ds for a sphere is r^2 sin $$\theta$$ d$$\theta$$ d$$\phi$$

But now sure what ds is for a cylinder?

Homework Equations

The Attempt at a Solution

 

[gabbagabbahey]

I know ds for a sphere is r^2 sin $$\theta$$ d$$\theta$$ d$$\phi$$

But now sure what ds is for a cylinder?

It depends on which surface you are talking about. A closed cylinder has 3 surfaces; one curved surface and two flat circular end-caps. For the end-caps, $dS=s ds d\phi$. While, for the curved surface, $dS=s d\phi dz$. (Using $\{s,\phi,z\}$ for the cylindrical coordinates)

Griffiths' Introduction to Electrodynamics derives the infinitesimal displacements ($dl_s=ds$, $dl_\phi=s d\phi$, $dl_z=dz$) in cylindrical coordinates in section 1.4.2. And the author gives a brief discussion of how to obtain area elements from these infinitesimal displacements at the end of page 40 (3rd edition).

 

[likephysics]

Great. I am going to go take a look at Griffith's right now.