Show the loop integral of Poynting vector is zero for plane waves

[stripes]

Homework Statement

Show that for plane waves, the following result holds:

$\oint \textbf{S}\cdot d \ell = 0.$

Homework Equations

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The Attempt at a Solution

$\oint \textbf{S}\cdot d \ell = \frac{1}{\mu_{0}}\oint (\textbf{E} \times \textbf{B})\cdot d \ell$

Now do I just use some vector identities and try and screw around with it? Or should I take a more intuitive approach? Either way I'm already lost...

 

[TSny]

Start by finding an explicit expression for S for plane waves.

 

[stripes]

As such?

 

[TSny]

As such?

Yes, except you need to include a unit vector indicating the direction of S.

[Edit: Also, S is a function of space and time, not just time.]

 

[stripes]

Alright, since the Poynting vector is arbitrary, can I use any old unit vector? And then how do I turn the differential into something I can work with?

 

[stripes]

In class, we did some derivations as such:

 

[TSny]

Alright, since the Poynting vector is arbitrary, can I use any old unit vector?

Do you know how the direction of S is related to the direction of the wavevector k?

You are dealing with vector equations, and vector equations are valid independent of orientation of the axes. Without loss of generality, you can orient your coordinate system in any way you want. So, think about a nice way to choose your axes.

However, if your instructor wants you to write it all out for arbitrary orientation of the axes, you can still do it.

And then how do I turn the differential into something I can work with?

Once you determine your direction of S, you can think about the dot product of S and dl.

Another approach is to invoke http://www.math.ufl.edu/~vatter/teaching/calcnotes/5-6-stokes.pdf