Classical electrodynamics -Good conductor

[cowrebellion]

Homework Statement

The question given is an electromagnetic wave incident on a vacuum metal interface. The wave is incident normally. We're given that the metal is a good conductor i.e. $$\omega \tau <<1$$ where $$\tau$$ is the collision time of the metal and omega is the angular frequency. The metal is also non-magnetic and the conductivity is of the order of $$10^8$$ Siemens per metre

The first part is easy enough it's just to show that T the transmissivity is equal to 2/n where n is the real part of N the refractive index.

The part that has me stumped is to find the value of $$\omega$$ so that the fraction of incident power deposited beyond a depth d is maximised

Homework Equations

I think the relevant equation is the poynting vector.

I'm taking the time averaged poynting vector for a wave in vacuum as $$S_{avg} = \frac{1}{2} \sqrt{\frac{\epsilon}{\mu}} E_{i}^{2}$$ and inside the metal I assume the form$$S_{avg} = \frac{1}{4} \sigma_{0} \delta E_{t}^{2} e^{\frac{-2 z}{\delta}$$

Where delta is given as$$\sqrt{\frac{2}{\mu \omega \sigma_{0}}}$$

I'm also taking$$n=\frac{c}{\omega \delta}$$

The Attempt at a Solution

I started by saying that since the time averaged poynting vectors is independent of x and y in both case we can say
$$\int S_{avg} dA=S_{avg} A$$

Using this I divided the power incident on the surface by the power incident on hte same area but at a distance d below the surface to obtain$$\frac{\delta \sigma_{0}}{2}\sqrt{\frac{\mu}{\epsilon}}\frac{E_{t}}{E_{i}}\frac{E_{t}}{E_{i}}e^{\frac{-2 d}{\delta}}$$but with this I differentiate w.r.t. omega and I can't obtain an answer? It's been bugging me for a while so I hope someone can help me out. Hope the format of the question is ok It's my first time posting here. =D

edit: I forgot to say that I replaced {E_{t}/E_{i}}^2 with 2/n I tried changing the latex code but it won't edit for some reason.

 

[marcusl]

You have already assumed that the field decays exponentially on the scale of the skin depth

$$\delta=\sqrt{\frac{2}{\mu \omega \sigma_{0}}}$$

But this already gives you the answer: to maximize power deposition beyond d, choose a frequency to make the skin depth greater than d.

 

[cowrebellion]

That makes sense but the question asks for a specific value of the frequency. Initially I thought I'd obtain an equation of the form $$\omega e^{-\omega}$$ which has a definite maximum value.