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CAF123
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Homework Statement
The following dispersion relation is obtained by substituting a plane wave solution for ##\mathbf{B}## into $$\nabla^2 \mathbf{B} = \mu \epsilon \frac{\partial^2 \mathbf{B}}{\partial^2 t} + \mu \sigma \frac{\partial \mathbf{B}}{\partial t}:$$ $$\hat{k} = (\mu \epsilon w^2 + i\mu \sigma w)^{1/2}$$
1)Expand this to obtain the limiting expressions for the skin depth in poor and good conductors.
2)Show that in a good conductor the magnetic field lags the electric field by π/4.
Homework Equations
Binomial expansion
The Attempt at a Solution
1)##\hat{k}## can be rexpressed as ##\sqrt{\mu \epsilon w^2} \sqrt{\left(1+\frac{i\sigma}{\epsilon w}\right)}## For a poor conductor, ##\sigma/ \epsilon w \rightarrow 0## so I can binomial expand. Doing this gives me a correct answer but when I consider expanding in ##\epsilon w/ \sigma## for a good conductor I don't. Is there a better way to do this expansion? Even though I obtained one correct answer, I feel that it is incorrect because of the imaginary unit. On an argand plane, i has magnitude 1, so via some 'hand-waved' argument overall ##i \sigma/\epsilon w## is <<1.
2)To derive this, we consider writing ##E = E_o e^{i \delta_E}## and ##B = B_oe^{i\delta_B}## where ##\delta_{E,B}## are characteristic skin depths of E and B (see Griffiths) but I am not sure how these come about.