In a conductor, the relation between the phase of the magnetic field $\delta_B$ and that of the electric field $\delta_E$ is given by $$\delta_B-\delta_E=\tan^{-1}(\frac{\beta}{\alpha})\tag{1}$$ where $\alpha$ and $\beta$ are the real and imaginary parts of the complex wavevector $k=\alpha+i\beta$.
EDIT: From equation (1), as I understand, the phase of magnetic field is ahead of the electric field only when $\tan^{-1}(\frac{\beta}{\alpha})$ is positive. However, in Griffith's electrodynamics book, he uses this equation to claim the opposite i.e., the magnetic field lags the electric field (without saying anything about the sign of $\tan^{-1}(\frac{\beta}{\alpha})$).
Can someone explain this assertion? In particular, what is wrong in my reasoning?
Best Answer
I'm not sure about that particular equation. There's always issues with how quantities are defined, and signs of particular values. To understand what that equation is saying exactly, you have to look at the derivation. So to solve this problem, I'm going back to some basic electromagnetic relations for a plane wave.
$$E = c B$$
$$c = \frac{1}{\sqrt{\epsilon \mu}}$$
We use the generalized $\epsilon$ that includes conductivity. If we use the time dependence of $e^{i\omega t}$, then $\epsilon = \epsilon'-i\sigma/\omega$
$$\frac{B}{E} = \sqrt{\epsilon \mu}$$
$$phase \left(\frac{B}{E}\right) = phase \left(\sqrt{\epsilon'-i\sigma/\omega}\right) $$
For good conductors at low frequencies, $\sigma/\omega >> \epsilon'$, and the phase difference between the magnetic and electric fields is very close to -45 degrees. This implies the magnetic field phase lags the electric field, just like Griffiths says.