Phase difference between electric and magnetic fields of EM wave in a conductor?

When an EM wave travels inside a conductor , we see that there is a phase difference between the Electric and magnetic fields within the conductor. The magnetic field lags behind the E field and this lag is roughly equal to 45 degrees within a good conductor (all from Griffiths chp 9.4). What is the cause of this phase difference though? If an EM wave is incident on a lossless dielectric, then the transmitted electric and magnetic waves are in phase with each other but the presence of a conductor introduces a phase difference

I know that in a driven LR circuit there is a phase difference between the driving voltage and the current through the inductor. Since the magnetic field produced by an inductor is proportional to the current, we know that there is an equal phase difference between the driving voltage and the magnetic field produced by the circuit. Is this phase difference in any way related to the phase difference we get in when an EM wave travels in a conductor? I can account for the phase difference in a driven LR circuit easily: The non-zero inductance of the inductor creates a back emf which "fights" against a change in current which ultimately leads to the current lagging behind the voltage. For large inductances and/or high frequencies and/or low resistance (ie good conductance), the phase lag approaches 90 degrees though. Not the 45 degrees we get for a good conductor when a EM wave travels within it. I can't seem to account for these seemingly conflicting behaviours. Is the phase delay between the electric and magnetic components of an EM wave even caused by a back emf at all or does it arise because of a different reason entirely?

Any help on this issue would be most appreciated as it has been driving me mad recently.

In equation 9.133 of Griffiths it is stated that

$$\phi = \tan^{-1}\frac{\kappa}{k}$$

where (according to equation 9.125):

• $$\\phi\$$ is the phase angle between the electric and magnetic fields,
• $$\\kappa\$$ is the imaginary part of the complex wave-number
• $$\k\$$ is the real part of the complex wave-number

According to Equation 9.126, one should have.

$$(\frac{\kappa}{k})^2 = \frac{\sqrt{1+\frac{\sigma}{\epsilon\omega}}-1}{\sqrt{1+\frac{\sigma}{\epsilon\omega}}+1}$$

And,

$$\lim_{\sigma \rightarrow \infty}\frac{\sqrt{1+\frac{\sigma}{\epsilon\omega}}-1}{\sqrt{1+\frac{\sigma}{\epsilon\omega}}+1} = 1$$

So, when conductivity $$\\sigma\$$ is very high, $$\\phi\$$ tends to 45 degrees.

• Thanks for the response. When I use symbolab to calculate that limit as $\sigma$ tends to infinity I get that the limit is 1? I find it such a weird "coincidence" (for a lack of a better word) that the phase difference should be exactly $\pi/4$ without any physical explanation available anywhere I look Apr 21, 2021 at 14:47