I am studying electrodynamics using Zangwill [1]. I have a narrow question regarding a proposition found there in. In the section on the Drude model of conducting matter that applies to particles with mean travel time $\tau$, a characteristic plasma frequency $\omega_p$, the Fourier transform of the permittivity, $\hat{\epsilon}(\omega)$, the Fourier transform of the permiability, $\hat{\mu}(\omega)$, and frequency parameter $\omega$. Adjusting from what Zangwill writes in [1], the proposition is that

In the high frequency limit when $$\frac{\hat{\epsilon}(\omega)}{\epsilon_o}\approx 1 - \frac{\omega_p^2}{\omega^2} \quad (\omega\,\tau \gg 1)$$ is valid, the dispersion relation for transverse waves $$k(\omega) = \omega\sqrt{\hat{\mu}(\omega)\,\hat{\epsilon}(\omega)} = \frac{\omega}{c}\,\hat{n}(\omega)$$ reads $$ \boxed{\omega^2 = \omega^2_p + c^2\,k^2 .}\tag{1} $$ I have made several attempts to maniupulate these and other equations to obtain (1). Yet, I come up short.


The narrow question here is, how does one derive the high-frequency (or collisionless) dispersion relation, i.e., Eq.(1), for transverse waves using the Drude model?


Zangwill, Modern Electrodynaics, 2013 pp. 630-632 .


1 Answer 1


The dispersion relation follows from a simple substitution of your first equation into the second. The square of the 2nd equation gives $$ k^2 = {\omega^2 \over c^2}{\epsilon \over \epsilon_0} $$ Substituting the 1st equation $$ k^2 = {\omega^2 \over c^2}{(1-{\omega_p^2 \over \omega^2})} $$

$$ \omega^2 = {\omega_p^2+k^2 c^2} $$ Edit based on comments about refractive index:

I am not sure what is troubling you regarding refractive index. Look at your 2nd equation. If you eliminate $\omega$, you have the definition of refractive index. $$n(\omega) = c \sqrt{\epsilon(\omega) \mu(\omega)}$$ $$n(\omega) = \sqrt{{\epsilon(\omega) \mu(\omega)} \over {\epsilon_0 \mu_0}}$$ With the usual assumption of non-magnetic media, $\mu(\omega)=\mu_0$ , then $$n(\omega) = \sqrt{{\epsilon(\omega)} \over {\epsilon_0}}$$ Also note that the ratio, $\epsilon(\omega) / \epsilon_0$ is called the relative permittivity or dielectric constant.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.