The relationship between material properties and EM wave frequency

Question

Assuming an EM wave traveling inside an electrically neutral dielectric material. The following electric field describes the EM.

$\vec{E}(t,x)=20\cos(\omega t-50x)\vec{u_y}$.

Using the 3rd Maxwell equation, the associated magnetic field is derived as

$\vec{B}(t,x)=\frac{10^3}{\omega}\cos(\omega t-50x)\vec{u_z}$

then, using the 4th Maxwell equation, I got the equality

$\mu\varepsilon=\frac{2500}{\omega^2}$.

Does it mean that there is a dependency between the material properties ($\mu$ and $\varepsilon$) and the wave frequency? It seems to be a little bit strange since that $\mu$ and $\varepsilon$ usually are constants.

Answer 1

You have got the usual dispersion relation for an EM field $\boldsymbol{E}=\boldsymbol{E}_{0} e^{i\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega t\right)}$ in a linear material

$$\omega=ck$$

with $c=\frac{1}{\sqrt{\mu\varepsilon}}$ the speed of the wave. In your case $k=50$ is given, and therefore $\omega$ can be calculated.