# Do electromagnetic waves travel in the same speed only in vacuum? Why so?

I've read that all electromagnetic waves travel in the same speed in vacuum. But what about air, water and other materials, is their speed different in those materials, if yes then why?

Your question can be answered in a variety of ways that are quite different from each other depending on whether a classical or quantum type answer is most meaningful to you. However, I will take a very simple Classical argument to answer your question.

Wave propagation in Free space is the Speed of Light usually denoted by the letter $c$. Maxwell's equations tell us that this value of $c$ can be computed from the following relationship with the permeability of free space $\mu_0$ and the permittivity of free space $\epsilon_0$: $$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$$ However, in material mediums, both of these values of permeability and permittivity can change in such a manner that the resultant speed computed is less than the standard speed of light in a vacuum.

These values of permeability and permittivity are sort of like an impedance to the flow of the electromagnetic waves. As a result, the medium is sometimes denoted as having a velocity factor for electromagnetic waves which is a percentage of the speed of light for that medium.

The phase and group velocity of light in vacuum is identical and constant $$c_{ph}=\frac {\omega}{k}=c_{gr}=\frac {d\omega}{dk}=c$$ due to the linear dispersion relationship $$k(\omega) =\frac {2\pi}{\lambda}=\frac {\omega}{c}$$ Therefore , in vacuum all electromagnetic waves travel at the same speed. In usual dielectric materials (with $\mu_r=1$) the electromagnetic field interacts with the molecules producing a dielectric polarization which leads to (frequency dependent) a relative dielectric permittivity $\epsilon_r \gt 1$. This, in general, changes the phase velocity to a value smaller than the vacuum light velocity $$c_{ph}=\frac {c}{n} \lt c$$ where the (frequency dependent) refractive index is $n =\sqrt \epsilon_r$. Thus the phase velocity of light in a dielectric medium is (with exceptions) smaller than the speed of light in vacuum. It usually also depends on the frequency of the light, which is called dispersion.

Yes, all EM waves travel with same speed $c$ only in vacuum. In other media, their velocity varies w.r.t their frequency.

In an EM wave the teo characteristic properties are frequency an wavelength. Here, frequency can be considered a constant as is decided by the source producing the EM wave. On the other hand, the wavelength does npt remain constant and is susceptible to change when travelling between media. So, on the whole the frequency is constant whereas the wavelength changes. Now, the speed of an EM wave = frequency $\times$ wavelength.

From the above facts, we understand that when travelling between media, the wavelength changes and therefore the speed of EM wave also changes. Also this change is different for different frequency. So, different EM waves possess different velocity of propagation in different medium.