Electromagnetic Waves

One can show (we won't) that Maxwell's Equations have waves as their solutions under certain conditions. In this case, one can write the electric and magnetic field in space, x, and time, t, as

here, the dot between the vectors and indicates a dot product of these two vectors.
The amplitudes of the electric and magnetic fields are both perpendicular to the wave vector (direction of propagation):

This means that the electric and magnetic fields are oscillating perpendicular to the direction of propagation. We call these types of waves transverse waves. (We also know of examples for longitudinal waves, in which the oscillations occur along the direction of propagation: sound waves).

The speed of propagation of the electromagnetic wave in vacuum is:

\[ \rm \mathbf{c = \frac{ \omega}{k} = ( \varepsilon_{0} \mu_{0})^{-1/2} } \]

The last step is the result of inserting the expressions for into Maxwell's Equations. The numerical value of c turns out to be:

c = 299,792,458 m/s,

about 300 000 km/s, exactly the speed of light! This lead Maxwell to postulate:

Light consists of electromagnetic waves!

The absolute value, k, of the the wave vector, , is related to the wavelength, $\lambda$, via

\[ \rm \mathbf{k = \frac{2 \pi}{ \lambda}} \]

and the angular frequency, $\omega$, and the frequency, f, are related via:

\[ \rm \mathbf{f = \frac{ \omega}{2 \pi}} \]

Thus we can also write for the speed of light:

c = $\lambda$ f

(Remark: often the symbol $\nu$ (Greek letter nu) is also used for the frequency. Thus you may also encounter the equation:

c = $\lambda\nu$

Light can also travel through some media, such as water or glass. (Through which media light can travel actually depends strongly on the wavelength of the light.) In these media the speed of propagation of the wave, v, is lower than c and is given by the index of refraction, n:

\[ \rm \mathbf{v = \frac{c}{n}} \]

where n is a number larger or equal to 1. More on the index of refraction will follow.