Nature of Electromagnetic Waves

Nature of Electromagnetic Waves

Electromagnetic waves are self-sustaining oscillations of electric and magnetic fields. From Maxwell’s equations, a key result emerges: in an electromagnetic wave, the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ are perpendicular to each other, and both are perpendicular to the direction of wave propagation. This is a general feature.

Why are $\mathbf{E}$ and $\mathbf{B}$ perpendicular?

Consider a charging capacitor (Fig. 8.2). The electric field inside the plates is perpendicular to the plates. The displacement current produces a magnetic field that circles around the plates (parallel to them). Thus, $\mathbf{E}$ and $\mathbf{B}$ are perpendicular in this case. This property holds for all electromagnetic waves.

Mathematical Description of a Plane Wave

A typical plane electromagnetic wave propagating along the $z$-direction is shown in Fig. 8.3. At a given time $t$, the electric field oscillates along the $x$-axis, and the magnetic field oscillates along the $y$-axis. Both vary sinusoidally with $z$.

The electric field $E_x$ and magnetic field $B_y$ are given by:

Here:

• $E_0$ and $B_0$ are the amplitudes of the electric and magnetic fields.
• $k$ is the magnitude of the wave vector (or propagation vector) $\mathbf{k}$. Its direction gives the direction of propagation.
• $\omega$ is the angular frequency.
• The wave propagates with speed $v = \omega/k$.

Relation Between $k$ and $\lambda$

The wave number $k$ is related to the wavelength $\lambda$ by:

Speed of Electromagnetic Waves in Vacuum

Using Maxwell’s equations with the wave equations for $E_x$ and $B_y$, we find that $\omega = ck$, where $c$ is the speed of light in vacuum:

This is a fundamental constant. In terms of frequency $\nu = \omega/2\pi$ and wavelength $\lambda = 2\pi/k$, the relation becomes:

Relation Between Electric and Magnetic Field Amplitudes

From Maxwell’s equations, the magnitudes of the electric and magnetic fields in an electromagnetic wave are related by:

This means that at any instant, the ratio of the electric field magnitude to the magnetic field magnitude is $c$.

Electromagnetic Waves in a Material Medium

In a material medium with permittivity $\varepsilon$ and magnetic permeability $\mu$, the speed of light becomes:

Thus, the speed of light depends on the electric and magnetic properties of the medium. The refractive index of one medium relative to another is the ratio of the speeds of light in the two media. …