Physics Bootcamp

The wave nature of light is essential for many phenomena exhibited by light. In 1865, James Clerk Maxwell identified light as intertwined waves of electric and magnetic fields. These waves are called electromagnetic waves. Before Maxwell, optics and electromagnetism were two separate branches of study in physics. Maxwell showed that optics is actually a subfield of electromagnetism.

According to the Maxwell's equations of electricity and magnetism, an oscillating electric field induces an oscillating magnetic field, which in turn induces oscillating electric field. This explains how light can travel in vacuum without needing any material medium. Maxwell also showed that the electric and magnetic fields of an electromagnetic wave in vacuum are perpendicular to each other, and they in turn are perpendicular to the direction of the propagation of the wave.

A particularly simple form of electromagnetic wave is a plane harmonic wave in which electric field oscillates along a single axis, and magnetic field in an axis perpendicular to that as shown in Figure 49.1.1. For instance, an electromagnetic wave traveling towards positive \(x\) axis with electric field oscillating along \(y\) axis and magnetic field oscillating \(z\) axis is an example of a plane wave.

We can write the plane electromagnetic wave shown in Figure 49.1.1 analytically by two vector functions, one the electric field and the other for the magnetic field. The two waves have the same angular frequency \(\omega\text{,}\) same wavenumber \(k\) , and same phase constant \(\phi\text{.}\) Let us denote their amplitudes by \(E_0\) and \(B_0\text{,}\) respectively. With unit vectors \(\hat u_y\) towards positive \(y\) axis and \(\hat u_z\) towards positive \(z\) axis, the following functions will describe oscillating electric and magnetic fields propagating towards positive \(x\) axis.

The amplitudes of the waves, viz. \(E_0\) and \(B_0\) are related by the wave speed \(v\) in the medium. We will use symbol \(c\) for the speed in vacuum.

With \(\hat u_x\) as unit vector towards positive \(x\) axis, the direction of the wave is given by the wave propagation vector, \(\vec k = k \hat u_x\text{,}\) which is the product of the wavenumber and the direction unit vector.

The argument of the trigonometric functions on the right of Eqs. (49.1.1) and (49.1.2) is called the phase of the wave. As shown in these formulas, the waves of these fields march with the same phase: the crest of the electric field occurs at the same location and time as the crest of the magnetic field wave and the trough with the trough.

The angular frequency \(\omega\) (radians per second) and wave number \(k\) (radians per meter) are related to the frequency \(f\) (cycles per second or Hertz, Hz) and wavelength \(\lambda\) (meter) respectively.

The ratio \(\omega/k\) is therefore equal to the wavespeed \(v\) in the medium by the following fundamental equation of waves.

Maxwell showed that the speed of electromagnetic wave in the vacuum is simply related to the electrromagnetic properties of vacuume, namely, electric susceptibility \(\epsilon_0\) and the magnetic permeability \(\mu_0\text{.}\)

In other linear media, a similar formula relates speed to electric susceptibility \(\epsilon\) and the magnetic permeability \(\mu\) of the medium.

It turns out that, for nonmagnetic media, \(\mu\approx \mu_0\text{.}\) Therefore,

The ratio \(\frac{\epsilon}{\epsilon_0}\) is called the dieelectric constant of the medium, which is denoted by \(\epsilon_r\text{.}\)

Refractive index \(n\) of the medium is related to the dielectric constant by

Therefore, \(v\) in a medium can also be written as

Since \(\epsilon_0\) and \(\mu_0\) are universal values, speed of light in vacuum, giben by Eq. (49.1.5), has the following exact alue.

Since electric and magnetic fields of an electromagnetic wave are interdependent, it often suffices to examine only the electric field of the wave. You should, however, keep in mind that magnetic field wave is also present.

The points in space with equal phase form a surface called the wavefront. Evidently, wave function has the same value at all points of the wavefront. We see that the wavefronts of the plane waves given in Eqs. (49.1.1) and (49.1.2) are planes perpendicular to the direction of the wave. That is why these waves are also called plane waves.

Let us examine the crest of the waves at \(t=0\text{.}\) The crests will occur when the argument of the cosine is equal to \(2n\pi\) for \(n\) an integer.

Thus, a plane wave moving along \(x\) axis crests simultaneously at all points of planes parallel to the \(yz\) plane separated by one wavelength.

Not all waves are plane waves. For instance, an isotropic light emitted by a point source will spread out as spherical wave fronts centered at the source. These waves are called spherical waves. The traveling spherical waves are given by a wave function whose amplitude drops off inversely with the distance \(r\) from the source. With \(\hat u_r\) a unit vector pointed radially outward from the origin, we get the following for a spherical wave.

where \(V_0\) is a constant amplitude with units of electric potential. Pictorially, a traveling electromagnetic wave of a single frequency can be represented by drawing wavefronts that are a single wavelength apart. Rather than draw a three-dimensional plot it is customary to draw a cross-section. Thus, for a plane wave we draw parallel lines, each line representing a plane wavefront, while for a spherical wave we draw arcs or circles, each representing a spherical wavefront (Figure 49.1.2). Note that the wave fronts are perpendicular to the direction of wave.