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2.4: Classical Light-Waves

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Consider a classical, monochromatic, linearly-polarized, plane light-wave, propagating through a vacuum in the x-direction. It is convenient to characterize a light-wave (which is, of course, a type of electromagnetic wave) by specifying its associated electric field. Suppose that the wave is polarized such that this electric field oscillates in the y-direction. (According to standard electromagnetic theory, the magnetic field oscillates in the z-direction, in phase with the electric field, with an amplitude which is that of the electric field divided by the velocity of light in vacuum. ) Now, the electric field can be conveniently represented in terms of a complex wavefunction:

ψ(x,t)=ˉψei(kxωt).

Here, i=1, k and ω are real parameters, and ˉψ is a complex wave amplitude. By convention, the physical electric field is the real part of the previous expression. Suppose that

ˉψ=|ˉψ|eiφ,

where φ is real. It follows that the physical electric field takes the form

Ey(x,t)=Re[ψ(x,t)]=|ˉψ|cos(kxωt+φ),

where |ˉψ| is the amplitude of the electric oscillation, k the wavenumber, ω the angular frequency, and φ the phase angle. In addition, λ=2π/k is the wavelength, and ν=ω/2π the frequency (in hertz).

According to standard electromagnetic theory , the frequency and wavelength of light-waves are related according to the well-known expression

c=νλ,

or, equivalently,

ω=kc,

where c=3×108m/s is the velocity of light in vacuum. Equations (2.4.3) and (2.4.5) yield

Ey(x,t)=|ˉψ|cos(k[x(ω/k)t]+φ)=|ˉψ|cos(k[xct]+φ).

Note that Ey depends on x and t only via the combination xct. It follows that the wave maxima and minima satisfy xct=constant.

Thus, the wave maxima and minima propagate in the x-direction at the fixed velocity

dxdt=c.

An expression, such as Equation (2.4.5), that determines the wave angular frequency as a function of the wavenumber, is generally termed a dispersion relation. As we have already seen, and as is apparent from Equation (2.4.6), the maxima and minima of a plane-wave propagate at the characteristic velocity

vp=ωk,

which is known as the phase-velocity. Hence, the dispersion relation (2.4.5) is effectively saying that the phase-velocity of a plane light-wave, propagating through a vacuum, always takes the fixed value c, irrespective of its wavelength or frequency.

From standard electromagnetic theory , the energy density (i.e., the energy per unit volume) of a plane light-wave is

U=E2yϵ0,

where ϵ0=8.85×1012F/m is the electrical permittivity of free space. Hence, it follows from Equations (2.4.1) and (2.4.3) that

U|ψ|2.

Furthermore, a light-wave possesses linear momentum, as well as energy. This momentum is directed along the wave’s direction of propagation, and is of density

G=Uc.

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 2.4: Classical Light-Waves is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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