9.1: Introduction to Plane Waves
( \newcommand{\kernel}{\mathrm{null}\,}\)
An electric dipole directed along z, located at the origin, and oscillating with the circular frequency ω produces electric and magnetic fields far from the origin that have the form (see equations (7.4.5)):
Eθ=−ω24πϵ0p0sinθc2Rexp(−iω[t−R/c]),Bϕ=cEθHϕ=−ω24πp0sinθcRexp(−iω[t−R/c])
where pz=p0exp(−iω[t−R/c]), and t is the time at which the observer at →R measures the fields. It must always be kept in mind that the fields are represented by real numbers; the notation of complex numbers is simply a convenient book-keeping device for dealing with sinusoidal functions. The notation exp(−iωt) “the real part of exp(−iωt)” i.e. cos(ωt). It is particularly important to remember this when calculating the Poynting vector or the energy densities which involve the product of two field amplitudes. For example, the Poynting vector corresponding to the fields of Equations (9.1.1) is given by
Sr=EθHϕ=14πϵ0ω44πp20sin2θc3R2cos2(ω[t−R/c])
Note that the time factor is not the same as
Real (exp(−2iω[t−R/c]))=cos(2ω[t−R/c]).
The time average of Equation (???) is zero, whereas the time average of the correct expression, Equation (???), is given by
<Sr>=(18π)(c4πϵ0)(ωc)4p20sin2θR2,
since the time average of the cosine squared function is 1/2. At distances far removed from the dipole radiator the surface of constant R can be approximated locally by a plane perpendicular to ˆur, a unit vector parallel with →R. This suggests that Maxwell’s equations ought to have plane wave solutions of the form
→E(→r,t)=→E0exp(i[→k⋅→r−ωt]),
→B(→r,t)=→B0exp(i[→k⋅→r−ωt]),
where →k is a vector whose magnitude is ω/c and whose direction lies along the direction of propagation of the wave, and where →E0 and →B0 are constant vectors that are perpendicular to each other and to the wave-vector →k (see Figure (9.1.1)).
Equations (???) can be written in component form using some convenient co-ordinate system, and using Real(exp(i[→k⋅→r−ωt]))=cos(→k⋅→r−ωt):
Ex=E0xcos(kxx+kyy+kzz−ωt),Ey=E0ycos(kxx+kyy+kzz−ωt),Ez=E0zcos(kxx+kyy+kzz−ωt),Bx=B0xcos(kxx+kyy+kzz−ωt),By=B0ycos(kxx+kyy+kzz−ωt),Bz=B0zcos(kxx+kyy+kzz−ωt),
Using these expressions it is easy to show that
curl(→E)=−(→k×→E0)sin(→k⋅→r−ωt),
div(→E)=−(→k⋅→E0)sin(→k⋅→r−ωt),curl(→B)=−(→k×→B0)sin(→k⋅→r−ωt),div(→B)=−(→k⋅→B0)sin(→k⋅→r−ωt),
In free space Maxwell’s equations become
curl(→E)=−∂→B∂t,curl(→B)=ϵ0μ0∂→E∂t,div(→E)=0,div(→B)=0.
Substitution of Equations (???) into Maxwell’s Equations (9.1.11) gives
→k×→E0=ω→B0,
→k×→B0=−ϵ0μ0ω→E0,
→k⋅→E0=0,
→k⋅→B0=0.
The last two equations state that for plane wave solutions of Maxwell’s equations in free space both the electric and magnetic field vectors must be perpendicular to the direction of propagation specified by the vector →k; i.e. →E0 and →B0 must be parallel with the surfaces of constant phase. The first two equations of (9.1.8) state that the fields →E0 and →B0 must be mutually perpendicular; thus the three vectors →E0, →B0, and →k form an orthogonal right handed triad. In order to satisfy Maxwell’s equations the magnitude of the wave-vector must be given by
k2=ϵ0μ0ω2=(ωc)2,
and the amplitudes of the electric and magnetic fields must be related by
|→E0|=c|→B0|,
see Figure (9.1.2). Notice that E and B oscillate in phase: ie. they have exactly the same sinusoidal dependence on space and on time. These relations are the same as those which were earlier associated with the wave produced by an oscillating dipole, Equations (7.4.5).
In free space the displacement vector, →D, is related to the electric field by →D=ϵ0→E so that the time rate of change of the energy density stored in the electric field, Equation (8.2.6), becomes
∂WE∂t=ϵ0→E⋅∂→E∂t=∂∂t(ϵ0E22).
Using (9.1.10), the energy density stored in the electric field of a plane wave is given by
WE=ϵ0E202cos2(→k⋅→r−ωt), Joules/m3,
This energy density oscillates in both space and time, in particular at a fixed point in space the energy density periodically vanishes. However, the average energy density measured at any point in space is independent of both position and time:
<WE>=ϵ04E20, Joules/m3,
Similarly, the time rate of change of the energy density stored in the magnetic field is given by (8.7)
∂WB∂t=→H⋅∂→B∂t=∂∂t(B22μ0).
Therefore one can write
WB=B22μ0=B202μ0cos2(→k⋅→r−ωt) Joules/m3.
The time averaged energy density stored in the magnetic field is independent of position and since B=E/c is given by
<WB>=B204μ0=E204μ0c2=ϵ0E204=<WE> Joules/m3.
The average energy density stored in the magnetic field is exactly the same, in free space, as the average energy density stored in the electric field. The total time averaged energy density stored in the electromagnetic field is
<W>=<WE>+<WB>=ϵ0E202, Joules/m3.
The average rate at which energy in the electromagnetic field is transported across a unit area normal to the direction of propagation, i.e. normal to →k, can be obtained by multiplying Equation (???) by the speed of light: this rate is also just the time average of the Poynting vector
<S>=cϵ0E202=12√ϵ0μ0E20, Watts /m2.
The quantity Z0=√μ0/ϵ0 has the units of a resistance; it is called the impedance of free space, and Z0=377 Ohms. From the equations for the space and time variation of a plane wave, Equations (???), it follows that for a fixed time the electric and magnetic fields vary in space with a period along the direction of veck given by 2π/|→k|. By definition, this spatial period is the wavelength, λ, therefore |→k|=2π/λ. Similarly, at a fixed position in space the fields oscillate in time with the period 2π/ω; by definition, this period, T, is the inverse of the frequency,f, therefore ω=2πf. In order to satisfy Maxwell’s equations, the frequency and wavelength of a plane wave are related by Equation (???)
ω=c|→k|;
this can be written in the more familiar form fλ=c.