Chapter 8

Inhomogeneous Solutions

We have shown that, in vacuum,Maxwell’s equations reduce to the wave

equation

∇2E =µoεo∂2E

∂t2(8.1)

which needs to be solved subject to

∇ · E = 0 (8.2)

Such a solution isE = Eo cos(k · r− ωt) (8.3)

where Eo · k = 0; substitution into the wave equation gives

−k2E = −µoεoω2E (8.4)

ork2 = µoεoω

2 (8.5)

which determines the magnitude of k (that is, the wavelength, since k =2π/λ). This is called the ’dispersion relation’, relating the wave vector to thefrequecy. From this, we can get the velocity;

v =k

ω=

1√εoµo

= c (8.6)

Once E is known, the other fields can be easily obtained. For example,

∇× E = −∂B∂t

(8.7)

43

44 CHAPTER 8. INHOMOGENEOUS SOLUTIONS

gives

−k× Eo sin(k · r− ωt)= −µo

∂H

∂t(8.8)

or

H =1

Zok̂×E (8.9)

If we choose the z-axis to be along k and the x-axis along Eo, the solutionbecomes

E =Eox̂ cos(kz − ωt) (8.10)

withk2 = µoεoω

2 (8.11)

and

v =k

ω=

1√εoµo

= c (8.12)

In addition to this simple plane wave solution, consider a solution of theform

E =Eoe−αyx̂ cos(kz − ωt) (8.13)

then

∇2E =µoεoω2∂

2E

∂t2(8.14)

gives(α2 − k2)E = −µoεoω

2E (8.15)

ork2 − α2 = µoεoω

2 (8.16)

Now we see that, for a given frequency, a range of k values are allowed (anddifferent wavelengths can occur) depending on α. These solutions, whichvary exponentially in the direction perpendicular to k, are called inhomo-geneous solutions. Note that, if α �= 0, the wavelength is shorter than in aplane wave (α = 0) at the same frequency. These solutions occur in nature,they are usually called ’evanescent waves’, and we will encounter them againwhen we discuss total internal reflection.