(631) |

ensures that the wave propagates at the speed of light. Note that this dispersion relation is very similar to that of sound waves in solids [see Eq. (508)]. Electromagnetic waves always propagate in the direction perpendicular to the coupled electric and magnetic fields (

Consider an enclosure whose walls are maintained at fixed
temperature . What is the nature of the
steady-state electromagnetic radiation inside
the enclosure? Suppose that the enclosure is a parallelepiped with
sides of lengths , , and . Alternatively, suppose that the
radiation field inside the enclosure is periodic in the -, -, and -directions,
with periodicity lengths , , and , respectively. As long as the
smallest of these lengths, , say, is much greater than the longest wavelength
of interest in the problem,
, then these assumptions should not
significantly
affect the nature of the radiation inside the enclosure. We find, just as in
our earlier discussion of sound waves (see Sect. 7.12), that the periodicity constraints ensure that
there are only a discrete set of allowed wave-vectors (*i.e.*, a discrete
set of allowed modes of oscillation of the electromagnetic field inside the
enclosure).
Let
be the number of allowed modes *per unit volume*
with wave-vectors in the range to
. We know,
by analogy with
Eq. (514), that

(633) |

(634) |

(635) |

Let us consider the situation classically. By analogy with sound waves, we can treat
each allowable mode of oscillation of the electromagnetic field as an independent
harmonic oscillator. According to the equipartition theorem
(see Sect. 7.8), each mode possesses a
mean energy in thermal equilibrium at temperature . In fact,
resides with the oscillating electric field, and another with
the oscillating magnetic field.
Thus,
the classical *energy density* of electromagnetic radiation (*i.e.*, the
energy per unit volume associated with modes whose frequencies lie in the
range to
) is

(636) |

According to Debye theory (see Sect. 7.12),
the energy density of sound waves in a solid is
analogous to the Rayleigh-Jeans law, with one very important difference.
In Debye theory
there is a cut-off frequency (the Debye frequency) above which
no modes exist. This cut-off comes about because of the discrete nature of
solids (*i.e.*, because solids
are made up of atoms instead of being continuous).
It is, of course,
impossible to have sound waves whose wavelengths are much less than
the inter-atomic spacing.
On the other hand, electromagnetic waves propagate through a vacuum,
which possesses no discrete structure. It follows that
there is no cut-off frequency for electromagnetic waves, and so the Rayleigh-Jeans
law holds for all frequencies. This immediately poses a severe problem. The total
classical energy density of electromagnetic radiation is given by

(637) |

So, how do we obtain a sensible answer? Well, as usual, quantum mechanics comes
to our rescue. According to quantum mechanics,
each allowable mode of oscillation of the
electromagnetic field corresponds to a *photon state* with energy and
momentum

(638) | |||

(639) |

respectively. Incidentally, it follows from Eq. (632) that

(640) |

(641) |

(642) |

(643) |

(644) |

According to the above discussion, the true energy density of
electromagnetic radiation inside an enclosed cavity is written

(645) |

(646) |