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Henceforth, we omit the s dependence in u ν , since the RHS of equation
                           (23) does not depend on any specific direction, as expected from the
                           isotropic nature of thermal radiation.

                        2. The energy density per unit solid angle u ν must follow the well-known
                           Rayleigh–Jeans radiation law:
                                                               2ν 2
                                                         u ν =    kT,                          (24)
                                                               c 3
                           in the classical limit hν  kT. Therefore, we must require in equation
                           (23)
                                                         α m  =  2hν 3  .                      (25)
                                                           n
                                                         β n m   c 3

                      The above discussions thus lead to Planck’s formula of blackbody radiation:

                                                        2hν 3    1
                                                  u ν =               .                        (26)
                                                               hν
                                                         c 3  e kT − 1
                         For the intensity I ν , we obtain
                                                            2hν 3   1
                                                I ν = cu ν =             ,                     (27)
                                                                  hν
                                                             c 2  e kT − 1
                      (see Figure 20). For the energy density U ν , we have

                                                                 8πhν      1
                                              I                       3
                                        U ν =   u ν dΩ = 4πu ν =                .              (28)
                                                                         hν
                                                                   c 3  e kT − 1

                      9.1    Two Extreme Cases of the Planck Spectrum

                                                                           hν
                         • Rayleigh–Jeans region (hν  kT, and hence e kT ' 1 +      hν  ):
                                                                                     kT
                                                              2ν 2
                                                         I ν =    kT.                          (29)
                                                               c 2

                           Note that thermal radiation in the radio frequency range is mostly in
                           the Rayleigh–Jeans region.

                         • Wien region (hν  kT):

                                                             2hν 3   hν
                                                       I ν =      e − kT .                     (30)
                                                              c 2





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