28.

Let us now consider briefly the phenomenon of scattering at thermodynamic equilibrium. Every ray meeting the volume-element $v$ suffers there, apart from absorption, a certain weakening of its intensity because a certain fraction of its energy is diverted in different directions. The value of the total energy of scattered radiation received and diverted, in the time $dt$ by the volume-element $v$ in all directions, may be calculated from expression [eqn:(3)] (3) in exactly the same way as the value of the absorbed energy was calculated in [sect:26.] Sec. 26. Hence we get an expression similar to [eqn:(25)] (25), namely, $dtv8\pi {\int }_0^{\infty }{\beta }_{\nu }{𝖪}_{\nu }d\nu .\mathrm{\backslash Label}[eqn](28)\text{upshape (28)}$ The question as to what becomes of this energy is readily answered. On account of the isotropy of the medium, the energy scattered in $v$ and given by [eqn:(28)] (28) is radiated uniformly in all directions just as in the case of the energy entering $v$. Hence that part of the scattered energy received in $v$ which is radiated out in a cone of solid angle $d\Omega$ is obtained by multiplying the last expression by ${\displaystyle \frac{d\Omega }{4\pi }}$. This gives $2dtvd\Omega {\int }_0^{\infty }{\beta }_{\nu }{𝖪}_{\nu }d\nu ,$ and, for monochromatic plane polarized radiation, $dtvd\Omega {\beta }_{\nu }{𝖪}_{\nu }d\nu .\mathrm{\backslash Label}[eqn](29)\text{upshape (29)}$

Here it must be carefully kept in mind that this uniformity of radiation in all directions holds only for all rays striking the element $v$ taken together; a single ray, even in an isotropic medium, is scattered in different directions with different intensities and different directions of polarization. (See end of [sect:8.] Sec. 8.)

It is thus found that, when thermodynamic equilibrium of radiation exists inside of the medium, the process of scattering produces, on the whole, no effect. The radiation falling on a volume-element from all sides and scattered from it in all directions behaves exactly as if it had passed directly through the volume-element without the least modification. Every ray loses by scattering just as much energy as it regains by the scattering of other rays.