= Solution
{c}
Specify the radiation normalization first. Let $u_\nu$ be <radiation energy density> per ordinary <frequency>, with SI units $\mathrm{J\,m^{-3}\,Hz^{-1}}$. Then $A_{21}$ is the <spontaneous emission> rate per upper-state <chemical atom>, while $B_{12}u_\nu$ and $B_{21}u_\nu$ are the induced <absorption> and <stimulated emission> rates, evaluated at a sharp transition <frequency>. For a finite <spectral line profile>, replace $u_\nu$ by its profile-weighted average.
The <Einstein coefficients> obey
$$
\boxed{g_1B_{12}=g_2B_{21},\qquad
\frac{A_{21}}{B_{21}}=\frac{8\pi h\nu_0^3}{c^3}.}
$$
Here $g_i$ are the <statistical weights> of the two <energy levels>, and the field is unpolarized and isotropic. The units and <physical dimensions> are
$$
[A]=\mathrm{s^{-1}},\quad [A]=\mathsf T^{-1};\qquad
[B]=\mathrm{m^3\,J^{-1}\,s^{-2}},\quad [B]=\mathsf L\mathsf M^{-1}.
$$
The apparently time-independent dimension of $B$ follows because a per-<hertz> density includes an extra unit of time. If instead $B$ multiplies the <mean intensity> $J_\nu$, use $B^{(J)}=(4\pi/c)B$ and $A/B^{(J)}=2h\nu_0^3/c^2$. These distinct normalizations must not be mixed in the <radiative transfer equation>.
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