Derivation of Einstein Coefficient Relation→
Let us consider the $N_{1}$ and $N_{2}$ is the mean population of lower energy state and upper energy state respectively. If the energy density of incident light is $\rho(\nu)$ then

The rate of transition of number of atoms due to absorption process:

$R_{abs}=B_{12} \: \rho(v) \: N_{1} \qquad(1)$

The above equation shows the number of atoms absorbing the photon per second per unit volume

Where $B_{12}$= Einstein Absorption Coefficent

The rate of transition of number of atoms due to sponteneous emission process:

$R_{sp}=A_{21} \: N_{2} \qquad(2)$

The above equation shows the number of atoms emitting the photon per second per unit volume due to spontaneous emission

Where $A_{21}$= Einstein Spontaneous Emission Coefficient

The rate of transition of the number of atoms due to stimulated emission process:

$R_{st}=B_{21} \: \rho(v) \: N_{2} \qquad(3)$

The above equation shows the number of atoms emitting the photon per second per unit volume due to stimulated emission

Where $B_{21}$= Einstein Stimulated Emission Coefficient

Under the thermal equilibrium, the mean population $N_{1}$ and $N_{2}$ in lower and upper energy states respectively must remain constant. This condition requires that the transition of the number of atoms from $E_{2}$ to $E_{1}$ must be equal to the transition of the number of atoms from $E_{1}$ to $E_{2}$. Thus

$\left.\begin{matrix}The \: number \: of \: atoms \: absorbing \\ photons \: per \: second \: per \: unit \: volume
\end{matrix}\right\} \\ = \left.\begin{matrix} The \: number \: of \: atoms \: emitting \\ photons \: per \: second \: per \: unit \: volume
\end{matrix}\right\}$

i.e $R_{abs}= R_{sp}+R_{st}$

$B_{12} \: \rho(v) \: N_{1}= A_{21} \: N_{2} + B_{21} \: \rho(v) \: N_{2}$

$B_{12} \: \rho(v) \: N_{1} - B_{21} \: \rho(v) \: N_{2} = A_{21} \: N_{2} $

$ \rho(v) (B_{12} \: N_{1} - B_{21} \: N_{2} ) = A_{21} \: N_{2} $

$\rho(v)=\frac{A_{21} \: N_{2}}{(B_{12} \: N_{1} - B_{21} \: N_{2} )} \qquad(4)$

We know that

$\frac{N_{1}}{N_{2}}=e^{\frac{(E_{2}-E_{1})}{kT}}$

$\frac{N_{1}}{N_{2}}=e^{\frac{h\nu}{kT}}$

Now substitute the value of $\frac{N_{1}}{N_{2}}$ in equation $(4)$

$\rho(v)=\frac{A_{21}}{B_{12}} \left [ \frac{1}{e^{\frac{h\nu}{kT}}- \frac{B_{21}}{B_{12}}} \right ] \qquad(5)$

According to Planck's Radiation Law

$\rho(v)=\frac{8\pi h \nu^{3}}{c^{3}} \left [ \frac{1}{e^{\frac{h\nu}{kT}}- 1} \right ] \qquad(6)$

Now comparing the equation $(5)$ and equation $(6)$

$\frac{B_{21}}{B_{12}}=1$ and $\frac{A_{21}}{B_{12}}=\frac{8\pi h \nu^{3}}{c^{3}}$

From the above equation, we get

$B_{21}=B_{12}$

$B_{12}=B_{21}=\frac{c^{3}}{8\pi h \nu^{3}}A_{21}$