A peak emission from a black body at a certain temperature occurs at a wavelength of $9000\stackrel{°}{A}$. On increasing the temperature, the total radiation emitted is increased $81$ times. At the initial temperature when the peak radiation from the black body is incident on a metal surface, it does not cause any photoemission from the surface. After the increase of temperature the peak radiation from the black body caused photoemission. To bring these photoelectrons to rest, a potential equivalent to the excitation energy between then $n=2$ and $n=3$ Bohr levels of hydrogen is required. Find the work function of the metal.

Let $T$ be the initial temperature of the black body. The total energy emitted by the body per unit area per second is given by

$E=\sigma T^4$

Where $\sigma$ is the Stefan’s constant. When the temperature of the black body is raised to $T\text{'}$, we have

$E\text{'}=\sigma T{\text{'}}^4$

Now, $\frac{E\text{'}}{E}=81$  (given).

Hence, $81={\left(\frac{T\text{'}}{T}\right)}^4$

Which gives $T\text{'}=3T$

It is given that peak emission at temperature $T$ occurs at a wavelength ${\lambda }_m=9000\stackrel{°}{A}$. If $\lambda {\text{'}}_m$ is the wavelength for peak emission at temperature $T\text{'}$, then from Wien’s displacement law ${\lambda }_{m}T=cons\tan t$, we have 

$\lambda {\text{'}}_{m}T\text{'}={\lambda }_{m}T$

or $\lambda {\text{'}}_m={\lambda }_m\frac{T}{T\text{'}}=9000\stackrel{°}{A}\times \frac{T}{3T}=3000\stackrel{°}{A}$

According to the problem, the kinetic energy of the emitted photo-electrons $=$ excitation energy for transition $n=2$ to $n=3$, which is

$K{E}_{max}=E_{2\to 3}=\left(13.6eV\right)\left(\frac{1}{2^2}-\frac{1}{3^2}\right)=13.6\times \frac{5}{36}=1.89eV$

The energy of the photon of wavelength $\lambda \text{'}=3000\stackrel{°}{A}$ in $eV$ is

$E=\frac{12400}{3000}eV=4.13eV$

Now, from Einstein’s photoelectric equation, the work function for the metal surface is given as 

$$\begin{gathered} \varphi =E-K{E}_{max}=\left(4.13-1.89\right)eV \\ =2.24eV \end{gathered}$$