The de Broglie wavelength
Question:

The de Broglie wavelength $(\lambda)$ associated with a photoelectron varies with the frequency $(v)$ of the incident radiation as, $\left[v_{0}\right.$ is threshold frequency $]$ :

1. $\lambda \propto \frac{1}{\left(v-v_{0}\right)}$

2. $\lambda \propto \frac{1}{\left(v-v_{0}\right) \frac{1}{4}}$

3. $\lambda \propto \frac{1}{\left(v-v_{0}\right)^{\frac{3}{2}}}$

4. $\lambda \propto \frac{1}{\left(v-v_{0}\right)^{\frac{1}{2}}}$

Correct Option: , 4

Solution:

According to de-Broglie wavelength equation,

$\lambda=\frac{h}{m v} \Rightarrow \lambda \propto \frac{1}{v}$

According to photoelectric effect,

$\mathrm{h} v-\mathrm{h} v_{0}=\frac{1}{2} \mathrm{mv}^{2} ; v-v_{0}=\frac{1}{2} \frac{\mathrm{mv}^{2}}{\mathrm{~h}}$

$v-v_{0} \propto v^{2}$

$v \propto\left(v-v_{0}\right)^{1 / 2}$

$\therefore \lambda \propto \frac{1}{\left(v-v_{0}\right)^{1 / 2}}$