Question

If a photon and an electron have equal energy $E$, then $\frac{{\lambda }_{\text{photon}}}{{\lambda }_{\text{electron}}}$ is proportional to,

• A. $\sqrt{E}$
• B. $\frac{1}{\sqrt{E}}$
• C. $\frac{1}{E}$
• D. Does not depend upon $E$.

Answer 1

The correct option is B $\frac{1}{\sqrt{E}}$

$\text{Energy of photon}E=\frac{hc}{{\lambda }_{photon}}$
$\text{Energy of electron}E=\frac{1}{2}m{v}^2$

Now, $E=\frac{p^2}{2m}\Rightarrow \sqrt{2mE}=p$

The de-Broglie wavelength is,
${\lambda }_e=\frac{h}{p}=\frac{h}{\sqrt{2mE}}$

So,$\frac{{\lambda }_p}{{\lambda }_e}=\frac{\frac{hc}{E}}{\frac{h}{\sqrt{2mE}}}=c\sqrt{\frac{2m}{E}}$

$\therefore \frac{{\lambda }_p}{{\lambda }_e}\propto \frac{1}{\sqrt{E}}$

Hence, $\left(B\right)$ is the correct answer.