Question

An electron of mass $m$ and a photon have the same energy $E$. The ratio of wavelength of electron to that of photon is -

$[c=$ speed of light$]$

• A. ${\left(\frac{E}{2m}\right)}^{\frac{1}{2}}$
• B. $\frac{1}{c}{\left(\frac{E}{2m}\right)}^{\frac{1}{2}}$
• C. $c{\left(2mE\right)}^{\frac{1}{2}}$
  
• D. $\frac{1}{c}{\left(\frac{2m}{E}\right)}^{\frac{1}{2}}$

Answer 1

The correct option is B $\frac{1}{c}{\left(\frac{E}{2m}\right)}^{\frac{1}{2}}$

For electron,
${\lambda }_e=\frac{h}{\sqrt{2mE}}...\left(1\right)$

For photon,

${\lambda }_p=\frac{hc}{E}...\left(2\right)$

So,
$\frac{{\lambda }_e}{{\lambda }_p}=\frac{\frac{h}{\sqrt{2mE}}}{\frac{hc}{E}}\left[\text{From}\right(1\left)\text{and}\right(2\left)\right]$

$\Rightarrow \frac{{\lambda }_e}{{\lambda }_p}=\sqrt{\frac{E}{2m{c}^2}}=\frac{1}{c}{\left(\frac{E}{2m}\right)}^{\frac{1}{2}}$