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  4. The kinetic energy of a particle is equal to the energy of a photon. The particle moves at 5 % of the speed of light. The ratio of the photon wavelength to the de Broglie wavelength of the particle is [No need to use relativistic formula for the particle.]

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Q. The kinetic energy of a particle is equal to the energy of a photon. The particle moves at $5 \%$ of the speed of light. The ratio of the photon wavelength to the de Broglie wavelength of the particle is [No need to use relativistic formula for the particle.]

Dual Nature of Radiation and Matter

Solution:

Let $m$ be the mass of particle.
$\frac{m v^{2}}{2}=\frac{h c}{\lambda_{\text {photon }}}$,
where symbols have their usual meanings.
$\frac{p^{2}}{2 m}=\frac{h c}{\lambda_{\text {photon }}}$
and $p=\frac{h}{\lambda_{\text {particle }}}$
$\Rightarrow \frac{h^{2}}{2 m \lambda_{\text {particle }}^{2}}=\frac{h c}{\lambda_{\text {photon }}}$
$\Rightarrow \frac{\lambda_{\text {photon }}}{\lambda_{\text {particle }}}=\frac{2 m c}{h} \times \lambda_{\text {particle }}$
$=\frac{2 m c}{h} \times \frac{h}{m v}$
$=\frac{2 c}{0.05 c}=40$