The ratio of the wavelength of a photon of energy ' $E$ ' to that of the electron of same energy is (…

The ratio of the wavelength of a photon of energy ' $E$ ' to that of the electron of same energy is ( $\mathrm{m}=$ mass of an electron, $\mathrm{c}=$ speed of light, $\mathrm{h}=$ Planck's constant)
  1. $\sqrt{\frac{\mathrm{m}}{\mathrm{cE}}}$
  2. $\sqrt{\frac{2 \mathrm{~m}}{\mathrm{cE}}}$
  3. $\mathrm{c} \sqrt{\frac{\mathrm{m}}{\mathrm{E}}}$
  4. $\mathrm{c} \sqrt{\frac{2 \mathrm{~m}}{\mathrm{E}}}$

Solution

We know, Energy of a photon $E=\frac{h c}{\lambda_p}$ $\therefore \quad \lambda_{\mathrm{p}}=\frac{\mathrm{hc}}{\mathrm{E}}$
Wavelength of an electron $\lambda_e=\frac{h}{m v}$ We also know, $E=\frac{1}{2} m v^2 \Rightarrow v=\sqrt{\frac{2 E}{m}}$ $\begin{array}{ll} \therefore \quad & m v=\sqrt{2 E m} \\ \therefore \quad & \lambda_e=\frac{h}{\sqrt{2 E m}} \\ & \Rightarrow \frac{\lambda_p}{\lambda_e}=\frac{h c}{E} \times \frac{\sqrt{2 E m}}{h}=c \sqrt{\frac{2 m}{E}} \end{array}$ .

Asked in: MHT CET 2024 (03 May Shift 1)

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