The wavelength $\lambda_e$ of an electron and $\lambda_p$ of a photon of same energy $E$ are related by

The wavelength $\lambda_e$ of an electron and $\lambda_p$ of a photon of same energy $E$ are related by
  1. $\lambda_p \propto \lambda_e^2$
  2. $\lambda_p \propto \lambda_e$
  3. $\lambda_p \propto \sqrt{\lambda_e}$
  4. $\lambda_\rho \propto \frac{1}{\sqrt{\lambda_e}}$

Solution

Wavelength of electron, $\begin{aligned} & \lambda_e=\frac{h}{\sqrt{2 m E}} \\ & \text { and proton } \lambda_p=\frac{h c}{E} \\ & \Rightarrow \quad \lambda_e^2=\frac{h^2}{2 m E} \text { or } E=\frac{h c}{\lambda_p} \\ & \therefore \quad \lambda_e^2=\frac{h^2}{2 m \cdot \frac{h c}{\lambda_p}} \Rightarrow \lambda_e^2=\frac{h^2}{2 m h C} \lambda_p \\ & \Rightarrow \quad \lambda_e^2 \propto \lambda_p \end{aligned}$

Asked in: NEET 2013 (All India)

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