GATE 2025 CY – Question 60
The kinetic energies of an electron ($e$) and a proton ($p$) are $E$ and $3E$, respectively. Given that mass of a proton is 1836 times that of an electron, the ratio of their de Broglie wavelengths $(\lambda_e/\lambda_p)$ is _____ (rounded off to two decimal places).
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Correct answer: 73.85 to 74.59
Explanation
The **de Broglie wavelength** of a particle with kinetic energy $E$ is
$$\lambda=\frac{h}{p}=\frac{h}{\sqrt{2mE}} .$$
**Ratio** for the electron (mass $m_e$, energy $E$) and the proton (mass $m_p=1836\,m_e$, energy $3E$):
$$\frac{\lambda_e}{\lambda_p}=\sqrt{\frac{m_p\,E_p}{m_e\,E_e}}=\sqrt{\frac{1836\,m_e\times3E}{m_e\times E}}=\sqrt{5508}=\mathbf{74.22}.$$