Question

A photon and an electron have the same energy \( E \). If \( \lambda_p \) is the wavelength of the photon and \( \lambda_e \) is the de Broglie wavelength of the electron, then the ratio \( \frac{\lambda_p}{\lambda_e} \) is:

• A. \( \frac{E}{mc^2} \)
• B. \( \frac{\sqrt{2mE}}{c^2} \)
• C. \( \frac{\sqrt{2mE}}{c} \)
• D. \( \frac{\sqrt{2mE}}{R} \)

Hint:

Use the relations \( E = \frac{hc}{\lambda_p} \) for photons and \( \lambda_e = \frac{h}{\sqrt{2mE}} \) for electrons to find the wavelength ratio.

Answer 1

The Correct Option is C

For a photon, the energy is related to the wavelength by: \[ E = \frac{hc}{\lambda_p} \] Where: - \( h \) is Planck's constant, - \( c \) is the speed of light, - \( \lambda_p \) is the wavelength of the photon. For an electron, the de Broglie wavelength is given by: \[ \lambda_e = \frac{h}{\sqrt{2mE}} \] Where: - \( m \) is the mass of the electron, - \( E \) is the energy of the electron. Now, the ratio of the wavelengths is: \[ \frac{\lambda_p}{\lambda_e} = \frac{\frac{hc}{E}}{\frac{h}{\sqrt{2mE}}} = \frac{c}{\sqrt{2mE}} \] Thus, the ratio is: \[ \frac{\lambda_p}{\lambda_e} = \frac{\sqrt{2mE}}{c} \]