The de Broglie wavelengths for an electron and a photon are λ _ e...

The de Broglie wavelengths for an electron and a photon are

{\lambda}_{e}

and

{\lambda}_{p}

respectively. For the same kinetic energy of electron and photon, which of the following presents the correct relation between the de Broglie wavelengths of two?

Options

A ${\lambda}_{p} ∝ {\lambda}_{e}^{2}$
B ${\lambda}_{p} ∝ {\lambda}_{e}$
C ${\lambda}_{p} ∝ \sqrt{{\lambda}_{e}}$
D ${\lambda}_{p} ∝ \sqrt{\dfrac{1}{{\lambda}_{e}}}$

Correct answer

A. ${\lambda}_{p} ∝ {\lambda}_{e}^{2}$

Step-by-step solution

De-Broglie wavelength for electron is given by, ${\lambda}_{e} = \dfrac{h}{\sqrt{2mk}}$

For photon $k = \dfrac{hc}{{\lambda}_{p}} ⇒ {\lambda}_{p} = \dfrac{hc}{k}$ . Therefore,

${\lambda}_{e} = \dfrac{h\sqrt{{\lambda}_{p}}}{\sqrt{2mhc}}$

${\lambda}_{p} ∝ {{\lambda}_{e}}^{2}$

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