Question 75: Dual Nature of Matter and Radiation

Question

A proton of mass ' $m_P$ ' has same energy as that of a photon of wavelength ' $\lambda$ '. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is.

Accepted Answer

To find the ratio of the de Broglie wavelength of a proton to the wavelength of a photon when they have the same energy, we follow these steps:
Energy Equivalence
:
Both the proton and the photon have the same energy, denoted by
E
.
Energy of the Photon
:
The energy of a photon with wavelength
ÃÂ»
is given by:
$    E = \frac{hc}{\lambda}    $
Kinetic Energy of the Proton
:
Since the proton moves at a non-relativistic speed, its kinetic energy can be expressed as:
$    	ext{KE}_{	ext{proton}} = \frac{1}{2}m_P v^2 = E    $
Momentum of the Proton
:
The momentum
p
of the proton is derived from its kinetic energy:
$    p = \sqrt{2m_P E}    $
de Broglie Wavelength of the Proton
:
The de Broglie wavelength
ÃÂ»_{proton}
is:
$    \lambda_{	ext{proton}} = \frac{h}{p} = \frac{h}{\sqrt{2m_P E}}    $
Ratio of Wavelengths
:
The ratio of the de Broglie wavelength of the proton to the wavelength of the photon is:
$    \frac{\lambda_{	ext{proton}}}{\lambda_{	ext{photon}}} = \left( \frac{h}{\sqrt{2m_P E}} ight) 	imes \left( \frac{E}{hc} ight)    $
Simplifying, we get:
$    \frac{\lambda_{	ext{proton}}}{\lambda_{	ext{photon}}} = \frac{1}{c} \sqrt{\frac{E}{2m_P}}    $
Thus, the ratio of the de Broglie wavelength of the proton to the wavelength of the photon is $\frac{1}{c} \sqrt{\frac{E}{2m_P}}$.

Answer

$\frac{1}{c} \sqrt{\frac{E}{m_p}}$

Answer

$\frac{1}{\mathrm{c}} \sqrt{\frac{2 \mathrm{E}}{\mathrm{m}_{\mathrm{p}}}}$

Answer

$\frac{1}{\mathrm{2c}} \sqrt{\frac{ \mathrm{E}}{\mathrm{m}_{\mathrm{p}}}}$

Answer

$\frac{1}{\mathrm{c}} \sqrt{\frac{\mathrm{E}}{2 \mathrm{~m}_{\mathrm{p}}}}$

Dual Nature of Matter and Radiation

A proton of mass ' m_P ' has same energy as that of a photon of wavelength ' \lambda '. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is.

Choose the correct answer: