Question 318: Dual Nature of Matter and Radiation

Question

The ratio of the wavelength of a photon of energy ' $E$ ' to that of the electron of same energy is ( $\mathrm{m}=$ mass of an electron, $\mathrm{c}=$ speed of light, $\mathrm{h}=$ Planck's constant)

Accepted Answer

To determine the ratio of the wavelength of a photon to that of an electron, both having the same energy $E$, we'll use the following expressions:
Wavelength of a photon:
The energy $E$ of a photon can be expressed using the equation:
$$ E = \frac{hc}{\lambda_{	ext{photon}}} $$
where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda_{	ext{photon}}$ is the wavelength of the photon. Solving for the wavelength:
$$ \lambda_{	ext{photon}} = \frac{hc}{E} $$
Wavelength of an electron:
Using de Broglie's wavelength formula, the wavelength $\lambda_{	ext{electron}}$ of an electron is:
$$ \lambda_{	ext{electron}} = \frac{h}{p} $$
The energy $E$ of the electron is given by:
$$ E = \frac{p^2}{2m} $$
Solving for $p$, we get:
$$ p = \sqrt{2mE} $$
Substituting $p$ into the expression for the wavelength:
$$ \lambda_{	ext{electron}} = \frac{h}{\sqrt{2mE}} $$
Ratio of the wavelengths:
The ratio of the wavelength of the photon to that of the electron is:
$$ \frac{\lambda_{	ext{photon}}}{\lambda_{	ext{electron}}} = \frac{\frac{hc}{E}}{\frac{h}{\sqrt{2mE}}} $$
Simplifying this expression:
$$ \frac{\lambda_{	ext{photon}}}{\lambda_{	ext{electron}}} = \frac{hc}{E} 	imes \frac{\sqrt{2mE}}{h} $$
Cancel out $h$:
$$ = \frac{c}{E} 	imes \sqrt{2mE} $$
Simplify further:
$$ = c \sqrt{\frac{2m}{E}} $$
Hence, the correct option is
Option D
: $c \sqrt{\frac{2m}{E}}$.

Answer

$\sqrt{\frac{\mathrm{m}}{\mathrm{cE}}}$

Answer

$\sqrt{\frac{2 \mathrm{~m}}{\mathrm{cE}}}$

Answer

$c \sqrt{\frac{\mathrm{~m}}{\mathrm{E}}}$

Answer

$c \sqrt{\frac{2 m}{E}}$

Dual Nature of Matter and Radiation

The ratio of the wavelength of a photon of energy ' E ' to that of the electron of same energy is ( \mathrm{m}= mass of an electron, \mathrm{c}= speed of light, \mathrm{h}= Planck's constant)

Choose the correct answer: