Question

Two sources of light emit $X-\text{rays}$ of wavelength $1\text{nm}$ and visible light of wavelength $500\text{nm}$, respectively. Both the sources emit light of the same power $200\text{W}$. The ratio of the number density of photons of $X-\text{rays}$ to the number density of photons of the visible light of the given wavelengths is:

• A. $\frac{1}{500}$
• B. $250$
• C. $\frac{1}{250}$
• D. $500$

Answer 1

The correct option is A $\frac{1}{500}$

Given,
Wavelength of $X-\text{rays}$,
${\lambda }_1=1\text{nm}=1\times {10}^{-9}\text{m}$

Wavelength of visible light,
${\lambda }_2=500\times {10}^{-9}\text{m}$

The number of photons emitted per second from a source of monochromatic radiation of wavelength $\lambda$ and power $P$ is given as

$n=\frac{P}{E}$

$\because E=hv\text{and}v=\frac{c}{\lambda }$

$\Rightarrow n=\frac{P}{hv}=\frac{P\lambda }{hc}$

$\Rightarrow n\propto \lambda$

$\Rightarrow \frac{n_1}{n_2}=\frac{{\lambda }_1}{{\lambda }_2}=\frac{1}{500}$

Hence, option $\left(A\right)$ is correct.