Two sources of light emit X-rays of wavelength

Question:

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

  1. (1) $\frac{1}{500}$

  2. (2) 250

  3. (3) $\frac{1}{250}$

  4. (4) 500


Correct Option: 1

Solution:

(1) Given,

Wavelength of X-rays, $\lambda_{1}=1 \mathrm{~nm}=1 \times 10^{-9} \mathrm{~m}$

Wavelength of visible light, $\lambda_{2}=500 \times 10^{-9} \mathrm{~m}$

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

$n=\frac{P}{E}=\frac{P}{h v}=\frac{P \lambda}{h c}$                          $\left(\because \mathrm{E}=\right.$ h $v$ and $\left.v=\frac{\mathrm{c}}{\lambda}\right)$

$\Rightarrow$ Clearly $n \propto \lambda$ 

$\Rightarrow \frac{n_{1}}{n_{2}}=\frac{\lambda_{1}}{\lambda_{2}}=\frac{1}{500}$

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