Question

A point source of light of power ‘P’ and wavelength '$\lambda$' is emitting light in all directions. The number of photons present in a spherical region of radius ‘r’ to radius ‘2r’ with centre at source, is

• A. $\frac{P\lambda }{4\pi r^{2}hc}$
• B. $\frac{P\lambda r}{h{c}^2}$
• C. $\frac{P\lambda }{4\pi rh{c}^2}$
• D. $\frac{P\lambda r}{2h{c}^2}$

Answer 1

The correct option is B $\frac{P\lambda r}{h{c}^2}$

Intensity at a distance x from point source is given by,
$I=\frac{P}{4\pi x^2}$
Consider an elemental shell of radius 'x' and thickness dx, in this region, the energy contained is,
$dE=\frac{I}{C}\left(4\pi x^{2}dx\right)=\frac{Pdx}{C}$
Let dn be the number of photons in this elemental shell,
Then $dn\times \frac{hc}{\lambda }=dE=\frac{Pdx}{C}$
$\Rightarrow dn=\frac{P\lambda dx}{h{c}^2}$
Total no. of photons in the shell of inner radius 'r' and outer radius 2r is
$n=\int dn={\int }_r^{2r}\frac{P\lambda dx}{h{c}^2}=\frac{P\lambda r}{h{c}^2}$