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A sensor is exposed for time t to a lamp of power P placed at a distance l . The sensor has a circular opening that is 4d in diameter. Assuming all energy of the lamp is given off as light, the number of photons entering the sensor if the wavelength of light is $λ$ is : $(l > > d)$

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$\frac{4 {Pλd}^{2} t}{{hcl}^{2}}$

$\frac{{Pλd}^{2} t}{{hcl}^{2}}$

$\frac{{Pλd}^{2} t}{4 {hcl}^{2}}$

$\frac{{Pλd}^{2} t}{16 {hcl}^{2}}$

answer is A.

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Detailed Solution

Energy emitted by the lamp in time t = Pt

where P is the power of the lamp

If 2d is radius of the sphere and l is the distance of source, then the energy reaching the sphere.

$E = \frac{Pt}{4 {πl}^{2}} π (2 d)^{2}$

$E = \frac{{Ptd}^{2}}{l^{2}}$

$\frac{nhc}{λ} \Rightarrow \frac{{Ptd}^{2}}{l^{2}}$

$n = \frac{{Ptd}^{2}}{l^{2}} \times \frac{λ}{hc} \Rightarrow n = \frac{{Ptλd}^{2}}{{hcl}^{2}}$

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