91Ó°ÊÓ

The energy Eof a photon of wavelength $\mathrm{λ}$is given by,

$E=\frac{hc}{\mathrm{λ}}$

Here, h is the Planck’s constant, and c is the speed of light.

(a)

Assume that the protons are emitted by a rate R from the sodium lamp. Then, the power P of the sodium lamp is equal to the product of rate R and the energy of each photon E.

$\begin{array}{c}P=RE\\ P=R\left(\frac{hc}{\mathrm{λ}}\right)\\ R=\frac{P\mathrm{λ}}{hc}\end{array}$…… (1)

Substitute the below values in eq 1.

$$\begin{gathered} \mathrm{P}=100\mathrm{W} \\ \mathrm{λ}=590\mathrm{nm} \\ \mathrm{h}=6.626×{10}^{-34}\mathrm{J}.\mathrm{s} \\ \mathrm{c}=3×{10}^8\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, the rate of emitted protons from the lamp is$2.96×{10}^{20}\mathrm{photons}/\mathrm{s}.$

(b)

The expression for intensity I at a distance r from the lamp is given by,

$\begin{array}{c}I=\frac{R}{4\mathrm{Ï€}{r}^2}\\ r=\sqrt{\frac{R}{4\mathrm{Ï€}I}}......\left(2\right)\end{array}$

Substitute the below values in eq 2,

$$\begin{gathered} \mathrm{R}=2.96×{10}^{20}\mathrm{photons}/\mathrm{s} \\ \mathrm{I}=1\mathrm{photons}/{\mathrm{cm}}^2.\mathrm{s} \end{gathered}$$

Therefore, the required distance from the lamp is$4.86×{10}^7\mathrm{m}.$

Substitute $2.96×{10}^{20}\mathrm{photons}/\mathrm{s}$for R, and 2m for r in equation $I=\frac{R}{4\mathrm{π}{r}^2}$

$$\begin{gathered} I=\frac{2.96×{10}^{20}\mathrm{photons}/\mathrm{s}}{4\mathrm{π}{\left(2\mathrm{m}\right)}^2} \\ =5.89×{10}^{18}\mathrm{photons}/{\mathrm{m}}^2.\mathrm{s} \end{gathered}$$

Therefore, the photon flux per unit time is$5.89×{10}^{18}\mathrm{photons}/{\mathrm{m}}^2.\mathrm{s}.$