Question

A beam of white light is incident normally on a plane surface absorbing 70% of the light and reflecting the rest. If the incident beam carries 10 W of power, find the force exerted by it on the surface.

Answer 1

Power of the incident beam, P = 10 watt
Relation between wavelength $\left(\lambda \right)$ and momentum (p):
$$\begin{gathered} \lambda =\frac{h}{p}, \\ \text{w}\mathrm{here}h\mathrm{is}\text{P}\mathrm{lanck}\text{'}\text{s}\mathrm{constant} \\ \Rightarrow p=\frac{h}{\lambda } \\ \mathrm{On}\mathrm{dividing}\mathrm{both}\mathrm{side}\text{s}\mathrm{by}t,\mathrm{we}\mathrm{get}: \\ \frac{p}{t}=\frac{h}{\lambda t}...\left(1\right) \end{gathered}$$
Energy,
$$\begin{gathered} E=\frac{hc}{\lambda } \\ \Rightarrow \frac{E}{t}=\frac{hc}{\lambda t} \end{gathered}$$
Let P be the power. Then,
$$\begin{gathered} P=\frac{E}{t}=\frac{hc}{\lambda t} \\ P=\frac{pc}{t}[\text{U}\mathrm{sing}\mathrm{equation}\left(1\right)] \\ \Rightarrow \frac{P}{c}=\frac{p}{t} \\ \text{F}\mathrm{orce}, \\ F=\frac{p}{t}=\frac{P}{c}\left(\mathrm{Since}F=\frac{\text{M}\mathrm{omentum}}{\text{T}\mathrm{ime}}\right) \\ \mathrm{Force},F=\frac{7}{10}\left(\mathrm{absorbed}\right)+2\times \frac{3}{10}\left(\mathrm{reflected}\right) \\ F=\frac{7}{10}\times \frac{P}{c}+2\times \frac{3}{10}\times \frac{P}{c} \\ F=\frac{7}{10}\times \frac{10}{3\times {10}^8}+2\times \frac{3}{10}\times \frac{10}{3\times {10}^8} \\ F=\frac{13}{3}\times {10}^{-8}=4.33\times {10}^{-8}\mathrm{N} \end{gathered}$$