A flashlight beam produces 2.5 W of electromagnetic radiation in a narrow beam. Although the light it produces is white (all visible wavelengths), make the simplifying assumption that the wavelength is 550 nm, the middle of the visible spectrum. (a) How many photons per second emanate from the flashlight (b) What force would the beam exert on a “perfect mirror”?

The photon energy can be expressed as:-

$\mathit{E}\mathbf{=}\frac{\mathbf{h}\mathbf{c}}{\mathbf{\lambda }}$

Here c is the speed of light h is the Plank’s constant and λ is the wavelength.

Substituting the values and solving the above equation for E, we get,

$\begin{array}{rcl}\mathit{E}& \mathbf{=}& \left(6.63\times {10}^{-34}J·s\right)\left(\frac{3\times {10}^{8}m/s}{550\times {10}^{-9}m}\right)\\ & \mathbf{=}& \mathbf{3}\mathbf{.}\mathbf{61}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{19}}\mathit{J}\end{array}$

And then to get the number of photons per second, we decide the power P by the energy per photon E.

$\begin{array}{rcl}\frac{no.ofphotons}{time}& =& \frac{P}{E}\\ n& =& \frac{2.5J/s}{3.6\times {10}^{-19}J}\\ n& =& 6.9\times {10}^{18}photons/second\end{array}$

To find that we can start with Newton’s relation:-

$F=\frac{dp}{dt}$………………..(1)

For a given momentum p and using the expression for the momentum of a photon for a given energy E.

$P=\frac{E}{C}$

This can be used to rewrite (1)

$$\begin{gathered} F=\frac{dp}{dt} \\ F=\frac{d}{dt}\left(\frac{E}{c}\right) \\ F=\frac{1}{c}\frac{dE}{dt} \\ F=\frac{1}{c}P \end{gathered}$$

That though is the force for a surface that’s perfectly absorbing the force on a perfect mirror would be double that, since the mirror is effectively stopping, then re-emitting them back the way they came:-

$F=\frac{2P}{c}$

substitute 2.5 J/s for P and 3x10⁸m/s for the speed of light in the above equation and solve for F

$\begin{array}{rcl}F& =& \frac{2\times \left(2.5J/s\right)}{3\times {10}^{8}m/s}\\ & =& 1.7\times {10}^{-8}N\end{array}$

Hence, the total force exerted on the perfect mirror by the flashlight would be

$F=1.7\times {10}^{-8}N$