Question

A laser beam of wavelength 500 nm and power 5 mW strikes normally on a perfectly reflecting surface of area 1 mm\(^2\) of a body. It rebounds back from the surface. Find the force exerted by the laser beam on the body.

Hint:

For a perfectly reflecting surface, the force exerted by the laser beam is related to the rate of change of momentum of the photons, which is proportional to the power of the beam.

Answer 1

The force exerted by the laser beam on the body can be found using the relation between the power of the laser beam and the rate of change of momentum. Step 1: Power of the laser beam The power \( P \) of the laser beam is given as: \[ P = 5 \, \text{mW} = 5 \times 10^{-3} \, \text{W} \] Step 2: Momentum change due to reflection The momentum \( p \) of a photon is related to its energy \( E \) by: \[ p = \frac{E}{c} \] where: - \( E \) is the energy of a single photon, - \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)). The energy \( E \) of a photon is related to its wavelength \( \lambda \) by: \[ E = \frac{hc}{\lambda} \] where: - \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \)), - \( \lambda \) is the wavelength of the laser beam (\( 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)). Substituting the value of \( \lambda \): \[ E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{500 \times 10^{-9}} = 3.9756 \times 10^{-19} \, \text{J} \] Step 3: Rate of momentum transfer The total power \( P \) is the rate of energy transfer. The number of photons per second \( N \) can be found by dividing the total power by the energy per photon: \[ N = \frac{P}{E} = \frac{5 \times 10^{-3}}{3.9756 \times 10^{-19}} = 1.257 \times 10^{16} \, \text{photons/s} \] Since the laser beam is perfectly reflected, the change in momentum of each photon is \( 2p \) (because the direction of motion changes, doubling the momentum change). Therefore, the total force \( F \) is the rate of change of momentum, given by: \[ F = \Delta p \times N = 2 \times \frac{E}{c} \times N \] Substitute the values: \[ F = 2 \times \frac{3.9756 \times 10^{-19}}{3 \times 10^8} \times 1.257 \times 10^{16} = 2.8 \times 10^{-6} \, \text{N} \] Thus, the force exerted by the laser beam on the body is \( 2.8 \times 10^{-6} \, \text{N} \).