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 a laser beam on a body is determined by the relationship between the beam's power and its rate of momentum change. Step 1: Laser Beam Power The power ($P$) of the laser beam is provided as: \[ P = 5 \, \text{mW} = 5 \times 10^{-3} \, \text{W} \] Step 2: Momentum Change from Reflection The momentum ($p$) of a photon is defined by its energy ($E$) and the speed of light ($c$): \[ p = \frac{E}{c} \] 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 also 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 laser beam's wavelength ($500 \, \text{nm} = 500 \times 10^{-9} \, \text{m}$). Substituting the wavelength value: \[ E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{500 \times 10^{-9}} = 3.9756 \times 10^{-19} \, \text{J} \] Step 3: Rate of Momentum Transfer Total power ($P$) represents the rate of energy transfer. The number of photons per second ($N$) is calculated by dividing the total power by the energy per photon: \[ N = \frac{P}{E} = \frac{5 \times 10^{-3}}{3.9756 \times 10^{-19}} = 1.257 \times 10^{16} \, \text{photons/s} \] Due to perfect reflection, the momentum change for each photon is $2p$ (as the direction reverses, doubling the momentum change). Consequently, the total force ($F$) equals the rate of momentum change: \[ F = \Delta p \times N = 2 \times \frac{E}{c} \times N \] Substituting the values: \[ F = 2 \times \frac{3.9756 \times 10^{-19}}{3 \times 10^8} \times 1.257 \times 10^{16} = 2.8 \times 10^{-6} \, \text{N} \] The force exerted by the laser beam on the body is $2.8 \times 10^{-6} \, \text{N}$.