Energy & Momentum of a Photon (Cambridge (CIE) A Level Physics): Revision Note

Light of wavelength 490 nm is incident normally on a surface, as shown in the diagram.

The power of the light is 3.6 mW. The light is completely absorbed by the surface.

Calculate the number of photons incident on the surface in 2.0 s.

Answer

Step 1: Write down the known quantities

• Wavelength, λ = 490 nm = 490 × 10^-9 m

• Power, P = 3.6 mW = 3.6 × 10^-3 W

• Time, t = 2.0 s

Step 2: Write the equation for photon energy in terms of wavelength

$E=\frac{hc}{\lambda }$

Step 3: Calculate the energy of one photon

$E=\frac{\left(6.63\times {10}^{-34}\right)\times \left(3.0\times {10}^8\right)}{490\times {10}^{-9}}=4.06\times {10}^{-19}\mathrm{J}$

Step 4: Calculate the number of photons hitting the surface every second

$\frac{Poweroflightsource}{Energyofonephoton}=\frac{3.6\times {10}^{-3}}{4.06\times {10}^{-19}}=8.9\times {10}^{15}{\mathrm{s}}^{-1}$

Step 5: Calculate the number of photons that hit the surface in 2 s

$\left(8.9\times {10}^{15}\right)\times 2=1.8\times {10}^{16}$

The values of Planck’s constant and the speed of light are both given on your data sheet, however, it helps to memorise them to speed up calculation questions.

Remember this equation for E is only for the energy of a photon, not the energy for any other particle!

A 5.0 mW laser beam is incident normally on a fixed metal plate. The cross-sectional area of the beam is 8.0 × 10^-6 m². The light from the laser has a frequency of 5.6 × 10¹⁴ Hz.

Assuming that all the photons are absorbed by the plate, calculate the momentum of the photon, and the pressure exerted by the laser beam on the metal plate.

Answer:

Step 1: Write down the known quantities

• Power, P = 5.0 mW = 5.0 × 10^-3 W

• Frequency, f = 5.6 × 10¹⁴ Hz

• Cross-sectional area, A = 8.0 × 10^-6 m²

Step 2: Write the equations for photon energy and momentum

$p=\frac{E}{c}\Rightarrow p=\frac{hf}{c}$

Step 3: Calculate the photon momentum

$p=\frac{hf}{c}=\frac{\left(6.63\times {10}^{-34}\right)\times \left(5.6\times {10}^{14}\right)}{3.0\times {10}^8}=1.24\times {10}^{-27}\mathrm{N}\mathrm{s}$

Step 4: Calculate the number of photons incident on the plate every second

$\frac{Poweroflightsource}{Energyofonephoton}=\frac{5.0\times {10}^{-3}}{hf}=\frac{5.0\times {10}^{-3}}{\left(6.63\times {10}^{-34}\right)\times \left(5.6\times {10}^{14}\right)}=1.35\times {10}^{16}{\mathrm{s}}^{-1}$

Step 5: Calculate the force exerted on the plate in a 1.0 s time interval

• Force is the rate of change in momentum

$F=\frac{∆p}{∆t}$

F  = number of photons per second × momentum of each photon

$F=(1.35\times {10}^{16})\times (1.24\times {10}^{-27})=1.67\times {10}^{-11}\mathrm{N}$

 Step 6: Calculate the pressure

• Pressure (P) is the force per unit area

$P=\frac{F}{A}=\frac{1.67\times {10}^{-11}}{8.0\times {10}^{-6}}=2.1\times {10}^{-6}\mathrm{Pa}$

It's not unusual for multiple equations to be required for a question involving the photon momentum. 

Always watch out of the units! For the momentum p to be in kg m s^–1, the energy must be in J and the speed of light in m s^–1.