Radiation Pressure: Explanation, Formula, and Derivation

Radiation pressure is the pressure exerted by electromagnetic radiation on matter.

• The pressure is given by the force exerted on the object per unit area perpendicularly over which the force is distributed.
• The emission or transmission of energy in the form of waves or particles across space or a material medium is referred to as radiation.
• The concept of radiation pressure arises from the momentum transfer between the electromagnetic field and the object.
• The SI unit of radiation pressure is Pascal (Pa) or N/m²
• The dimensional formula of radiation pressure is [M L^-1 T^-2]

The radiation pressure formula for a surface having a surface reflection coefficient α is given by

\(P_r = (1+ \alpha) \frac{I}{c}\)

Where

• I is the intensity of the radiation
• α is the surface reflection coefficient
• c is the speed of the light

For a perfectly reflecting surface, α = 1, therefore

\(P_r = \frac{2I}{c}\)

For a perfectly absorbing surface, α = 0, therefore

\(P_r = \frac{I}{c}\)

Key Terms: Electromagnetic radiation, Photon energy, Momentum, Radiation pressure, Force, Intensity of light, Waves, Speed of light

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What is Radiation?

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Radiation is the emission or transmission of energy in the form of waves or particles through space or a material medium. 

• Radiation does not require a medium, and the transition speed is the same as the speed of light in that medium. 
• It is a method of transferring heat through a vacuum.
• Radiation consists of packets of radiant energy called photons, or quanta. 
• Electromagnetic radiation is an electric and magnetic disturbance that travels at the speed of light through space.
• There are two types of radiation: Ionizing and Non-ionizing radiation.
• The various forms of radiation are electromagnetic radiation, acoustic radiation, particle radiation, and gravitational radiation.

Electromagnetic waves include radio waves, X-rays, UV rays infrared, visible light, microwaves, and gamma rays. The entire distribution of electromagnetic radiation according to frequency or wavelength is known as the Electromagnetic spectrum.

Electromagnetic Spectrum

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Radiation Pressure

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Electromagnetic waves carry momentum with them. When these waves strike a surface, momentum is transferred between the electromagnetic waves and the surface. This results in a force, which is known as radiation pressure.

Radiation pressure is defined as the mechanical pressure exerted on any surface as a result of the momentum exchange between the electromagnetic field and the object.

If the total energy transmitted to a surface is E in time t, the magnitude of the total momentum delivered to this surface is given by

p = E/c

Depending on the surface and the intensity of light used, radiation pressure varies on any surface.

• In 1903, American physicists Nicols and Hull determined the radiation pressure of visible light. It was discovered to be 7 x 10^-6 N/m²
• This is a very small force, but it was the first direct measurement of radiation pressure, and it confirmed the predictions of Maxwell's theory of electromagnetism.
• A basic example of radiation pressure is when the rays of the sun incident in our hands, it causes them to get heated. 
• Because the speed of light (c) is very large, the amount of momentum imparted by electromagnetic waves is extremely small, hence we cannot feel the pressure.

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Radiation Pressure Formula

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Radiation pressure affects celestial objects. We may not feel pressure in all instances, but photons are extremely significant for astronomical objects like stars because vast amounts of photons are emitted in the form of radiation. 

Consider the blackbody condition seen in a star; the radiation pressure is proportional to the fourth power of absolute temperature and can be calculated as follows:

\(P= \frac{4 \sigma}{3c}T^4\)

Where

• P is the radiation pressure
• σ is the Stefan-Boltzmann constant
• T is the absolute temperature
• c is the speed of the light

Outside the star, the radiation pressure formula is

\(P = \frac{xLcos^2 \alpha}{4 \pi R^2c}\)

Where

• L is the luminous intensity
• α is the angle made by the surface of a reflecting or absorbing surface and a light beam
• R is the distance of the star
• c is the speed of light
• x is the absorption constant of the surface

The radiation pressure formula for a surface having a surface reflection coefficient α is given by

\(P_r = (1+ \alpha) \frac{I}{c}\)

Where

• α is the surface reflection coefficient
• I is the intensity of the radiation
• c is the speed of the light

For a perfectly reflecting surface, α = 1, therefore

\(P_r = \frac{2I}{c}\)

For a perfectly absorbing surface, α = 0, therefore

\(P_r = \frac{I}{c}\)

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Derivation of Radiation Pressure Formula

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Let a beam of light of intensity I be incident on a surface. The energy of each photon incident on the surface is given by

E’ = hɳ

Where

• h is Planck’s constant
• ɳ is the frequency of the light

For N number of photons, the total energy is given by

E = Nhɳ

The intensity of incident light is given by the total energy incident on a unit area of the surface in unit time i.e.

I = E/AΔt

⇒ I = Nhɳ/AΔt

⇒ N/AΔt = I/hɳ …(i)

Now, momentum delivered to the unit area in unit time is given by

Δp/AΔt = (N/AΔt) x (Change in the momentum of each photon)

Using equation (i), we get

Δp/AΔt = (I/hɳ) x (Change in the momentum of each photon) …(ii)

But the rate of change of momentum (Δp/Δt) is equal to the force (F) exerted on the surface i.e.

Δp/Δt = F

Using equation (ii), we get

F/A = (I/hɳ) x (Change in the momentum of each photon)

But force exerted on a unit area of the surface by the light or any radiation is known as radiation pressure (P_r), therefore

P_r = (I/hɳ) x (Change in the momentum of each photon) …(iii)

For a perfectly reflecting surface

The momentum of each incident photon = hɳ/c

The momentum of each photon after reflection = - hɳ/c

Where c is the speed of light.

Change in the momentum of each photon = (hɳ/c) - (- hɳ/c) = 2hɳ/c

On substituting the value in equation (iii), we get

P_r = (I/hɳ) x (2hɳ/c) = 2I/c

Hence the radiation pressure for a perfectly reflecting surface is given by

P_r = 2I/c

For a perfectly absorbing surface

The momentum of each incident photon= hɳ/c

Since the surface is perfectly absorbing, therefore

The momentum of each photon after reflection = 0

Change in the momentum of each photon = (hɳ/c) - 0 = hɳ/c

On substituting the value in equation (iii), we get

P_r = (I/hɳ) x (hɳ/c) = I/c

Hence the radiation pressure for a perfectly absorbing surface is given by

P_r = I/c

For a surface, having surface reflection coefficient α

The momentum of each incident photon= hɳ/c

Since the surface has a surface reflection coefficient α, therefore

The momentum of each photon after reflection = -(α x hɳ/c)

Change in the momentum of each photon = (hɳ/c) - [-(α x hɳ/c)] = (1 + α)hɳ/c

On substituting the value in equation (iii), we get

P_r = (I/hɳ) x (1 + α)hɳ/c = (1 + α)I/c

Hence the radiation pressure for a surface having a surface reflection coefficient α is given by

P_r = (1 + α)I/c

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Applications of Radiation Pressure

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The following are the applications of radiation pressure

• Lasers and other optical technologies rely heavily on radiation pressure forces as the underlying technology.
• Radiation pressure from stars and other astronomical objects plays an important role in shaping the structure and evolution of galaxies and other celestial systems.
• Radiation pressure from the Sun is responsible for the tails of comets and the solar wind.
• Radiation pressure can be used to manipulate small particles, such as atoms and molecules.
• Radiation pressure can be used to create nanoscale features in materials.

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Things to Remember

• The mechanical pressure exerted by electromagnetic radiation on a matter is known as the radiation pressure.
• The transmission or emission of energy from one place to another in the form of waves is known as Radiation.
• The momentum carried out by the photons of the radiation is responsible for the radiation pressure.
• The momentum carried out by the photons is given by, p = E/c
• Radiation pressure for a surface is given by, P_r = (1 + α)I/c
• For a perfectly reflecting surface, α = 1 and for a perfectly absorbing surface α = 0.

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