Radiation Pressure

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Radiation Pressure

 

 

Radiation Pressure :- Maxwell predicted that an electromagnetic wave carries momentum(\displaystyle p=\frac{h}{\lambda }) and energy(\displaystyle E=\frac{hc}{\lambda }) and when this electromagnetic wave is absorbed/reflected by a surface, it would experience a force in a direction perpendicular to the surface. Maxwell’s prediction was confirmed in 1903 by Nichols and Hull by precisely measuring radiation pressures using a torsion balance.

Let us consider that a light of intensity I falls on a surface A, having absorption and reflection coefficient ‘a’ and ‘r’ and assuming no transmission. For calculating the force exerted by the beam on the surface, we consider following cases :-

CASE I :- Radiation pressure in case of complete absorption of radiation and no reflection i.e., a = 1, r = 0.

Initial momentum of the photon, \displaystyle p=\frac{h}{\lambda } (in downward direction)

Final momentum of photon = 0

Change in momentum of photon, \displaystyle \Delta p=\frac{h}{\lambda } (perpendicular to the surface & upward)
Energy incident per unit time = IA
No. of photons incident per unit time, \displaystyle n=\frac{IA}{h\nu }=\frac{IA\lambda }{hc}
So total change in momentum per unit time, \displaystyle \Delta P=n\Delta p=\frac{IA\lambda }{hc}\times \frac{h}{\lambda }=\frac{IA}{c} (perpendicular to the surface & upward)
Force on photons = total change in momentum per unit time = \displaystyle \frac{IA}{c} (perpendicular to the surface & upward)
∴ Force on the surface due to photons, \displaystyle F=\frac{IA}{c} = (perpendicular to the surface & downward)
Hence radiation pressure on the surface, \displaystyle {{P}_{rad}}=\frac{F}{A}=\frac{IA}{c}\times \frac{1}{A}=\frac{I}{c}
\displaystyle \therefore \underbrace{{{P}_{rad}}=\frac{I}{c}}_{\text{Perfect absorber}}
Also note that (I/c) represents energy density (energy per unit volume) u,

Hence,

\displaystyle \therefore \underbrace{{{P}_{rad}}=\frac{I}{c}=u}_{\text{Perfect absorber}}

 

CASE II :- Radiation pressure in case of complete reflection of radiation and no absorption i.e., r = 1, a = 0.

Initial momentum of the photon, \displaystyle p=\frac{h}{\lambda } (in downward direction)
Final momentum of the photon, \displaystyle p=\frac{h}{\lambda } (in upward direction)
Change in momentum of photon, \displaystyle \Delta p=\frac{h}{\lambda }+\frac{h}{\lambda }=\frac{2h}{\lambda } (perpendicular to the surface & upward)
Energy incident per unit time = IA
No. of photons incident per unit time, \displaystyle n=\frac{IA}{h\nu }=\frac{IA\lambda }{hc}
So total change in momentum per unit time, \displaystyle \Delta P=n\Delta p=\frac{IA\lambda }{hc}\times \frac{2h}{\lambda }=\frac{2IA}{c} (perpendicular to the surface & upward)
Force on photons = total change in momentum per unit time = \displaystyle \frac{2IA}{c} (perpendicular to the surface & upward)
∴ Force on the surface due to photons, \displaystyle F=\frac{2IA}{c} = (perpendicular to the surface & downward)
Hence radiation pressure on the surface, \displaystyle {{P}_{rad}}=\frac{F}{A}=\frac{2IA}{c}\times \frac{1}{A}=\frac{2I}{c}
\displaystyle \underbrace{{{P}_{rad}}=\frac{2I}{c}=2u}_{\text{Perfect reflector}}

CASE III :- Radiation pressure in case of partial reflection and partial absorption of radiation, i.e., o < r < 1 and  a + r = 1

Change in momentum of reflected photon, \displaystyle \Delta {{p}_{1}}=\frac{2h}{\lambda } (perpendicular to the surface & upward)
Change in momentum of absorbed photon, \displaystyle \Delta {{p}_{2}}=\frac{h}{\lambda } (perpendicular to the surface & upward)
No. of photons incident per unit time, \displaystyle n=\frac{IA}{h\nu }=\frac{IA\lambda }{hc}
No. of photons reflected per unit time = \displaystyle \frac{IA\lambda }{hc}\times r
No. of photon absorbed per unit time = \displaystyle \frac{IA\lambda }{hc}\times (1-r)
Force due to reflected photon,
\displaystyle {{F}_{r}}=\frac{IA\lambda }{hc}\times (r)\times \frac{2h}{\lambda }=\frac{2IAr}{c} (perpendicular to the surface & downward)
Force due to absorbed photon,
\displaystyle {{F}_{a}}=\frac{IA\lambda }{hc}\times (1-r)\times \frac{h}{\lambda }=\frac{IA}{c}(1-r) (perpendicular to the surface & downward)
Total Force,
\displaystyle F={{F}_{a}}+{{F}_{r}}=\frac{IA}{c}(1-r)+\frac{2IAr}{c}
\displaystyle \Rightarrow F=\frac{IA}{c}(1+r) (perpendicular to the surface & downward)
Hence radiation pressure on the surface,
\displaystyle P=\frac{IA}{c}(1+r)\times \frac{1}{A}
\displaystyle \underbrace{{{P}_{rad}}=\frac{I}{c}(1+r)=u(1+r)}_{\text{Partial absorption }\!\!\And\!\!\text{ Partial reflection}}
CASE IV :- Radiation Pressure in case of incidence of light beam at an angle θ on the surface
CASE (A) :- a = 1, r = 0
Radiation Pressure

Initial momentum of photon (at an angle θ with the normal) = \displaystyle \frac{h}{\lambda }

Final momentum of photon = 0

Change in momentum of photon perpendicular to surface and upward = \displaystyle \frac{h}{\lambda }\cos \theta

Energy incident per unit time normal to surface(of area A) = IA cosθ

No. of photons incident per unit time (on area A) = \displaystyle \frac{IA\text{ }cos\theta }{hc}.\lambda

Total change in momentum of photons per unit time perpendicular to surface(of area A) and upward = \displaystyle \frac{IA\text{ }cos\theta \lambda }{hc}.\frac{h}{\lambda }\cos \theta =\frac{IA\text{ }co{{s}^{2}}\theta }{c}

Total momentum imparted to the surface (of area A) per unit time and downward = \displaystyle \frac{IA\text{ }co{{s}^{2}}\theta }{c}

Force on the surface (F) = total change in momentum per unit time = \displaystyle \frac{IA\text{ }co{{s}^{2}}\theta }{c} (perpendicular to the surface & downward)

Radiation Pressure = \displaystyle \frac{IA\text{ }co{{s}^{2}}\theta }{c}.\frac{1}{A}

\displaystyle {{P}_{rad}}=\frac{I\text{ }co{{s}^{2}}\theta }{c}=uco{{s}^{2}}\theta

 

CASE (B) :- a = 0, r = 1

Radiation Pressure

In this case total change in momentum of one photon perpendicular to surface and upward = \displaystyle \frac{2h}{\lambda }\cos \theta

No. of photons incident per unit time (on area A) = \displaystyle \frac{IA\text{ }cos\theta }{hc}.\lambda

Total change in momentum of photons per unit time perpendicular to surface(of area A) and upward = \displaystyle \frac{IA\text{ }cos\theta \lambda }{hc}.\frac{2h}{\lambda }\cos \theta =\frac{2IA\text{ }co{{s}^{2}}\theta }{c}

Total momentum imparted to the surface (of area A) per unit time and downward = \displaystyle \frac{2IA\text{ }co{{s}^{2}}\theta }{c}

Force on the surface (F) = total change in momentum per unit time = \displaystyle \frac{2IA\text{ }co{{s}^{2}}\theta }{c} (perpendicular to the surface & downward)

Radiation Pressure = \displaystyle \frac{2IA\text{ }co{{s}^{2}}\theta }{c}.\frac{1}{A}

\displaystyle {{P}_{rad}}=\frac{2I\text{ }co{{s}^{2}}\theta }{c}=2uco{{s}^{2}}\theta

 

CASE (C) :- Partial reflection and partial absorption, i.e., o < r < 1 and a + r = 1

Change in momentum of reflected photon(perpendicular to surface and upward) = \displaystyle \frac{2h}{\lambda }\cos \theta

Change in momentum of absorbed photon(perpendicular to surface and upward) = \displaystyle \frac{h}{\lambda }\cos \theta

No. of photons incident per unit time (on area A) = \displaystyle \frac{IA\text{ }cos\theta }{hc}.\lambda

No. of photons reflected per unit time = \displaystyle \frac{IAcos\theta \lambda }{hc}\times r

No. of photon absorbed per unit time = \displaystyle \frac{IAcos\theta \lambda }{hc}\times (1-r)

Force due to reflected photon,

\displaystyle {{F}_{r}}=\frac{IAcos\theta \lambda }{hc}\times (r)\times \frac{2hcos\theta }{\lambda }=\frac{2IAco{{s}^{2}}\theta }{c}.r (perpendicular to the surface & downward)

Force due to absorbed photon,

\displaystyle {{F}_{a}}=\frac{IAcos\theta \lambda }{hc}\times (1-r)\times \frac{hcos\theta }{\lambda }=\frac{IAco{{s}^{2}}\theta }{c}\times (1-r) (perpendicular to the surface & downward)

Total Force,

\displaystyle F={{F}_{a}}+{{F}_{r}}=\frac{IAco{{s}^{2}}\theta }{c}\times (1-r)+\frac{2IAco{{s}^{2}}\theta }{c}.r

\displaystyle \Rightarrow F=\frac{IAco{{s}^{2}}\theta }{c}(1+r)

Hence radiation pressure on the surface,

\displaystyle P=\frac{IAco{{s}^{2}}\theta }{c}(1+r)\times \frac{1}{A}

\displaystyle {{P}_{rad}}=\frac{Ico{{s}^{2}}\theta }{c}(1+r)=u(1+r)co{{s}^{2}}\theta

 

 

Next Topic :- Davisson and Germer Experiment

Previous Topic :- de Broglie wavelength associated with an electron

Class 12th NCERT

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