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 light of intensity I falls on a surface of area 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 }{hc}\lambda .\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 }{hc}\lambda .\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 }{hc}\lambda \times r

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

Force due to reflected photon,

\displaystyle {{F}_{r}}=\frac{IAcos\theta }{hc}\lambda \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 }{hc}\lambda \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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