Question

An $EM$ wave of intensity $I$ falls on a surface kept in vacuum and exerts radiation pressure $p$ on it. Whether the following statements are true or not?

i) Radiation pressure is $I/c$ if the wave is totally absorbed.

ii) Radiation pressure is $I/c$ if the wave is totally reflected.

iii) Radiation pressure is $2I/c$ if the wave is totally reflected.

iv) Radiation pressure is in the range $I/c<p<2I/c$

Answer 1

Definition:
Radiation pressure $\left(p\right)$ is the force exerted by electromagnetic wave on unit area of the surface, i.e., rate of change of momentum per unit area of the surface.

Let us consider a surface exposed to electromagnetic radiation as shown in figure.


The radiation is falling normally on the surface. Further, intensity of radiation is $I$ and area of surface exposed to radiation is $A$.

$E=$ Energy received by surface per second $=I.A$

$N=$ Number of photons received by surface per second

$N=\frac{E}{E_{photon}}=\frac{E\lambda }{hc}=\frac{IA\lambda }{hc}$


i) Now, if the surface is perfectly absorbing

$\Delta P_{onephoton}=\frac{h}{\lambda }$

$F=N\times \Delta P_{onephoton}=\frac{IA}{c}$

Also,

Pressure, $P=\frac{F}{A}=\frac{I}{c}$

Hence (i) is TRUE.



ii) Now, if the surface is perfectly reflecting,

$\Delta P_{onephoton}=$ Change in momentum $=\frac{2h}{\lambda }$

$\therefore$ Total force experienced
$F=N\times \Delta P_{onephoton}=\frac{2IA}{c}$

Also, Pressure $P=\frac{F}{A}=\frac{2I}{c}$

Hence (ii) is FALSE.



iii) Now, if the surface is perfectly reflecting then,

$\Delta P_{onephoton}=$ Change in momentum $=\frac{2h}{\lambda }$

$\therefore$ Total force experienced
$F=N\times \Delta P_{onephoton}=\frac{2IA}{c}$

Also, Pressure $P=\frac{F}{A}=\frac{2I}{c}$

Hence (iii) is TRUE.



iv) If the surface is perfectly reflecting then,

$\Delta P_{onephoton}=$ Change in momentum $=\frac{2h}{\lambda }$

Therefore,

Total force experienced
$F=N\times \Delta P_{onephoton}=\frac{2IA}{c}$

Also,

Pressure $P=\frac{F}{A}=\frac{2I}{c}$


If the surface is perfectly absorbing then,

$\Delta P_{onephoton}=\frac{h}{\lambda }$

$F=N\times \Delta P_{onephoton}=\frac{IA}{c}$

Also,

Pressure $P=\frac{F}{A}=\frac{I}{c}$

Hence radiation pressure is in the range $\frac{I}{c}<p<\frac{2I}{c}$ for real surfaces.

Hence (iv) is TRUE.