Monochromatic light of wavelength `632.8nm` is produced by a helium-neon laser. The power emitted is `9.42 mW`. (a) Find the energy and momentum of each photon in the light beam. (b) How many photons per second, on the average, arrive at a target irradiated by this beam? (Assume the beam to have uniform cross-section which is less than the target area). (c) How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon. `h=6.663xx10^(-34)Js,1 a.m.u. =1.66xx10^(-27)kg`.

Explanation

Wavelength of the monochromatic light, `lambda=632.8 nm = 632.8xx10^(-9) m`
Power emitted by the laser, P=9.42 mW=`9.42xx10^(-3)W`
Planck's constant , `h=6.626xx10^(-34)` Js
Speed of light , `c=3xx10^8` m/s
Mass of a hydrogen atom, m=`1.66xx10^(-27)` kg
(a)The energy of each photon is given as :
`E=(hc)/lambda`
`=(6.626xx10^(-34)xx3xx10^8)/(632.8xx10^(-9))=3.141xx10^(-19)` J
The momentum of each photon is given as : `P=h/lambda`
`=(6.626xx10^(-34))/632.8=1.047xx10^(-27) kg ms^(-1)`
(b)Number of photons arriving per second, at a target irradiated by the beam =n . Assume that the beam has a uniform cross-section that is less than the target area. Hence , the equation for power can be written as :
P=nE
`therefore n=P/E`
`=(9.42xx10^(-3))/(3.141xx10^(-19))~~3xx10^16` photon/s
(c )Momentum of the hydrogen atom is the same as the momentum of the photon, `p=1.047xx10^27 kg ms^(-1)`
Momentum is given as : p=mv
Where, v = Speed of the hydrogen atom
`therefore v=p/m`
`=(1.047xx10^(-27))/(1.66xx10^(-27))=0.621 m//s`