#### Question

Monochromatic light of wavelength 632.8 nm 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), and

**(c) **How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon?

#### Solution

Wavelength of the monochromatic light, *λ* = 632.8 nm = 632.8 × 10^{−9} m

Power emitted by the laser, *P* = 9.42 mW = 9.42 × 10^{−3} W

Planck’s constant, *h* = 6.626 × 10^{−34} Js

Speed of light, *c* = 3 × 10^{8} m/s

Mass of a hydrogen atom, *m* = 1.66 × 10^{−27} kg

**(a)**The energy of each photon is given as:

`E = (hc)/lambda`

`= (6.626 xx 10^(-34) xx 3xx 10^8)/632.8 xx 10^(-9) = 3.141 xx 10^(-19) J`

The momentum of each photon is given as:

`P = h/lambda`

`= (6.626 xx 10^(-34))/632.8 = 1.047 xx 10^(-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`

`:. n = P/E`

`= (9.42 xx 10^(-3))/3.141 xx 10^(-19) ~~ 3 xx 10^16 "photon/s"`

**(c)**Momentum of the hydrogen atom is the same as the momentum of the photon,

`p = 1.047 xx 10^(-27) kg ms^(-`)

Momentum is given as:

p = mv

Where,

*v* = Speed of the hydrogen atom

`:. v = p/m`

`= (1.047 xx 10^(-27))/(1.66 xx 10^(-27)) = 0.621 "m/s"`