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?

Wavelength of monochromatic light $\lambda=632.8\, nm$

Power of He-be laser $P=9.42\, mW$

$(a)$ Energy of a photon is given by $E = hv$

Or $E=h c / \lambda$

Which gives $E=3.14 \times 10^{-19}\, J$

Now momentum of a photon $p=h / \lambda$

$p=1.05 \times 10^{-27}\, kg m / s$

$(b)$ For a beam of uniform cross-section having cross-sectional area less than target area

$P=E x N$

where $P =$ power emitted

$E =$ energy of photon

$N =$ number of photons Therefore

$N = P / E$

Substitution gives

$N =3 \times 10^{16}$ photons / second

$(c)$ Momentum of $He-Ne$ laser $=1.05 \times 10^{-27} \,kg m / s$

For this much momentum of a hydrogen atom $mv =1.05 \times 10^{-27}$

$v=1.05 \times 10^{-27} / 1.6 \times 10^{-27}$

$v=0.63\, m / s$

The required speed for hydrogen atom is $0.63\, m / s$