Question

A) Monochromatic light of wavelength $632.8\text{nm}$ is produced by a helium-neon laser. The power emitted is $9.42\text{mW}$ .Find the energy and momentum of each photon in the light beam.

B) Monochromatic light of wavelength $632.8\text{nm}$ is produced by a helium-neon laser. The power emitted is $9.42\text{mW}$. 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) Monochromatic light of wavelength $632.8\text{nm}$ is produced by a helium − neon laser. The power emitted is $9.42\text{mW}$. How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon?"

Answer 1

A) Step 1: Find the energy of each photon.
Formula Used: $E=\frac{hc}{\lambda }$
Given,
Wavelength of the monochromatic light, $\lambda =632.8\text{nm}=632.8\times {10}^{-9}m$
Power emitted by the laser,
$P=9.42\text{nW}=9.42\times {10}^{-3}W$
Also,
Planck's constant, $h=6.626\times {10}^{-34}Js$
Speed of light, $C=3\times {10}^8{\text{ms}}^{-1}$
Mass of a hydrogen atom, $m=1.66\times {10}^{-27}\text{kg}$
The energy of each photon is given by,
$E=\frac{hc}{\lambda }$
$E=\frac{6.626\times {10}^{-34}\times 3\times {10}^8}{632.8\times {10}^{-9}}$
$E=3.141\times {10}^{-19}J$
Step 2: Find the momentum of each photon.
Formula Used: $p=\frac{h}{\lambda }$
The momentum of each photon is given by,
$p=\frac{h}{\lambda }$
$p=\frac{6.626\times {10}^{-34}}{632.8\times {10}^{-9}}$
$p=1.05\times {10}^{(-27)}\text{kgm/s}$
Final Answer: $E=3.141\times {10}^{-19}J$
$p=1.05\times {10}^{(-27)}\text{kgm/s}$

B) Step:1 Find the energy of each photon.
Formula Used : $E-\frac{hc}{\lambda }$
Given,
Wavelength of the monochromatic light,
$\lambda =632.8\text{nm}=632.8\times {10}^{-9}m$
Power emitted by the laser,
$P=9.42\text{mW}=9.42\times {10}^{-3}W$
Also,
Planck's constant, $h=6.626\times {10}^{-34}Js$
Speed of light , $c=3\times {10}^{8}m{s}^{-1}$
Mass of a hydrogen atom, $m=1.66\times {10}^{-27}\text{kg}$
Step : 1 Find the energy of each photon.
The energy of each photon is given by,
$E=\frac{hc}{\lambda }$
$E=\frac{6.626\times {10}^{-34}\times 3\times {10}^8}{632.8\times {10}^{-9}}$
$E=3.141\times {10}^{-19}J$
Step:2 Find the number of photons per second.
Let the number of photons arriving per second, at a target irradiated by the beam be n
Hence, the equation for power can be written as,
$p=nE$
$n=\frac{P}{E}$
$n=\frac{9.42\times {10}^{-3}}{3.141\times {10}^{-19}}$
$n\approx 3\times {10}^{16}\text{photons /s}$
Final Answer : $n\approx 3\times {10}^{16}\text{photons/s}$

C) Step:1 Find the Momentum of hydrogen atom.
Formula Used: $p=\frac{h}{\lambda }$
The momentum of each photon is given by,
$p=\frac{h}{\lambda }$
$p=\frac{6.626\times {10}^{-34}}{632.8\times {10}^{-9}}$
$p=1.05\times {10}^{-27}\text{kgm/s}$
Step:2 Find the velocity of each photon.
Formula Used: $p=mv$
Momentum of the hydrogen atom is the same as the momentum of the photon.
Therefore, if the velocity of hydrogen atom is v, then,
$p=mv$
$v=\frac{1.05\times {10}^{-27}}{1.66\times {10}^{-27}}$
$v=0.63\text{m/s}$
Final Answer: $v=0.63\text{m/s}$