Chapter 28, Problem 41A

a.

To determine

To calculate: Number of photons emitted per second.

Answer to Problem 41A

Number of photons per second that are emitted = $\underset{\_}{2}$ .

Explanation of Solution

Given:

A laser beam’s power equals the photon energy times the number of photons per second that are emitted.

Wavelength of first laser = $\text{840 nm}$

Wavelength of second laser = $\text{427 nm}$

Formula used:

Energy change is given by,

  $\Delta E=\frac{hc}{\lambda }$

Where $h$ is Planck’s constant, $c$ is speed of light and $\lambda$ is wavelength.

Calculation:

Energy change in $\text{840 nm}$ laser is calculated as:

  $\begin{array}{l}\Delta E=\frac{hc}{\lambda }\text{ [}hc=1240\text{ eV}\cdot \text{nm}]\\ \text{ =}\frac{1240\text{ eV}\cdot \text{nm}}{\text{840 nm}}\\ =1.48\text{ eV}\end{array}$

Energy change in $\text{427 nm}$ laser is calculated as:

  $\begin{array}{l}\Delta E=\frac{hc}{\lambda }\text{ [}hc=1240\text{ eV}\cdot \text{nm}]\\ \text{ =}\frac{1240\text{ eV}\cdot \text{nm}}{\text{427 nm}}\\ =2.90\text{ eV}\end{array}$

Ratio of energy change in each photon:

  $\begin{array}{l}=\frac{1.48\text{ eV}}{2.90\text{ eV}}\\ \text{= 0}\text{.5}\end{array}$

Number of photons per second = $\frac{1}{0.5}=2$

Conclusion:

Number of photons per second that are emitted = $\underset{\_}{2}$ .

b.

To determine

To calculate: Number of photons emitted per second.

Answer to Problem 41A

Number of photons per second = $\underset{\_}{2.1\times {10}^{16}}$

Explanation of Solution

Given:

Power of laser beam = $5.0\text{ mW}$

Wavelength of laser beam = $\text{840 nm}$

Formula used:

Energy change is given by,

  $\Delta E=\frac{hc}{\lambda }$

Where $h$ is Planck’s constant, $c$ is speed of light and $\lambda$ is wavelength.

Number of photons per second is given by,

  $n=\frac{P}{E}$

Where $P$ is power and $E$ is energy.

Calculation:

Energy change in $\text{840 nm}$ laser is calculated as:

  $\begin{array}{l}\Delta E=\frac{hc}{\lambda }\text{ [}hc=1240\text{ eV}\cdot \text{nm}]\\ \text{ =}\frac{(1240\text{ eV}\cdot \text{nm)(}1.60\times {10}^{-19}\text{ J/eV)}}{\text{840 nm}}\\ \text{ =}2.4\times {10}^{-19}\text{ J}\end{array}$

Number of photons per second is calculate as:

  $\begin{array}{l}n=\frac{P}{E}\\ \text{ =}\frac{5.0\times {10}^{-3}\text{ J/s}}{2.4\times {10}^{-19}\text{ J/photon}}\\ \text{ = }2.1\times {10}^{16}\text{ photon/s}\end{array}$

Conclusion:

Number of photons per second = $\underset{\_}{2.1\times {10}^{16}}$ .