Chapter 7, Problem 7A.1ST

Interpretation Introduction

Interpretation:

Under the given conditions, the number of photon has to be calculated.

Concept introduction:

The transmission or absorption of energy takes place in small- small bundles; these bundles are collectively called energy packets or photons.  Photon is the fundamental particles of visible light.  The energy of photon is directly proportional to the frequency of radiation.

Answer to Problem 7A.1ST

The number of photons is $\underset{\_}{5.03\times 10^{14}}$.

Explanation of Solution

The number of photons is calculated by using the formula,

  $N=\frac{\lambda E}{hc}$                                                                                                         (1)

Where,

• $N$ is the number of photon.
• $E$ is the energy of photon.
• $c$ is the velocity of light $\left(3\times {\text{10}}^{\text{8}}\text{m}{\text{s}}^{-1}\right)$.
• $\lambda$ is the wavelength of radiation.
• $h$ is the Planck’s constant $\left(6.626\times {10}^{-34}\text{J}\text{s}\right)$

Since, the energy of photon is calculated by the formula,

  $E=P\Delta t$                                                                                                         (2)

Where,

• $P$ is the power of lamp.
• $\Delta t$ is the time interval.

Substitute $E=P\Delta t$ in equation (1).

  $N=\frac{\lambda P\Delta t}{hc}$                                                                                                     (3)

It is given that,

The wavelength of radiation is $\text{1000}\text{nm}$.

The time interval is $\text{0}\text{.1}\text{s}$.

The power of monochromatic light is $\text{1}\text{mW}$.

The conversion of $\text{nm}$ to $\text{m}$ is done as,

  $\text{1}\text{nm}={10}^{-9}\text{m}$

Therefore, the conversion of $\text{1000}\text{nm}$ to $\text{m}$ is done as,

  $\begin{array}{c}\text{1000}\text{nm}=1000\times {10}^{-9}\text{m}\\ \text{=}{10}^{-6}\text{m}\end{array}$

The conversion of $\text{1}\text{mW}$ to $\text{W}$ is done as,

$\text{1}\text{mW}={\text{10}}^{-3}\text{W}$

The conversion of $\text{W}$ to $\text{J}{\text{s}}^{-1}$ is done as,

$1\text{W}=1\text{J}{\text{s}}^{-1}$

Therefore, the conversion of ${\text{10}}^{-3}\text{W}$ to $\text{J}{\text{s}}^{-1}$ is done as,

  ${\text{10}}^{-3}\text{W}={\text{10}}^{-3}\text{J}{\text{s}}^{-1}$

Substitute the values of $P$, $\Delta t$, $h$, $\lambda$, and $c$ in equation (3).

  $\begin{array}{c}N=\frac{{10}^{-6}\text{m}\times {\text{10}}^{-3}\text{J}{\text{s}}^{-1}\times 0.1\text{s}}{6.626\times {10}^{-34}\text{J}\text{s}\times 3\times {\text{10}}^{\text{8}}\text{m}{\text{s}}^{-1}}\\ =\frac{{10}^{16}}{19.878}\\ =\underset{\_}{5.03\times 10^{14}}\end{array}$

Therefore, the number of photon is $\underset{\_}{5.03\times 10^{14}}$.