Chapter 22, Problem 82P

To determine

Verify the equation $I=\langle u\rangle c$ dimensionally correct.

Answer to Problem 82P

The equation $I=\langle u\rangle c$ is dimensionally correct.

Explanation of Solution

Write an expression for average energy density in an EM wave.

  $\langle u\rangle ={\epsilon }_0{E}_{rms}^2$                                                                                                              (I)

Here, $\langle u\rangle$ average energy density in an EM wave, ${\epsilon }_0$ is the permittivity of free space and $E_{rms}$ is the rms energy.

Write dimensional expression for permittivity of free space.

  $\left[{\epsilon }_0\right]=\frac{{\text{C}}^2}{\text{N}\cdot {\text{m}}^{\text{2}}}$                                                                                                          (II)

Write dimensional expression for rms energy.

  $\left[E_{rms}\right]=\frac{\text{V}}{m}$                                                                                                          (III)

Write dimensional expression for average energy density in an EM wave.

  $\left[\langle u\rangle \right]=\left[{\epsilon }_0\right]\left[E_{rms}^2\right]$                                                                                                (IV)

Substitute equation (II) and equation (III) in equation (IV) to find $\left[\langle u\rangle \right]$.

  $\left[\langle u\rangle \right]=\frac{{\text{C}}^{\text{2}}}{\text{N}\cdot {\text{m}}^{\text{2}}}{\left(\frac{\text{V}}{\text{m}}\right)}^2$                                                                                             (V)

Write an expression for the intensity.

  $I=\langle u\rangle c$                                                                                                              (VI)

Here, $I$ is the intensity and $c$ is the speed of light.

Write dimensional expression for intensity.

  $\left[I\right]=\left[\langle u\rangle \right]\left[c\right]$                                                                                                  (VII)

Write dimensional expression for speed of light.

  $\left[c\right]=\frac{\text{m}}{\text{s}}$                                                                                                          (VIII)

Substitute equation (V) and equation (VIII) in equation (VII) to find $\left[I\right]$.

    $\begin{array}{c}\left[I\right]=\left(\frac{{\text{C}}^{\text{2}}}{\text{N}\cdot {\text{m}}^{\text{2}}}{\left(\frac{\text{V}}{\text{m}}\right)}^2\right)\left(\frac{\text{m}}{\text{s}}\right)\\ =\left(\frac{{\text{C}}^{\text{2}}}{\text{N}\cdot {\text{m}}^{\text{2}}}\left(\frac{{\text{V}}^2}{{\text{m}}^2}\right)\right)\left(\frac{\text{m}}{\text{s}}\right)\\ =\left(\left(\frac{{\text{C}}^{\text{2}}}{\text{J}\cdot \text{m}}\right)\left(\frac{{\text{J}}^2}{{\text{C}}^{\text{2}}\cdot {\text{m}}^2}\right)\right)\left(\frac{\text{m}}{\text{s}}\right)\\ =\left(\frac{\text{J}}{{\text{m}}^3}\right)\left(\frac{\text{m}}{\text{s}}\right)\\ =\frac{\text{J/s}}{{\text{m}}^2}\\ =\frac{\text{W}}{{\text{m}}^{\text{2}}}\end{array}$

The dimension of intensity is $\text{W}/{\text{m}}^2$. Thus, the equation $I=\langle u\rangle c$ is dimensionally correct.