DATA The company where you work has obtained and stored five lasers in a supply room. You have been asked to determine the intensity of the electromagnetic radiation produced by each laser. The lasers are marked with specifications, but unfortunately different information is given for each laser A: power = 2.6 W; diameter of cylindrical beam = 2.6 mm Laser B: amplitude of electric field = 480 V/m Laser C: amplitude of magnetic field =$\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{-}\mathbf{6}\mathbf{)}$ T Laser D: diameter of cylindrical beam = 1.8 mm; force on totally reflecting surface =$\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}\mathbf{N}$ Laser E: average energy density in beam =$\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{}\mathbf{J}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$ Calculate the intensity for each laser, and rank the lasers in order of increasing intensity. Assume that the laser beams have uniform intensity distributions over their cross sections.

Given that power P = 2.6 W and the diameter of its cylinder is d = 2.6 mm.. The intensity I is proportional to the area A and it represents the incident power P per area A.

${\mathrm{l}}_{\mathrm{A}}=\frac{\mathrm{P}}{\mathrm{A}}=\frac{\mathrm{P}}{4{\mathrm{\tau \tau r}}^2}$

The diameter of the laser is d =2.6 mm Hence, the radius is r = 1.3 mm and its area is calculated by

$\mathrm{A}={\mathrm{\pi r}}^2=\mathrm{\pi }(1.3\times {10}^{-3}\mathrm{m}{)}^2=5.31\times {10}^{-6}{\mathrm{m}}^2$

Substitute the values we have,

${\mathrm{l}}_{\mathrm{A}}=\frac{\mathrm{P}}{\mathrm{A}}=\frac{2.6\mathrm{W}}{5.31\times {10}^{-6}\mathrm{m}}=48.97\times {10}^4\mathrm{W}/{\mathrm{m}}^2$

Therefore, the intensity of Laser A is${\mathrm{I}}_{\mathrm{A}}=48.97\times {10}^4\mathrm{W}/{\mathrm{m}}^2$

The intensity of a sinusoidal electromagnetic wave in a vacuum is related to the

electric-field amplitude E and it is given by equation${\mathrm{I}}_{\mathrm{B}}=\frac{1}{2}{\mathrm{\epsilon }}_0{{\mathrm{cE}}_{\max }}^2$Where${\epsilon }_0$is the electric constant, c is the speed of light. The intensity is proportional to${\mathrm{E}}_{\max }^2$

Substitute the values we have,

$$\begin{gathered} {\mathrm{l}}_{\mathrm{B}}=\frac{1}{2}\left(8.854\times {10}^{-12}\right)\left(3\times {10}^8\right)\left(480\right) \\ =306\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Therefore, the intensity of Laser B is$306\mathrm{W}/{\mathrm{m}}^2$

The maximum electric field is related to the maximum magnetic field and the relationship between both of them is given by equation in the form ${\mathrm{B}}_{\max }=\frac{{\mathrm{E}}_{\max }}{\mathrm{c}}$We know from the above equation that $\mathrm{l}=\frac{1}{2}{\mathrm{\epsilon }}_0{\mathrm{cE}}_{\max }^2=\frac{1}{2}{\mathrm{\epsilon }}_0\mathrm{c}{\left({\mathrm{cB}}_{\max }\right)}^2=\frac{1}{2}{\mathrm{\epsilon }}_0{\mathrm{c}}^3{\mathrm{B}}_{\max }^2$Substitute the values we have,

$$\begin{gathered} {\mathrm{l}}_{\mathrm{c}}=\frac{1}{2}\left(8.854\times {10}^{-12}\right)\left(3\times {10}^8\right)\left(8.7\times {10}^{-6}\right) \\ =904\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Therefore, the intensity of Laser C is$9047\mathrm{W}/{\mathrm{m}}^2$

Given that the diameter by d = 1.8 mm (r =0.9 mm), the force of radiation for totally reflected is $F_{rad}6\times {10}^{-6}$To calculate the radiation pressure $p_{rad}$by

$$\begin{gathered} {\mathrm{p}}_{\mathrm{rad}}=\frac{{\mathrm{F}}_{\mathrm{rad}}}{\mathrm{A}} \\ =\frac{6\times {10}^{-6}}{\mathrm{\pi }{\left(0.9\times {10}^{-3}\right)}^2} \\ =0.0235\mathrm{Pa} \end{gathered}$$

The wave exerts an average force F per unit area and this is the radiation pressure Prad and it is the average value of dp/dt divided by the area. So, the radiation pressure of the wave that totally reflected is given by equation in the form$p_{rad}=\frac{2l_D}{c}$

Substitute the values we have

$$\begin{gathered} {\mathrm{l}}_{\mathrm{D}}=\frac{{\mathrm{cp}}_{\mathrm{rad}}}{2} \\ =\frac{\left(3\times {10}^8\right)\left(0.0235\right)}{2}=3.5\times {10}^6\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Therefore, the intensity of Laser D is$3.5\times {10}^{6}W/m^2$

The average energy density is given by u= 3 x 10¯7 J/m3. The intensity is related to $\mu$by$l_E=\mu c$Substitute the values we have,

$$\begin{gathered} {\mathrm{l}}_{\mathrm{E}}=\mathrm{\mu c} \\ =3\times {10}^{-7}\left(3\times {10}^8\right)=90\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Therefore, the intensity of Laser E=$90\mathrm{W}/{\mathrm{m}}^2$

From the results, the order of intensities of all lasers is$\mathrm{D}\succ \mathrm{A}\succ \mathrm{C}\succ \mathrm{B}\succ \mathrm{E}$