Question

Three ideal polarizing filters are stacked, with the polarizing axis of the second and third filters at 21 degrees and 61 degrees, respectively, to that of the first. If unpolarized light is incident on the stack, the light has intensity 60.0 w/cm^ 2 after it passes through the stack.
If the incident intensity is kept constant:
1) What is the intensity of the light after it has passed through the stack if the second polarizer is removed?
2) What is the intensity of the light after it has passed through the stack if the third polarizer is removed?

Answer 1

Answer:

1

When second polarizer is removed the intensity after it passes through the stack is    

                    $I_f_3 = 27.57 W/cm^2$

2 When third  polarizer is removed the intensity after it passes through the stack is    

                $I_f_2 = 102.24 W/cm^2$

Explanation:

  From the question we are told that

       The angle of the second polarizing to the first is  $\theta_2 = 21^o$  

        The angle of the third  polarizing to the first is     $\theta_3 = 61^o$

        The unpolarized light after it pass through the polarizing stack   $I_u = 60 W/cm^2$

Let the initial intensity of the beam of light before polarization be $I_p$

Generally when the unpolarized light passes through the first polarizing filter the intensity of light that emerges is mathematically evaluated as

                     $I_1 = \frac{I_p}{2}$

Now according to Malus’ law the  intensity of light that would emerge from the second polarizing filter is mathematically represented as

                    $I_2 = I_1 cos^2 \theta_1$

                       $= \frac{I_p}{2} cos ^2 \theta_1$

The intensity of light that will emerge from the third filter is mathematically represented as

                  $I_3 = I_2 cos^2(\theta_2 - \theta_1 )$

                          $I_3= \frac{I_p}{2}(cos^2 \theta_1)[cos^2(\theta_2 - \theta_1)]$

making $I_p$ the subject of the formula

                  $I_p = \frac{2L_3}{(cos^2 \theta [cos^2 (\theta_2 - \theta_1)])}$

    Note that $I_u = I_3$ as $I_3$ is the last emerging intensity of light after it has pass through the polarizing stack

         Substituting values

                      $I_p = \frac{2 * 60 }{(cos^2(21) [cos^2 (61-21)])}$

                      $I_p = \frac{2 * 60 }{(cos^2(21) [cos^2 (40)])}$

                           $=234.622W/cm^2$

When the second    is removed the third polarizer becomes the second and final polarizer so the intensity of light would be mathematically evaluated as

                      $I_f_3 = \frac{I_p}{2} cos ^2 \theta_2$

$I_f_3$ is the intensity of the light emerging from the stack

                     

substituting values

                     $I_f_3 = \frac{234.622}{2} * cos^2(61)$

                       $I_f_3 = 27.57 W/cm^2$

  When the third polarizer is removed  the  second polarizer becomes the

the final polarizer and the intensity of light emerging from the stack would be  

                  $I_f_2 = \frac{I_p}{2} cos ^2 \theta_1$

$I_f_2$ is the intensity of the light emerging from the stack

Substituting values

                  $I_f_2 = \frac{234.622}{2} cos^2 (21)$

                     $I_f_2 = 102.24 W/cm^2$