Question

Ordinary light passes through two polarizing filters. The filters have been rotated so that their polarizing axes are oriented at ${90}^{\circ }$ to each other, and no light gets through both of them.
By adding a third polarizing filter so that there are three in a row, how might one cause light to pass through the three filters?

• A. Orient the third filter so that its polarizing axis is rotated ${45}^{\circ }$ clockwise relative to the first and place it in front of the first
• B. Orient the third filter so that its polarizing axis is rotated ${45}^{\circ }$ counter-clockwise relative to the seconds and place it in back of the second
• C. Orient the third filter so that its polarizing axis is rotated ${45}^{\circ }$ clockwise relative to the first and place it in between the two filters
• D. Orient the third filter so that its polarizing axis is rotated ${90}^{\circ }$ clockwise relative to the first and place it in front of the first
• E. Both A and B will work to allow light through

Answer 1

The correct option is A Orient the third filter so that its polarizing axis is rotated ${45}^{\circ }$ clockwise relative to the first and place it in front of the first

The intensity of the light passing through a polarizer is given by Malus' Law, $I=I_{0}co{s}^2\left(theta\right)$. When randomly polarized light (unpolarized light) passes through the first polarizer, the intensity is cut in half because there is an even distribution of incident angles, and the mean value of $co{s}^2\left(theta\right)$ is $\frac{1}{2}$ from 0 to $\frac{\pi }{2}$. When it passes through the second polarizer, our "new $I_0$" (for Malus' Law) is $\frac{1}{2}$ of the original$I_0$, and applying Malus' Law, we see that $cos\left(45°\right)=\frac{1}{2}$. In other words, the intensity is halved again and the orientation of the polarization is shifted by 45°. This then happens again on the final polarizer, resulting in a final intensity of $\frac{1}{8}{I}_0$.

If the polarizers had been set to anything other than 45°, this would not be the case.