Three polarizing sheets are placed in a stack with the polarizing directions of the first and third perpendicular to each other.What angle should the polarizing direction of the middle sheet make with the polarizing direction of the first sheet to obtain maximum transmitted intensity when unpolarized light is incident on the stack?
A)0°
B)30°
C)45°
D)60°
E)90°

Question

Quizplus · Accepted Answer

When unpolarized light is incident on the stack of polarizing sheets, the intensity of transmitted light is given by $I_	ext{T}=I_0\cos^2	heta$, where $I_0$ is the intensity of incident light and $	heta$ is the angle between the polarizing directions of the first and second sheets. The transmitted intensity through the third sheet will be maximum when the polarizing direction of the third sheet is parallel to the transmitted component of the light through the first two sheets.

Let $I_1$ be the intensity of the light transmitted through the first sheet and $I_2$ be the intensity of the transmitted component of the light through the second sheet that is parallel to the polarizing direction of the third sheet. Then we have, $I_2=I_1\cos	heta$, where $	heta$ is the angle between the polarizing directions of the first and second sheets.

Similarly, let $I_3$ be the intensity of the transmitted component of the light through the third sheet that is parallel to the polarizing direction of the first sheet. Then we have, $I_	ext{T}=I_3\cos\phi$, where $\phi$ is the angle between the polarizing directions of the second and third sheets.

Since the polarizing directions of the first and third sheets are perpendicular to each other, we have $\phi=90^\circ-	heta$. Therefore, $I_	ext{T}=I_3\sin^2	heta$. To maximize $I_	ext{T}$, we need to maximize $I_3$.

It can be shown that the maximum value of $I_3$ is obtained when the polarizing direction of the middle sheet makes an angle of $45^\circ$ with the polarizing direction of the first sheet. Therefore, the answer is (C), 45°.

Three Polarizing Sheets Are Placed in a Stack with the Polarizing