An ordinary light of intensity $I_0$ is passed through a polarised $P_1$ which is crossed with another polaroid $P_2$. Now a third polaroid is placed in between $P_1$ and $P_2$ with its pass axis making an angle $\beta$ with the pass axis of $P_1$. The value of $\beta$ for which the intensity of light from $P_2$ is $\frac{I_0}{8}$ is:

45°

The correct answer is Option (4) → 45°

Light intensity after passing through $P_1$,

$I_1=\frac{I_0}{2}$

Light intensity after passing through $P_3$,

$I_2=I_1\cos^2β=\frac{I_0}{2}\cos^2β$

Light intensity after passing through $P_3$,

$I_3=I_2\cos^2(90-β)$

$=\frac{I_0}{2}\cos^2β\sin^2β$

and,

$\frac{I_0}{2}\cos^2β\sin^2β=\frac{I_0}{8}$

$⇒\sin^2(2β)=1$

$⇒2β=90°$

$⇒β=45°$