An unpolarized light of certain intensity passes through a combination of two polarizers whose transmission axes are at $30^\circ$ and $90^\circ$, respectively, with respect to the horizontal axis. A third polarizer with its transmission axis at $60^\circ$ with the horizontal axis is placed between the two existing polarizers. The ratio of the output intensities with and without the third polarizer is ______.
Answer: (C) $\dfrac{9}{4}$
After the first polarizer: $\dfrac{I_0}{2}$.
Without the third: the axes differ by $60^\circ$, so $I = \dfrac{I_0}{2}\cos^2 60^\circ = \dfrac{I_0}{8}$.
With the third at $60^\circ$: two steps of $30^\circ$, so $I' = \dfrac{I_0}{2}\cos^2 30^\circ\cos^2 30^\circ = \dfrac{I_0}{2} \times \dfrac{9}{16} = \dfrac{9I_0}{32}$.
Ratio $= \dfrac{9/32}{1/8} = \dfrac{9}{4}$.
Solution by Sreeraj P, M.Sc Physics