An ordinary light of intensity \( I_0 \) is passed through a polaroid \( P_1 \) 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:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 45°
Unpolarised light loses half its intensity in the first polaroid, and each later polaroid transmits according to Malus’ law, \( I = I_1\cos^{2}\theta \), where \( \theta \) is the angle between the pass axes. \( P_2 \) is crossed with \( P_1 \), so the angle between the middle polaroid and \( P_2 \) is \( 90^\circ - \beta \).
\( \displaystyle I_1 = \frac{I_0}{2} \) (after \( P_1 \))
\( \displaystyle I_2 = I_1\cos^{2}\beta = \frac{I_0}{2}\cos^{2}\beta \) (after the middle polaroid)
\( \displaystyle I_3 = I_2\cos^{2}(90^\circ - \beta) = \frac{I_0}{2}\cos^{2}\beta\,\sin^{2}\beta = \frac{I_0}{8}\sin^{2}2\beta \)
Setting \( I_3 = \frac{I_0}{8} \) gives \( \sin^{2}2\beta = 1 \), so \( 2\beta = 90^\circ \) and \( \beta = 45^\circ \).