As shown in figures, three identical polaroids $$P_1$$, $$P_2$$ and $$P_3$$ are placed one after another. The pass axis of $$P_2$$ and $$P_3$$ are inclined at angle of $$60°$$ and $$90°$$ with respect to axis of $$P_1$$. The source $$S$$ has an intensity of $$256$$ W m$$^{-2}$$. The intensity of light at point $$O$$ is ______ W m$$^{-2}$$.

Solution :

According to Malus law :

$$I = I_0\cos^2\theta$$

Given initial intensity :

$$I_0 = 256\text{ W m}^{-2}$$

After passing through first polaroid $$P_1$$ :

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

$$= \frac{256}{2}$$

$$= 128\text{ W m}^{-2}$$

Angle between axes of $$P_1$$ and $$P_2$$ :

$$\theta = 60^\circ$$

After passing through $$P_2$$ :

$$I_2 = I_1\cos^2 60^\circ$$

$$= 128 \times \left(\frac{1}{2}\right)^2$$

$$= 32\text{ W m}^{-2}$$

Angle between axes of $$P_2$$ and $$P_3$$ :

$$90^\circ - 60^\circ = 30^\circ$$

After passing through $$P_3$$ :

$$I_3 = I_2\cos^2 30^\circ$$

$$= 32 \times \left(\frac{\sqrt3}{2}\right)^2$$

$$= 32 \times \frac{3}{4}$$

$$= 24\text{ W m}^{-2}$$

Final Answer :

$$24\text{ W m}^{-2}$$