Unpolarised light of intensity 32 W m$$^{-2}$$ passes through the combination of three polaroids such that the pass axis of the last polaroid is perpendicular to that of the pass axis of first polaroid. If intensity of emerging light is 3 W m$$^{-2}$$, then the angle between pass axis of first two polaroids is _______ $$^\circ$$.

Given: Unpolarised light intensity $$I_0 = 32$$ W/m², final intensity $$I = 3$$ W/m², three polaroids with last perpendicular to first.

After the first polaroid: $$I_1 = \frac{I_0}{2} = 16$$ W/m²

Let the angle between first and second polaroid be $$\theta$$.

After second polaroid: $$I_2 = I_1 \cos^2\theta = 16\cos^2\theta$$

The angle between second and third polaroid is $$(90° - \theta)$$ (since third is perpendicular to first).

After third polaroid: $$I_3 = I_2\cos^2(90° - \theta) = 16\cos^2\theta \cdot \sin^2\theta$$

$$16\cos^2\theta\sin^2\theta = 3$$

$$4\sin^2 2\theta = 3$$ (using $$\sin 2\theta = 2\sin\theta\cos\theta$$)

$$\sin^2 2\theta = \frac{3}{4}$$

$$\sin 2\theta = \frac{\sqrt{3}}{2}$$

$$2\theta = 60°$$

$$\theta = 30°$$

The angle is 30°.