1. Light through first polaroid $A$

   The incident light is unpolarised with intensity $I_0$.

   When unpolarised light passes through an ideal polaroid, its intensity becomes half: $I_1 = \frac{I_0}{2}$

2. Light through second polaroid $B$

   The second polaroid is inclined at $45^\circ$ to the first.

   By Malus' law, $I_2 = I_1 \cos^2\theta$ where $\theta = 45^\circ$.

   So, $I_2 = \frac{I_0}{2} \cos^2 45^\circ$

   Since, $\cos 45^\circ = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad \cos^2 45^\circ = \frac{1}{2}$

   Therefore, $I_2 = \frac{I_0}{2} \cdot \frac{1}{2} = \frac{I_0}{4}$

3. Option checking

   • A: $I_0/2$ ❌
   • B: $I_0/8$ ❌
   • C: $I_0/4$ ✅
   • D: $I_0$ ❌

Hence, the intensity of emergent light is: $\boxed{\frac{I_0}{4}}$