A beam of unpolarized light of intensity I_0 passes through a...

When unpolarized light of intensity $I_0$ passes through $P_1$, the transmitted intensity is $I_1 = \dfrac{I_0}{2}$.

When this light passes through $P_2$, the emergent intensity is given by Malus's law: $I = I_1 \cos^2(\alpha)$, where $\alpha$ is the angle between the pass axes of $P_1$ and $P_2$.

The maximum intensity occurs when $\alpha = 0^{\circ}$ (i.e., $P_1$ and $P_2$ are parallel). Thus, $I_{\text{max}} = I_1 = \dfrac{I_0}{2}$.

Given $I_{\text{max}} = 10 \text{ W/m}^2$, we have $\dfrac{I_0}{2} = 10 \Rightarrow I_0 = 20 \text{ W/m}^2$.

Since the maximum occurs at $\theta = 30^{\circ}$, the pass axis of $P_1$ is oriented at $30^{\circ}$ from the reference axis.

When $P_2$ is set to $\theta = 90^{\circ}$, the angle between $P_1$ and $P_2$ is $\alpha = 90^{\circ} - 30^{\circ} = 60^{\circ}$.

The transmitted intensity at this position is $I = 10 \cos^2(60^{\circ}) = 10 \times \left(\dfrac{1}{2}\right)^2 = 2.5 \text{ W/m}^2$.

Answer: $20 \text{ W/m}^2, 2.5 \text{ W/m}^2$