Two polaroids are placed in the path of unpolarised beam of intensity $I_0$ such that no light is emitted from the second polaroid. If a third polaroid whose polarization axis makes an angle $\theta$ with the polarization axis of first polaroid, is placed between these polaroids then the intensity of light emerging from the last polaroid will be

No light is emitted from the second polaroid, so P₁ and P₂ perpendicular to each other

Let the initial intensity of light is $I_0.$ So Intensity of light after transmission from first polaroid $=\frac{I_0}{2}$.

Intensity of light emitted from P₃, ${\mathrm{I}}_1=\frac{{\mathrm{I}}_0}{2}{\cos }^2\mathrm{\theta }$

Intensity of light transmitted from last polaroid, i.e., from ${\mathrm{P}}_2={\mathrm{I}}_1{\cos }^2\left({90}^{\circ }-\mathrm{\theta }\right)=\frac{{\mathrm{I}}_0}{2}{\cos }^2\mathrm{\theta }\cdot {\sin }^2\mathrm{\theta }$

$=\frac{{\mathrm{I}}_0}{8}(2\mathrm{sin\theta cos\theta }{)}^2=\frac{{\mathrm{I}}_0}{8}{\sin }^{2}2\mathrm{\theta }$