Two polaroids are placed in the path of unpolarized beam of intensity I₀ 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 are perpendicular to each other 

Let the initial intensity of light is I₀. So intensity of light after transmission from first polaroid =$\frac{I_0}{2}$ . 

  Intensity of light emitted from  $P_3=I_1=\frac{I_0}{2}{\cos }^2\theta$

  Intensity of light transmitted from last polaroid i.e. from $P_2=I_1{\cos }^2({90}^0-\theta )=\frac{I_0}{2}{\cos }^2\theta .{\sin }^2\theta$$=\frac{I_0}{2\times 4}{\left(4{\sin }^2\theta {\cos }^2\theta \right)}^{ }=\frac{I_0}{8}{\left(2\sin \theta \cos \theta \right)}^2=\frac{I_0}{8}{\sin }^{2}2\theta \left(\because {\theta }_1+{\theta }_2=90 ,\sin 2\theta =2\sin \theta \cos \theta \right)$