• # question_answer (i) light passes through two polaroids ${{P}_{1}}$ and ${{P}_{2}}$  with pass axis of ${{P}_{2}}$ making an angle, $\theta$ with the pass axis of ${{P}_{1}}.$ For what value of $\theta$ is the intensity of emergent light zero? (ii) A third polaroid is placed between ${{P}_{1}}$ and ${{P}_{2}}$ with its pass axis making an angle, $\beta$ with the pass axis of ${{P}_{1}}.$ Find the value of $\beta$ for which the intensity of light from ${{P}_{2}}$ is $\frac{{{I}_{0}}}{8},$ where ${{I}_{0}}$ is the intensity of light on the Polaroid ${{P}_{1}}.$

(i) By law of Malus, intensity of emergent light from ${{P}_{2}}$ is $I={{I}_{0}}{{\cos }^{2}}\theta ,$ where $\theta$ is the angle between ${{P}_{1}}$ and ${{P}_{2}}.$ When $\theta =90{}^\circ \Rightarrow I={{I}_{0}}\times 0$        $[\because \cos \theta =0]$ $\therefore$ Intensity of emergent light, I = 0 (ii) Intensity of light from ${{P}_{3}}$             ${{I}_{3}}=\left( \frac{{{I}_{0}}}{2}{{\cos }^{2}}\beta \right)[co{{s}^{2}}(90{}^\circ -\beta )]$             Light passes through polaroids Similarly, intensity of light from ${{P}_{2}},$             ${{I}_{2}}=\frac{{{I}_{0}}}{2}{{\cos }^{2}}\beta {{\sin }^{2}}\beta =\frac{{{I}_{0}}}{8}{{\sin }^{2}}2\beta$ As,        $\frac{{{I}_{0}}}{8}{{\sin }^{2}}2\beta =\frac{{{I}_{0}}}{8}$          [given] So,       ${{(\sin 2\beta )}^{2}}=1$ $\Rightarrow$   $2\beta =90{}^\circ$   $\Rightarrow$   $\beta =45{}^\circ$