question_answer

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

A) \[\left( \frac{{{I}_{0}}}{8} \right){{\sin }^{2}}2\theta \] 

B) \[\left( \frac{{{I}_{0}}}{4} \right){{\sin }^{2}}2\theta \]

C) \[\left( \frac{{{I}_{0}}}{2} \right){{\cos }^{4}}2\theta \]

D) \[{{I}_{0}}{{\cos }^{4}}\theta \]

Correct Answer: A

Solution :

[a] No light is emitted from the second polaroid, so \[{{P}_{1}}\]and \[{{P}_{2}}\] are 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 \[{{P}_{3}},{{I}_{1}}=\frac{{{I}_{0}}}{2}{{\cos }^{2}}\theta \] Intensity of light transmitted from last Polaroid i.e., from \[{{P}_{2}}={{I}_{1}}{{\cos }^{2}}(90{}^\circ -\theta )=\frac{{{I}_{0}}}{2}{{\cos }^{2}}\theta .{{\sin }^{2}}\theta \] \[=\frac{{{I}_{0}}}{8}{{(2\sin \theta \cos \theta )}^{2}}=\frac{{{I}_{0}}}{8}{{\sin }^{2}}2\theta \]