question_answer

Two polaroids are placed in the path of unpolarized beam of intensity \[{{I}_{0}}\]such that no light is emitted from the second Polaroid. If a third polarioid whose polarization axis makes an angle \[\theta \] with the polarization axis of first polaroids, 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) \[\left( \frac{{{I}_{0}}}{2} \right){{\cos }^{2}}\theta \]  

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

Correct Answer: A

Solution :

The intensity of light transmitted through third Polaroid, \[I=\frac{{{I}_{0}}}{2}{{\cos }^{2}}\theta \] \[\therefore \] intensity of light transmitted through the last polaroid \[I'=\left( \frac{{{I}_{0}}}{2}{{\cos }^{2}}\theta  \right).{{\cos }^{2}}(90-\theta )=\frac{{{I}_{0}}}{2}{{\cos }^{2}}\theta .{{\sin }^{2}}\theta \] \[=\frac{{{I}_{0}}\times 4{{\sin }^{2}}\theta .{{\cos }^{2}}\theta }{2\times 4}=\frac{{{I}_{0}}}{8}\times {{(2sin\theta .cos\theta )}^{2}}\]                         \[=\left( \frac{{{I}_{0}}}{8} \right){{\sin }^{2}}2\theta \]