Question

A beam of unpolarized light is passed first through a tourmaline crystal $A$ and then through another tourmaline crystal $B$ , oriented so that its principal plane is parallel to that of $A$. The intensity of the emergent light is $I$. The crystal $A$ is now rotated by ${45}^{\circ }$ in a plane perpendicular to the direction of the incident ray. Find the ratio of the final intensity of the emergent light, in the second case, to that in the first case.

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $1$
• D. $\frac{1}{4}$

Answer 1

The correct option is A $\frac{1}{2}$

According to Malu's law, the intensity of the light coming out of a polarizer is given by, $I=I_0{\cos }^2\theta$

Case $\left(I\right)$: The intensity of light coming out of first polarizer will be $\frac{I_0}{2}$. As the transmission axes of both the polarizers are aligned, the same intensity, i.e. $\frac{I_0}{2}$, will emerge from the second polarizer.


Case $\left(II\right)$: The intensity of light coming out of first polarizer will be $\frac{I_0}{2}$. As the transmission axes of both the polarizers are now inclined at ${45}^{\circ }$, the intensity of light emerging from the second polarizer can be obtained using Malu's law, i.e. $I=I_0{\cos }^2\theta$
$\therefore I=\frac{I_0}{2}{\cos }^2{45}^{\circ }=\frac{I_0}{4}$


So, the required ratio of final intensities, is equal to, $\frac{\left(\frac{I_0}{4}\right)}{\left(\frac{I_0}{2}\right)}=\frac{1}{2}$

Hence, $\left(A\right)$ is the correct answer.