Question

A beam of unpolarized light passes through a tourmaline crystal A and then it passes through a second tourmaline crystal B oriented so that its principal plane is parallel to that of A. The intensity of emergent light is $I_0$. Now B is rotated by $45^\circ$ about the ray. The emergent light will have intensity $(\cos 45^\circ = \frac{1}{\sqrt{2)$

• A. $\frac{I_0}{2}$
• B. $\frac{I_0}{\sqrt{2$
• C. $\frac{\sqrt{2{I_0}$
• D. $\frac{2}{I_0}$

Hint:

Logic Tip: Malus's Law ($I = I_{max} \cos^2 \theta$) creates a very predictable pattern: At $0^\circ \rightarrow I_{max}$ At $45^\circ \rightarrow I_{max}/2$ At $60^\circ \rightarrow I_{max}/4$ At $90^\circ \rightarrow 0$ (crossed polarizers).

Answer 1

The Correct Option is A

Concept:
When unpolarized light passes through the first polarizer (crystal A), it becomes completely plane-polarized. When this polarized light passes through an analyzer (crystal B), the transmitted intensity $I$ is governed by Malus's Law: $$I = I_{max} \cos^2 \theta$$ where $I_{max}$ is the intensity of the polarized light incident on the analyzer, and $\theta$ is the angle between the transmission axes of the polarizer and the analyzer.
Step 1: Determine the initial state (parallel alignment).
Initially, the principal plane of crystal B is parallel to that of crystal A ($\theta = 0^\circ$). According to Malus's law: $$I_{initial} = I_{max} \cos^2(0^\circ)$$ $$I_{initial} = I_{max} (1)^2 = I_{max}$$ The problem states that this emergent intensity is $I_0$. Therefore, $I_{max} = I_0$.
Step 2: Calculate the intensity after rotation.
Now, crystal B is rotated by $\theta = 45^\circ$ relative to crystal A. Apply Malus's Law again to find the new emergent intensity ($I_{new}$): $$I_{new} = I_{max} \cos^2(45^\circ)$$ Substitute $I_{max} = I_0$ and the given value $\cos 45^\circ = \frac{1}{\sqrt{2$: $$I_{new} = I_0 \left( \frac{1}{\sqrt{2 \right)^2$$ $$I_{new} = I_0 \left( \frac{1}{2} \right)$$ $$I_{new} = \frac{I_0}{2}$$