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Q.39) The intensity of transmitted light when a polaroid sheet, placed between two crossed polaroids at $22.5^{\circ}$ from the polarization axis of one of the polaroid, is $\left(I_0\right.$ is the intensity of polarised light after passing through the first polaroid):
 

A) $\frac{I_0}{16}$
 

B) $\frac{I_0}{2}$
 

C)  $\frac{I_0}{4}$
 

D)  $\frac{I_0}{8}$

 

Answers (1)

best_answer

We use Malus's Law:

$$
I=I_0 \cos ^2 \theta
$$


Let's calculate in two steps:
1. First polaroid to middle one (angle $=22.5^{\circ}$ ):

$$
I_1=I_0 \cos ^2\left(22.5^{\circ}\right)
$$

2. Middle to second polaroid (angle $=90^{\circ}-22.5^{\circ}=67.5^{\circ}$ ):

$$
I_2=I_1 \cos ^2\left(67.5^{\circ}\right)
$$


So total intensity:

$$
I=I_0 \cos ^2\left(22.5^{\circ}\right) \cdot \cos ^2\left(67.5^{\circ}\right)
$$


Use identities:
- $\cos \left(22.5^{\circ}\right)=\cos \left(67.5^{\circ}\right)=\sqrt{\frac{1+\cos \left(45^{\circ}\right)}{2}}=\sqrt{\frac{1+\frac{1}{\sqrt{2}}}{2}} \approx 0.924$
- So $\cos ^2\left(22.5^{\circ}\right) \cdot \cos ^2\left(67.5^{\circ}\right) \approx(0.853)^2 \approx 0.125$

$$
I \approx \frac{I_0}{8}
$$

Posted by

Saumya Singh

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