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Chapter 1: Problem 64

Two polarizing sheets \(P_{1}\) and \(P_{2}\) are placed together with their transmission axes oriented at an angle \(\theta\) to each other. What is \(\theta\) when only \(25 \%\) of the maximum transmitted light intensity passes through them?

Short Answer

Expert verified
The angle \( \theta \) between the transmission axes of the polarizing sheets when only \(25\%\) of the maximum transmitted light intensity passes through them is \(60^\circ\).

Step by step solution

01

Write Malus' Law formula for transmitted intensity

Malus' Law states that the transmitted intensity I of light through two polarizers is given by the formula: \[ I=I_{0} \cos^2{\theta} \] where: - \(I_0\) is the maximum intensity of the light (when the angle between the polarizers is 0), - θ is the angle between the transmission axes of the polarizers and - I is the transmitted light intensity.
02

Calculate the ratio between the transmitted intensity and the maximum intensity

We are given that 25% of the maximum transmitted light intensity passes through the polarizers. So, we can write the ratio of transmitted intensity (I) to the maximum intensity (\(I_0\)) as follows: \[ \frac{I}{I_0} = 0.25 \]
03

Substitute the intensity ratio into Malus' Law formula and solve for θ

Using the given ratio between the transmitted intensity and the maximum intensity, we can substitute it into the Malus' Law formula: \[ 0.25 = \cos^2{\theta} \] Now, we need to find θ. First, take the square root of both sides of the equation: \[ \sqrt{0.25} = \cos{\theta} \] \[ 0.5 = \cos{\theta}\] Finally, find the inverse cosine (arccos) of 0.5 to calculate θ: \[ \theta = \arccos{0.5} \]
04

Calculate the angle θ

Using a calculator or online tool, we can find the angle θ as follows: \[ \theta = \arccos{0.5} = 60^\circ\] So, the angle between the transmission axes of the polarizers is 60° when only 25% of the maximum transmitted light intensity passes through them.

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