Problem 47 - College Physics

Question

When unpolarized light is incident on a polarizer-analyzer pair, \(30 \%\) of the original light intensity passes the analyzer. What is the angle between the transmission axes of the polarizer and analyzer?

Step-by-Step Solution

Answer

The angle between the transmission axes is approximately \(39.23^\circ\).

Step 1: Understand Malus's Law

Malus's Law states that the intensity of polarized light after passing through a polarizer, or analyzer, is given by \( I = I_0 \cos^2\theta \), where \( I \) is the intensity after the analyzer, \( I_0 \) is the initial intensity (intensity after the polarizer), and \( \theta \) is the angle between the transmission axes of the polarizer and analyzer.

Step 2: Define the Initial Conditions

For unpolarized light passing through a polarizer, the intensity becomes half the original intensity due to the polarizer's action. Therefore, if we start with an intensity \( I_0' \), the intensity after the polarizer \( I_0 \) is \( \frac{1}{2}I_0' \). The intensity after the analyzer is given as \( 30\% \) of the original intensity \( I_0' \), which is \( 0.3 I_0' \).

Step 3: Apply Malus's Law to the System

Using Malus's Law, the intensity after the analyzer is \( I = I_0 \cos^2\theta \). Plugging in the known values, \( 0.3 I_0' = \frac{1}{2} I_0' \cos^2\theta \). Simplifying the equation gives \( \cos^2\theta = \frac{0.3}{0.5} = 0.6 \).

Step 4: Solve for \( \theta \)

Take the square root of both sides to find \( \cos\theta \), which gives \( \cos\theta = \sqrt{0.6} \). To determine \( \theta \), take the arccosine: \( \theta = \cos^{-1}(\sqrt{0.6}) \). This results in \( \theta \approx 39.23^\circ \).

Key Concepts

Malus's Law

Malus's Law is a fundamental principle in optics, specifically dealing with the behavior of polarized light. It describes how the intensity of light changes as it passes through a polarizer-analyzer pair.
This law is mathematically expressed as:

• \( I = I_0 \cos^2 \theta \)
• Where:
  • \( I \) represents the intensity of light after passing through the analyzer,
  • \( I_0 \) is the intensity of light after the polarizer,
  • and \( \theta \) is the angle between the transmission axes of the polarizer and the analyzer.

Malus's Law is crucial for understanding how the direction relative to the polarization affects the resulting intensity.
The law shows that if the transmission axes are perpendicular, no light passes, and as they align, maximum light transmission can be achieved.

Unpolarized Light

Unpolarized light is light that vibrates in multiple planes. Most common sources of light, such as sunlight and light bulbs, emit unpolarized light.
In unpolarized light, the electric field vectors are distributed evenly across all possible directions perpendicular to the direction of the light wave.

• When unpolarized light is passed through a polarizer, it becomes polarized. This means its electric field vibrates in only one plane.
• After passing through the first polarizer, the intensity of this light is reduced by half. This is because any polarization direction is equally probable.

The conversion of unpolarized light to polarized light is a key step in applying Malus's Law to determine subsequent light behavior.
Understanding this enables us to determine how much light intensity will be reduced after passing through a polarizer and further analyzed by an analyzer.

Polarizer-Analyzer Pair

The polarizer-analyzer pair is an arrangement used to study and manipulate light polarization.
This setup consists of two optical devices:

• Polarizer: Acts as the first filter the light encounters. It only allows light vibrating in a specific direction to pass through. The result is polarized light with intensity reduced to half of that of the unpolarized source.
• Analyzer: The second optical device that the polarized light passes through. It further filters the light based on the angle \( \theta \) relative to the polarizer's axis, as described by Malus's Law.

Together, these devices are crucial for experiments and applications involving light polarization.
The alignment or misalignment of their transmission axes directly affects the resulting light intensity, which can be calculated using Malus's Law.
This setup is widely used in scientific research and various technologies such as polarized sunglasses and optical instruments.

Light Intensity

Light intensity refers to the amount of energy that light waves deliver per unit area. In the context of polarization, it changes as light passes through polarizers and analyzers.
Key points about light intensity in this context:

• When unpolarized light first encounters a polarizer, its intensity is reduced by half. This is because only one plane of vibration is allowed to pass through.
• After passing through the analyzer, the intensity is further subjected to the cosine-square law, detailed by Malus's Law.
• The final intensity after the analyzer can be manipulated by changing the angle \( \theta \) between the polarizer and analyzer.

Understanding light intensity in these terms allows us to predict and control how bright or dim the light will be after undergoing polarization.
It is a practical aspect in designing lighting systems and performing precise optical measurements.