Problem 50 - College Physics

Question

Unpolarized light of intensity \(I_{\mathrm{o}}\) is incident on a polarizer- analyzer pair. (a) If the angle between the polarizer and analyzer increases in the range of \(0^{\circ}\) to \(90^{\circ}\) the transmitted light intensity will (1) also increase, (2) decrease, (3) remain the same. Explain. (b) If the angle between the polarizer and analyzer is \(30^{\circ},\) what light intensity would be transmitted through the polarizer and the analyzer, respectively? (c) What if the angle is \(60^{\circ}\) ?

Step-by-Step Solution

Answer

(1) Decrease; (b) \(\frac{3I_0}{8}\); (c) \(\frac{I_0}{8}\).

Step 1: Understanding Unpolarized Light

Unpolarized light has no specific direction of polarization, meaning its plane of vibration changes randomly. On passing through a polarizer, it becomes polarized with half of its original intensity because the polarizer only allows through the component of light along its axis. Therefore, the intensity of the light after passing through the first polarizer is \( \frac{I_0}{2} \).

Step 2: Light Intensity After Polarizer-Analyzer Pair

The intensity of light transmitted through an analyzer is given by Malus's Law: \( I = I_{ ext{polarizer}} \cos^2 \theta \), where \( \theta \) is the angle between the polarizer and the analyzer. Initial intensity after the polarizer is \( \frac{I_0}{2} \).

Step 3: (a) Changes in Transmitted Light Intensity

Using Malus's Law, with \( \theta \) going from \( 0^{\circ} \) to \( 90^{\circ} \), the transmitted intensity \( I = \frac{I_0}{2} \cos^2 \theta \) will decrease because \( \cos^2 \theta \) decreases from 1 to 0.

Step 4: (b) Intensity at 30 degrees

When \( \theta = 30^{\circ} \), the intensity through the analyzer is \( I = \frac{I_0}{2} \cos^2 30^{\circ} = \frac{I_0}{2} \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3I_0}{8} \).

Step 5: (c) Intensity at 60 degrees

When \( \theta = 60^{\circ} \), the intensity through the analyzer is \( I = \frac{I_0}{2} \cos^2 60^{\circ} = \frac{I_0}{2} \left( \frac{1}{2} \right)^2 = \frac{I_0}{8} \).

Key Concepts

Polarization

Polarization is a fundamental concept in wave optics that refers to the orientation of the oscillations in a light wave. Unpolarized light, such as sunlight or light from an incandescent bulb, consists of light waves vibrating in all possible directions perpendicular to the direction of propagation.

This chaotic vibration means that unpolarized light doesn’t have a single, fixed direction of vibration. However, when light passes through a polarizer, it emerges as polarized light. A polarizer is a special filter that permits only the component of light vibrating in a specific direction. As a result, light exiting the polarizer is polarized along that particular direction.

• Unpolarized light: Light waves vibrating in random directions.
• Polarizer: A device that allows light vibrating in only one direction to pass through.
• Polarized light: Light with waves oscillating in one direction.

Understanding how polarizers work is crucial for technologies such as LCD screens, sunglasses, and photographic lenses.

Malus's Law

Malus's Law is a key principle in understanding the behavior of polarized light as it passes through an analyzer, which is often the second polarizer. Named after Étienne-Louis Malus, the law mathematically describes how the intensity of polarized light changes depending on the angle between the light's initial polarization direction and the axis of the second filter (analyzer).

According to Malus's Law, the intensity of light I that emerges from the analyzer is calculated as:
\[ I = I_{\text{polarizer}} \cos^2 \theta \]where:

• \(I_{\text{polarizer}}\) is the intensity of light after passing through the first polarizer.
• \(\theta\) is the angle between the polarization direction of the polarizer and analyzer.

This equation shows that the transmitted intensity varies based on the cosine of the angle squared. When \(\theta = 0^{\circ}\), the intensity is maximized because \(\cos^2 0^{\circ} = 1\). However, as \(\theta\) approaches \(90^{\circ}\), \(\cos^2 90^{\circ} = 0\), so no light is transmitted.

Polarizer-Analyzer Pair

A polarizer-analyzer pair is a common setup used in optics to study polarized light. The polarizer is the first element that transforms unpolarized light into polarized light. The analyzer, the second element, acts as another filter to further modify the light.

When unpolarized light is incident on the polarizer, half of its intensity is lost because the polarizer only allows light vibrating in its orientation to pass. This is why the intensity after the polarizer is \(\frac{I_0}{2}\), where \(I_0\) is the initial intensity of the unpolarized light.

The analyzer then further reduces the intensity, based on its alignment with the polarized light. This setup illustrates the principles of polarization and is frequently used in experiments to demonstrate and explore polarization concepts.

• Polarizer: Filters unpolarized light to a single orientation.
• Analyzer: Further filters light, dependent on its angle with the polarized light’s path.

Understanding this pair helps in grasping the practical applications of polarized films and screens.

Light Intensity

Light intensity, in the context of polarization, refers to the amount of light energy passing through a surface per unit area after passing through polarizing elements. Intensity is measured as the power of the light per unit area and determines how bright or dim the light appears.

Initially, unpolarized light has an intensity, \(I_0\). Once it passes through a polarizer, its intensity is reduced by half to \(\frac{I_0}{2}\) because the polarizer blocks out half of the light's oscillations, letting through only those in one direction. When this polarized light encounters an analyzer — dependent on the angle \(\theta\) between the polarizer and analyzer — the light intensity is further modified according to Malus's Law:
\[ I = \frac{I_0}{2} \cos^2 \theta \]This equation highlights that as the angular difference \(\theta\) increases, the transmitted light intensity decreases, reaching zero at \(90^{\circ}\). Understanding this relationship is critical for manipulating light in various optical applications, such as adjusting transparency in sunglasses to controlling light in photography and display technologies.