Problem 46 - College Physics

Question

Unpolarized light is incident on a polarizeranalyzer pair that can have their transmission axes at an angle of either \(30^{\circ}\) or \(45^{\circ} .\) (a) The \(30^{\circ}\) angle will allow (1) more, (2) the same, or (3) less light to go through. (b) Calculate the percentage of light that goes through the polarizer-analyzer pair in terms of the incident light intensity.

Step-by-Step Solution

Answer

(a) The 30° angle allows more light through. (b) At 30°, 75% goes through; at 45°, 50% goes through.

Step 1: Understand the concept of polarized light

Unpolarized light has no specific direction of oscillation, but when it passes through a polarizer, it becomes polarized. The intensity of polarized light transmitted through a polarizer depends on the angle between the light's polarization direction and the polarizer's transmission axis.

Step 2: Use Malus's Law

Malus's Law states that the intensity of light passing through a polarizer is given by \( I = I_0 \cos^2(\theta) \), where \( I_0 \) is the initial intensity of the light and \( \theta \) is the angle between the light’s polarization direction and the transmission axis of the polarizer.

Step 3: Calculate light intensity for 30 degrees

For a transmission axis angle of \( 30^{\circ} \), the intensity of light that gets through is: \[ I = I_0 \cos^2(30^{\circ}) = I_0 \left( \frac{\sqrt{3}}{2} \right)^2. \] This simplifies to \[ I = I_0 \times \frac{3}{4} = 0.75I_0. \]

Step 4: Calculate light intensity for 45 degrees

For a transmission axis angle of \( 45^{\circ} \), the intensity of light that gets through is: \[ I = I_0 \cos^2(45^{\circ}) = I_0 \left( \frac{1}{\sqrt{2}} \right)^2. \] This simplifies to \[ I = I_0 \times \frac{1}{2} = 0.5I_0. \]

Step 5: Compare the outcomes

Compare the results from Step 3 and Step 4. At \( 30^{\circ} \), \( 75\% \) of light intensity passes through, while at \( 45^{\circ} \), only \( 50\% \) does. Thus, more light passes through at \( 30^{\circ} \).

Key Concepts

Malus's Law

Malus's Law is a foundational concept in the study of polarization in optics. Named after Étienne-Louis Malus, this law explains how the intensity of light changes as it passes through a polarizer and encounters an analyzer at a certain angle. The mathematical representation of Malus's Law is given by \( I = I_0 \cos^2(\theta) \), where \( I \) is the intensity of the transmitted light, \( I_0 \) is the initial intensity of the incoming light, and \( \theta \) represents the angle between the light’s polarization direction and the transmission axis of the polarizer.

This relation shows that as the angle \( \theta \) increases, the cosine of the angle decreases, and therefore, the transmitted light intensity also decreases.

• When \( \theta = 0^{\circ} \), all light that passes is aligned, and thus the intensity \( I = I_0 \).
• At \( \theta = 90^{\circ} \), no light is transmitted through the polarizer, making the intensity \( I = 0 \).

Understanding Malus's Law is crucial for any calculations involving polarized light, such as figuring out how much light will pass through a polarizer-analyzer pair, as seen in the original exercise.

Unpolarized Light

Unpolarized light is light that has waves vibrating in multiple planes. In essence, it has no specific direction of oscillation. When light is first emitted from sources like the sun, light bulbs, or flames, it typically starts as unpolarized. This means that the electric field of the light waves is oriented randomly in different directions.

Once unpolarized light passes through a polarizer, it becomes polarized. This is because the polarizer filters out all planes of vibration except for one. The result is light that oscillates in a single plane.

• As light passes through the first polarizer, typically around 50% of the intensity is reduced, assuming the polarizer is ideal.
• The remaining light is now polarized, possessing a specific direction of oscillation.

The transformation from unpolarized to polarized light is vital in understanding and measuring light intensity changes, such as those seen when calculating intensity percentages in exercises, relying on Malus's Law.

Light Intensity

In optics, light intensity refers to the amount of energy that light waves carry per unit area in the direction perpendicular to that area. It is generally measured in watts per square meter (W/m²). Light intensity is a crucial component when analyzing how light behaves through different media, especially in contexts involving polarization.

When dealing with polarization, especially using devices like polarizers and analyzers, analyzing light intensity helps us understand how much light is being transmitted, absorbed, or lost. As seen in the exercise, the intensity of light changes depending on the angle between the polarizer and analyzer.

• At angles like \( 30^{\circ} \), more light intensity is transmitted due to higher cosine squared values.
• Conversely, at angles like \( 45^{\circ} \), less intensity is transmitted because \( \cos^2(45^{\circ}) \) results in a smaller value.

Light intensity is also critical in practical applications, such as photography, LCD technology, and even scientific research, where exact light measurements are necessary.