Problem 33 - University Physics with Modern Physics

Question

Unpolarized light of intensity 20.0 \(\mathrm {W/cm}^2\) is incident on two polarizing filters. The axis of the first filter is at an angle of 25.0\(^\circ\) counterclockwise from the vertical (viewed in the direction the light is traveling), and the axis of the second filter is at 62.0\(^\circ\) counterclockwise from the vertical. What is the intensity of the light after it has passed through the second polarizer?

Step-by-Step Solution

Answer

The intensity after the second polarizer is approximately 6.37 W/cm².

Step 1: Calculate Intensity After First Polarizer

Use Malus's Law to determine the intensity of light after it passes through the first polarizer. Since the light is initially unpolarized, the intensity after the first filter is half of the initial intensity:\[ I_1 = \frac{I_0}{2} = \frac{20.0 \, \text{W/cm}^2}{2} = 10.0 \, \text{W/cm}^2 \]

Step 2: Determine Angle Between Polarizers

Find the angle between the transmission axes of the two polarizers. Since the angles are given counterclockwise from the vertical, the angle \(\theta\) between them is:\[ \theta = 62.0^\circ - 25.0^\circ = 37.0^\circ \]

Step 3: Calculate Intensity After Second Polarizer

Apply Malus's Law again to find the intensity after the second polarizer, using the angle between the polarizer axes:\[ I_2 = I_1 \cos^2(\theta) = 10.0 \, \text{W/cm}^2 \cos^2(37.0^\circ) \]First, calculate \(\cos(37.0^\circ)\), which is approximately 0.798, then:\[ I_2 = 10.0 \, \text{W/cm}^2 \times 0.798^2 \approx 10.0 \, \text{W/cm}^2 \times 0.637 = 6.37 \, \text{W/cm}^2 \]

Key Concepts

Malus's Law

Malus's Law is a fundamental principle in the study of optics. It describes how light intensity changes as polarized light passes through a polarizing filter. The law states that the intensity of polarized light after passing through a polarizer is given by the equation:
\[ I = I_0 \cos^2(\theta) \]
Here, \(I\) is the transmitted light intensity, \(I_0\) is the initial light intensity, and \(\theta\) is the angle between the light's polarization direction and the axis of the polarizer.

• This equation highlights the cosine squared dependency, showing that intensity decreases significantly when the light's direction is not aligned with the polarizer's axis.
• For an angle \(\theta = 90^\circ\), the transmitted intensity becomes zero, effectively blocking the light.

Understanding Malus's Law is crucial in applications like photography, where polarizing filters manage reflections and glare.

Unpolarized Light

Unpolarized light consists of waves vibrating in multiple planes. Common sources of light, like the sun or a light bulb, emit this kind of light. It lacks a specific orientation, meaning it vibrates in all possible directions perpendicular to the direction of travel.

• When unpolarized light encounters a polarizing filter, it becomes polarized.
• Only half of the original intensity is transmitted through the first polarizer. This is because the filter aligns the light waves into a single plane.

This transformation using polarized filters is key in reducing glare and enhancing visual clarity in various optical devices.

Polarizing Filters

Polarizing filters are materials designed to convert unpolarized light into polarized light. They work by allowing light waves of a specific polarization to pass through while blocking others.

• The filter's effectiveness depends on the angle at which the light enters the filter relative to its polarization axis.
• In practical use, these filters adjust the intensity and directionality of light passing through them.

These filters are used in photography to enhance image quality by reducing reflections from surfaces like water or glass. They are also used in sunglasses to help reduce glare, making them invaluable in everyday life.