Problem 31 - University Physics with Modern Physics

Question

A beam of polarized light passes through a polarizing filter. When the angle between the polarizing axis of the filter and the direction of polarization of the light is \(\theta\), the intensity of the emerging beam is \(I\). If you now want the intensity to be \(I/2\), what should be the angle (in terms of \(\theta\)) between the polarizing angle of the filter and the original direction of polarization of the light?

Step-by-Step Solution

Answer

The angle should be \(\theta \pm 45^\circ\).

Step 1: Understand Malus's Law

Malus's Law states that when a beam of polarized light with initial intensity \(I_0\) passes through a polarizing filter, the intensity \(I\) of the emerging light is given by \(I = I_0 \cos^2 \theta\), where \(\theta\) is the angle between the light's initial polarization direction and the filter's polarizing axis.

Step 2: Set Up the Equation for Half Intensity

We want the emerging intensity to be half of \(I\), which is \(I/2\). According to Malus's Law, this means \(\frac{I}{2} = I_0 \cos^2 \theta'\), where \(\theta'\) is the new angle between the light's initial polarization direction and the filter's polarizing axis.

Step 3: Analyze the Relationship

Given that \(I = I_0 \cos^2 \theta\), substitute the expression for \(I\) into the half-intensity equation: \(\frac{I_0 \cos^2 \theta}{2} = I_0 \cos^2 \theta'\). Divide both sides by \(I_0\) to eliminate it: \(\frac{\cos^2 \theta}{2} = \cos^2 \theta'\).

Step 4: Solve for the New Angle

Take the square root of both sides: \(\frac{\cos \theta}{\sqrt{2}} = \cos \theta'\). This gives \(\cos \theta' = \frac{1}{\sqrt{2}} \cos \theta\).

Step 5: Determine the Required Angle

The value \(\cos \theta' = \frac{1}{\sqrt{2}} \cos \theta\) implies that \(\theta' = \theta + 45^\circ\) or \(\theta' = \theta - 45^\circ\) because \(\cos 45^\circ = \frac{1}{\sqrt{2}}\).

Key Concepts

Polarized Light

Polarized light is a fascinating phenomenon where light waves vibrate mostly in a single plane. Typically, natural light consists of waves vibrating in multiple planes, but when it's polarized, this randomness is reduced. Understanding this helps explain how certain sunglasses work, or why the sky appears to change in color under different conditions.

Polarization can occur through reflection, refraction, or by passing light through a specific material called a polarizer. This changes how light behaves and is often used in various optical devices.

• Reduces glare - useful in optics and photography.
• Improves contrast and visibility.
• Used in LCD screens and 3D movies.

Recognizing polarized light is crucial when dealing with polarizing filters, as they interact to control the intensity of light that passes through them.

Polarizing Filter

A polarizing filter, also known as a polarizer, is a tool designed to filter light waves, allowing only waves aligned with its axis to pass through. It's like a gatekeeper for light, controlling which waves proceed and which do not, based on their alignment.

These filters are used in various applications, like cameras to reduce reflections or sunglasses to cut glare. Their function is based on the principle that light consists of waves oscillating in different directions.

• Blocks unwanted light directions.
• Enhances colors and contrast in images.
• Used in polarization microscopy and in managing light pollution.

By understanding how polarizing filters work, you can manipulate the light's intensity, which is particularly useful in scientific and practical applications.

Intensity of Light

The intensity of light refers to the amount of energy a light wave carries. It's closely tied with brightness or how we perceive the strength of light. When polarized light passes through a polarizing filter, its intensity can change depending on the angle between the light's polarization and the filter's axis.

When studying light intensity, Malus's Law comes into play. It describes how the intensity of polarized light changes as it passes through a polarizer.

• Intensity without a polarizer is the highest possible.
• Post-filter, intensity depends on the cosine of the angle, squared.
• Helps determine how bright the transmitted light will be.

In many practical applications, controlling the intensity of light is crucial, whether it's to enhance images or to adjust lighting levels.

Angle of Polarization

The angle of polarization is essential in determining how much light passes through a polarizing filter. When a beam of polarized light hits a polarizer, the angle between the light's initial direction and the filter's polarizing axis plays a critical role.

Malus's Law provides a mathematical model for this relationship, establishing how the light intensity changes depending on this angle. The law states that the intensity is proportional to the square of the cosine of the angle.\[I = I_0 \cos^2 \theta\]where:

• \(I\) is the intensity of transmitted light,
• \(I_0\) is the initial intensity,
• \(\theta\) is the angle between the light's polarization direction and the polarizer's axis.

When aiming to adjust the intensity, understanding this angle becomes invaluable in fields like photography and optics.