Problem 33 - 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 \(\pm 45^\circ\) (or \(\pm \frac{\pi}{4}\) radians).

Step 1: Understanding the Initial Problem

When polarized light passes through a polarizing filter, the intensity of the light that emerges is determined by Malus's Law. According to this law, the transmitted intensity is given by the formula:\[ I = I_0 \cos^2(\theta) \]where \(I_0\) is the initial intensity of the light, and \(\theta\) is the angle between the light's initial polarization direction and the filter's axis.

Step 2: Set Up the Target Intensity

We want to find the angle \(\theta'\) that makes the emerging intensity \(I' = \frac{I}{2}\). We use Malus's Law again for this desired condition:\[ \frac{I_0}{2} = I_0 \cos^2(\theta') \] We can cancel \(I_0\) from both sides since it is non-zero, leading to:\[ \frac{1}{2} = \cos^2(\theta') \]

Step 3: Solve for the New Angle

To find \(\theta'\), we solve the equation:\[ \cos^2(\theta') = \frac{1}{2} \]Taking the square root of both sides gives us:\[ \cos(\theta') = \pm \frac{1}{\sqrt{2}} \]Thus, \(\theta'\) can be either:\[ \theta' = \frac{\pi}{4} + k\pi \text{ or } \theta' = -\frac{\pi}{4} + k\pi \]where \(k\) is any integer, since cosine is periodic with period \(\pi\). However, typically we refer to the principal value for \(\theta'\), which is \(\theta' = \frac{\pi}{4}\) (45 degrees).

Key Concepts

Polarized Light

Light can be made up of waves vibrating in many directions. When these waves are restricted to a single plane, we have polarized light. This type of light travels with waves oscillating in one specific direction, making it different from typical unpolarized light, which vibrates in multiple directions.
Polarization is a common phenomenon in nature. For example, light reflecting off surfaces like water or glass often becomes polarized. Sunglasses use polarizing filters to block this specific plane of light, reducing glare. Understanding polarized light helps us grasp a wide range of optics applications.
In physics, polarizing light helps us explore wave behavior more deeply. It allows the study of phenomena like birefringence or the interaction of light with different materials.

Intensity

Intensity refers to the brightness or power of light traveling through a given area. When dealing with polarized light, intensity helps us understand how much light makes it through the polarizing filter.
The intensity of polarized light passing through a filter is not constant. It depends on the angle between the light's polarization direction and the filter's axis. The formula to express this intensity is defined by Malus's Law:

• \( I = I_0 \cos^2(\theta) \)

where \( I \) is the transmitted intensity, \( I_0 \) is the initial intensity, and \( \theta \) is the angle in question.
Intensity is crucial for applications like photography and vision sciences, as it affects how images are captured and perceived.

Polarizing Filter

A polarizing filter is a device used to block certain orientations of light waves. When unpolarized or partially polarized light passes through a polarizing filter, only the wave components aligned with the filter's axis are transmitted.
Polarizing filters are used in many areas, including:

• Cameras and sunglasses to reduce glare
• Optical instruments to control light
• Scientific experiments to study light properties

The effectiveness of a polarizing filter in blocking non-aligned light is observed in the intensity changes as indicated by Malus's Law. Filters help us analyze and measure polarization in experimental and practical settings.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between angles and sides in triangles. It becomes essential when working with polarized light and other physics concepts involving angles.
In our context, trigonometric functions, like the cosine function, play a key role. They help in calculating changes in light intensity as light passes through a polarizing filter:

• \( \cos(\theta) \) and \( \cos^2(\theta) \) are directly related to light's intensity.

By using trigonometry:

• We quantify the angle-dependency aspects of light behavior.
• We reach solutions that explain how polarizing angles affect light.

Understanding trigonometric relationships allows for precise manipulation and prediction of light behavior in complex optical systems.

Angle of Polarization

The angle of polarization is the angle between the direction of polarized light and the axis of a polarizing filter. This angle significantly influences the amount of light that can pass through the filter.
In practice, knowing this angle helps:

• Predict the behavior of light in experimental setups.
• Adjust equipment to optimize light intensity for specific applications.

For example, the exercise showed how different angles alter the intensity based on Malus's Law:

• When the angle is \( \theta' \), and we want half intensity \( (I/2) \), we solve for this angle using trigonometric equations.

By manipulating the angle of polarization, scientists can finely control light's interaction with materials, leading to innovative breakthroughs in technology and science.