Problem 63 - College Physics

Question

\(\bullet$$\bullet\) 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 instead 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 \( \theta - 45^\circ \).

Step 1: Understanding the concept

The intensity of light passing through a polarizing filter is described by Malus's Law, which states that the transmitted intensity \( I \) is given by \( I = I_0 \cos^2(\theta) \), where \( I_0 \) is the initial intensity of the beam, and \( \theta \) is the angle between the light's polarization direction and the filter's axis.

Step 2: Setting up the equation

We want the emerging intensity to be half of the initial intensity that already passed through the polarizer. Therefore, we set \( I = \frac{1}{2} I \) in the equation \( I = I_0 \cos^2(\theta) \), where \( I \) is the intensity given in the original problem for angle \( \theta \). This gives us \( \frac{1}{2} I = I_0 \cos^2(\phi) \), where \( \phi \) is the new angle required.

Step 3: Solving for the angle

Since \( I = I_0 \cos^2(\theta) \), we substitute it into \( \cos^2(\phi) \) equation: \( \frac{1}{2} I_0 \cos^2(\theta) = I_0 \cos^2(\phi) \). This simplifies to: \( \frac{1}{2} \cos^2(\theta) = \cos^2(\phi) \). We take the square root of both sides to get \( \cos(\phi) = \frac{1}{\sqrt{2}} \cos(\theta) \).

Step 4: Final calculation

Further simplifying \( \cos(\phi) = \frac{1}{\sqrt{2}} \cos(\theta) \), let's use the property of cosines. The solution for this requires rearranging and possibly recognizing standard angle values. Due to the trigonometric identity, \( \cos(\phi) = \cos(\theta - 45^\circ) \), implies \( \phi = \theta - 45^\circ \).

Key Concepts

Polarized Light

Polarized light is a type of light wave in which all the waves oscillate in the same direction. Normally, light waves vibrate in all directions perpendicular to the direction of their travel. This multi-directional vibration is called unpolarized light, like sunlight or light from a standard bulb. Polarized light, however, has waves that vibrate parallel to a single direction or plane.
The process of creating polarized light involves filtering out certain wave directions. There are various natural phenomena, like reflections or passing through certain transparent materials, that can cause light to become polarized. Polarized sunglasses are an everyday example that uses this concept to reduce glare by blocking certain orientations of light waves.

Polarizing Filter

A polarizing filter is a device used to transform unpolarized or partially polarized light into fully polarized light. It works by blocking waves of light that are not aligned with the filter's axis. Think of it as a gate that only allows certain vibrations of light to pass through.
When light passes through a polarizing filter, only the component of the light wave that aligns with the filter's axis is transmitted. This concept is crucial in various optical devices, and it mimics how polarized sunglasses lessen glare by filtering horizontally polarized light, which is typically reflected off surfaces like water or roads.

Light Intensity

Light intensity refers to the amount of light energy reaching a surface per unit area and is often measured in lumens or watts per square meter. In the context of polarized light, intensity can be mathematically altered using Malus’s Law.
Malus’s Law states that the intensity of polarized light passing through a polarizing filter is given by \( I = I_0 \cos^2(\theta) \), where \( I_0 \) is the original intensity of the light, and \( \theta \) is the angle between the polarization direction of the incoming light and the axis of the polarizer.
This relationship shows how the angle of incidence can significantly decrease the intensity of the light beam, as seen when the filter angle is adjusted to let half the original intensity pass through.

Angle of Polarization

The angle of polarization is the tilt relative to a reference direction or axis at which the light's electric field vector is oscillating. It is pivotal in the application of Malus's Law, which involves adjusting this angle to control the light's intensity.
When polarized light passes through a filter, the angle between the light's polarization and the filter's axis becomes crucial. As per Malus's Law, if you want the light's intensity to be halved, this requires setting the angle of polarization such that \( \phi \) becomes \( \theta - 45^\circ \). In this arrangement, the cosine of this new angle squared (\( \cos^2(\phi) \)) becomes half of the original, achieving the desired reduction in intensity.