Problem 26 - Algebra 2

Question

OPTICS You may have polarized sunglasses that eliminate glare by polarizing the light. When light is polarized, all of the wayes are traveling in parallel planes. Suppose horizontally-polarized light with intensity \(I_{0}\) strikes a polarizing filter with its axis at an angle of \(\theta\) with the horizontal. The intensity of the transmitted light \(I_{l}\) and \(\theta\) are related by the equation \(\cos \theta=\sqrt{\frac{1}{I_{0}}} \cdot\) If one fourth of the polarized light is transmitted through the lens, what angle does the transmission axis of the filter make with the horizontal?

Step-by-Step Solution

Answer

The angle \( \theta \) is 60 degrees.

Step 1: Understanding the Problem

We need to find the angle \( \theta \) at which the intensity of the transmitted light is one fourth of the initial intensity \( I_0 \). The equation relating \( \theta \) to the intensities is given as \( \cos \theta = \sqrt{\frac{I_l}{I_0}} \). Here, \( I_l = \frac{1}{4}I_0 \).

Step 2: Substitute and Simplify

Substituting \( I_l = \frac{1}{4}I_0 \) into the equation \( \cos \theta = \sqrt{\frac{I_l}{I_0}} \), we get: \[ \cos \theta = \sqrt{\frac{\frac{1}{4} I_0}{I_0}} \]. Simplifying inside the square root gives \( \cos \theta = \sqrt{\frac{1}{4}} \).

Step 3: Calculate the Trigonometric Function

Evaluate \( \sqrt{\frac{1}{4}} \), which simplifies to \( \sqrt{\frac{1}{4}} = \frac{1}{2} \). Therefore, \( \cos \theta = \frac{1}{2} \).

Step 4: Find the Angle \( \theta \)

We know \( \cos \theta = \frac{1}{2} \). The cosine of \( 60^\circ \) is \( \frac{1}{2} \), so \( \theta = 60^\circ \). Hence, the angle that the transmission axis of the filter makes with the horizontal is \( 60^\circ \).

Key Concepts

Intensity of Light

The intensity of light refers to the amount of energy a light wave carries per unit of time and area. It determines the brightness of the light that we observe.
Key factors influencing light intensity include:

• The power of the light source.
• The distance from the source.
• Characteristics of the medium through which it travels.

When discussing polarized light, intensity is particularly important because it changes when passing through materials like polarizing filters. Polaroid sunglasses use this principle to reduce glare by controlling the amount of light intensity that reaches our eyes.
In problems involving polarization, we compare the initial intensity of light (\( I_0 \)) with the transmitted intensity (\( I_l \)) after passing through a filter. The relationship between these intensities often involves trigonometric functions, as certain wave orientations allow more light to pass through.

Polarizing Filter

A polarizing filter is a special optical filter that allows light waves of a particular polarization to pass through while blocking others.
Here's how it works:

• Light, even from a single source, is not always polarized. The waves may vibrate in multiple directions.
• A polarizing filter has a specific direction called the transmission axis. Only light vibrating along this axis passes through.
• The intensity of the transmitted light depends on the angle between the incoming light's polarization and the filter's transmission axis.

When polarized light enters a filter, the resulting light intensity can be calculated using Malus's Law: \( I_l = I_0 \cos^2(\theta) \), where \( \theta \) is the angle between the light's initial polarization direction and the filter's axis.
If you find that only one-fourth of the light is transmitted, this implies a specific alignment, such as in this problem, where one-fourth transmission corresponds to a \( 60^{\circ} \) angle.

Trigonometric Functions

Trigonometric functions are a set of relationships in mathematics that involve angles and sides of triangles. They are crucial in describing periodic phenomena, including wave behavior such as light.
Key trigonometric functions include:

• \( \sin(\theta) \): describes the ratio of the opposite side to the hypotenuse in a right triangle.
• \( \cos(\theta) \): represents the ratio of the adjacent side to the hypotenuse.
• \( \tan(\theta) \): the ratio of the opposite side to the adjacent side.

In polarizing filters, \( \cos(\theta) \) becomes particularly important.
For example, when light with an intensity \( I_0 \) is passing through a filter at an angle \( \theta \), the transmitted intensity is determined by \( \cos(\theta) \).
In this context, we often utilize the equation \( \cos(\theta) = \frac{1}{2} \) to solve for the angle \( \theta \), leading us to recognize familiar angles such as \( 60^{\circ} \), important for determining light behavior after passing through a polarizing lens.