Problem 35 - University Physics with Modern Physics

Question

Unpolarized light of intensity 20.0 \(\mathrm{W} / \mathrm{cm}^{2}\) 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 6.38 W/cm².

Step 1: Use Malus's Law for the First Polarizer

When unpolarized light passes through the first polarizer, the intensity is reduced to half of its initial value. For unpolarized light, the intensity after the first polarizer, \( I_1 \), can be calculated as:\[ I_1 = \frac{I_0}{2} \]where \( I_0 = 20.0 \, \mathrm{W/cm^2} \). Thus, the intensity after the first polarizer is:\[ I_1 = \frac{20.0}{2} = 10.0 \, \mathrm{W/cm^2} \]

Step 2: Determine Relative Angle Between Polarizers

To find the relative angle between the first and second polarizers, subtract the angle of the first polarizer from the angle of the second polarizer:\[ \theta = 62.0^{\circ} - 25.0^{\circ} = 37.0^{\circ} \]

Step 3: Apply Malus's Law for the Second Polarizer

Malus's Law states that the intensity of light passing through a polarizer is given by the equation:\[ I_2 = I_1 \cdot \cos^2(\theta) \]Using the intensity from the first polarizer \( I_1 = 10.0 \, \mathrm{W/cm^2} \) and the angle \( \theta = 37.0^{\circ} \), we can calculate the final intensity as:\[ I_2 = 10.0 \cdot \cos^2(37.0^{\circ}) \]Since \( \cos(37.0^{\circ}) \approx 0.7986 \),\[ \cos^2(37.0^{\circ}) \approx 0.638 \]Thus,\[ I_2 = 10.0 \cdot 0.638 = 6.38 \, \mathrm{W/cm^2} \]

Key Concepts

Malus's Law

Malus's Law is a fundamental principle in optics, and it describes how light intensity changes as it passes through polarizing filters. This law is significant when analyzing how polarized light behaves. Malus's Law states that the intensity of polarized light after passing through a second polarizer is given by the equation: \[ I = I_0 \cdot \cos^2(\theta) \] Here, \( I \) is the intensity of the transmitted light, \( I_0 \) is the initial light intensity after the first polarizer, and \( \theta \) is the angle between the light's initial polarization direction and the axis of the polarizer. Mild Simplifications - When the light first passes through a polarizer, its intensity is halved. - As it goes through a second polarizer, use the angle between the polarizers to find the intensity after this filter. In the context of our exercise, Malus's Law helps us determine the intensity of the light first at an angle of 25 degrees, and then after it passes through the second polarizer, which is situated at a 62-degree angle.

Unpolarized Light

Unpolarized light consists of light waves vibrating in multiple planes. This is typical of light emitted from common sources such as sunlight, light bulbs, and many other conventional light sources. When such light encounters a polarizing filter, its intensity reduces by half since only vibrations aligned with the filter's axis pass through. Key Points to Understand - Unpolarized light is a mix of light waves oriented in different directions. - A polarizer, when exposed to unpolarized light, allows half the light intensity to pass through. In our specific exercise, the unpolarized light of intensity 20.0 \( \mathrm{W/cm^2} \) encounters the first polarizer, reducing its intensity to 10.0 \( \mathrm{W/cm^2} \). Understanding this initial reduction is crucial as it sets the basis for further intensity calculations using Malus's Law for subsequent polarizers.

Polarizing Filters

Polarizing filters are optical devices that remove certain orientations of light vibration. These filters are made of materials that only allow light waves oscillating in one specific direction to pass through. This characteristic makes them invaluable in reducing glare and in adjusting light intensity in various optical applications. Highlights about Polarizing Filters - Polarizing filters convert unpolarized light into polarized light by blocking specific orientations of vibrations. - The effectiveness of a polarizer in blocking light depends on its angle relative to the light's polarization direction. In the considered exercise, the role of two polarizing filters is evident. The first reduces the intensity of the unpolarized light, and the second further modulates the light using Malus's Law. The relative angle between the polarizers (37 degrees in this case) quantitatively informs us about the reduction in light intensity as it proceeds through the system of polarizers.