Problem 51 - University Physics with Modern Physics

Question

A 10.0 -mW vertically polarized laser beam passes through a polarizer whose polarizing angle is \(30.0^{\circ}\) from the horizontal. What is the power of the laser beam when it emerges from the polarizer?

Step-by-Step Solution

Answer

Answer: The power of the laser beam when it emerges from the polarizer is 7.50 mW.

Step 1: Understand Malus' Law

Malus' law states that the intensity of polarized light transmitted through a polarizer is reduced by a factor of the square of the cosine of the angle between the plane of polarization of the incident light and the polarizing axis of the polarizer. The equation is given as: \( I_{transmitted} = I_{incident} * \cos^{2}{\theta} \) Here, \(I_{transmitted}\) is the transmitted intensity of light, \(I_{incident}\) is the incident intensity of light, and \(\theta\) is the angle between the initial polarization plane and the polarizing axis of the polarizer.

Step 2: Convert given power and angle units

The given power of the laser beam is 10.0 mW. We need to convert it to watts before plugging it into the equation. Also, the angle given is in degrees, and we need to have it in radians as \(\cos\) function takes radians. Power in watts: \( I_{incident} = 10.0 \cdot 10^{-3} W\) Convert angle from degrees to radians: \( \theta = 30 \cdot \frac{\pi}{180} \)

Step 3: Apply Malus' Law and calculate transmitted power

Now we can apply Malus' law to find the transmitted power. \( I_{transmitted} = I_{incident} * \cos^{2}{\theta} \) \( I_{transmitted} = (10.0 \cdot 10^{-3}) * \cos^{2}\left(30 \cdot \frac{\pi}{180}\right) \)

Step 4: Calculate final transmitted power

After evaluating the above expression, we have: \( I_{transmitted} = (10.0 \cdot 10^{-3}) * (0.866) ^{2} \) \( I_{transmitted} = 7.50 \cdot 10^{-3} W \) So, the power of the laser beam when it emerges from the polarizer is 7.50 mW.

Key Concepts

Malus' Law

Malus' Law is a principle used to determine the intensity of polarized light after it passes through a polarizer. It helps explain how the orientation of light waves impacts their transmission. According to Malus' Law:

• The transmitted intensity (\( I_{transmitted} \)) is dependent on the initial intensity of the light (\( I_{incident} \)).
• The law uses the cosine squared of the angle (\( \theta \)) between the light's polarization and the polarizer's axis.
• The equation is: \[ I_{transmitted} = I_{incident} \cdot \cos^2{\theta} \]

This relationship demonstrates that as the angle increases, the transmitted light intensity decreases. Malus' Law is pivotal in optics, explaining behaviors like light diminishing through sunglasses or camera filters.

Laser Beam

A laser beam is a highly focused and coherent stream of light. Lasers are unique because they emit light of a specific wavelength, making them indispensable in a variety of applications such as

• communications
• medicine
• engineering

In our exercise, we dealt with a vertically polarized laser beam. This means all light waves oscillate in a single direction. Before passing through the polarizer, this laser has an intensity measured in milliwatts (mW). When using a laser, understanding its intensity and polarization is crucial, as these factors determine how the light behaves when interacting with materials or passing through filters.

Polarizing Angle

The polarizing angle is crucial in determining how much light is transmitted through a polarizer. It is the angle between the initial polarization direction of the light and the axis of the polarizer.

• In our example, the angle was given as \(30.0^\circ\) from the horizontal.
• This angle needs to be converted into radians (the standard unit for trigonometric functions in physics) using the formula: \( \theta = 30 \cdot \frac{\pi}{180} \).

The polarizing angle directly affects the amount of light that will be transmitted according to Malus' Law. Understanding how to measure and use this angle is crucial for accurately predicting light transmission.

Transmitted Intensity

Transmitted intensity refers to the amount of light remaining after passing through a polarizer. In our exercise, we started with an incident laser power of 10.0 mW. To find the transmitted intensity using Malus' Law:

• Convert the initial intensity to watts: \( I_{incident} = 10.0 \times 10^{-3} \text{ W} \).
• Use the polarizing angle (converted to radians) to calculate the cosine squared value: \( \cos^2{\left(30 \cdot \frac{\pi}{180}\right)} \approx 0.866^2 \).
• Multiply these values to find the transmitted intensity: \( I_{transmitted} = 10.0 \times 10^{-3} \times 0.866^2 \approx 7.50 \times 10^{-3} \text{ W} \).

Thereby, the transmitted intensity shows how much power exits the polarizer, essential for understanding energy conservation in optical systems.