Problem 36 - Physics

Question

For each of the three sheets of polarizing material shown in the drawing, the orientation of the transmission axis is labeled relative to the vertical. The incident beam of light is unpolarized and has an intensity of \(1260.0 \mathrm{~W} / \mathrm{m}^{2}\). What is the intensity of the beam transmitted through the three sheets when \(\theta_{1}=19.0^{\circ}, \theta_{2}=55.0^{\circ},\) and \(\theta_{3}=100.0^{\circ} ?\)

Step-by-Step Solution

Answer

The transmitted intensity is approximately 254.84 W/m².

Step 1: Polarization by First Sheet

When unpolarized light passes through the first polarizer, its intensity is reduced by half. This follows from Malus's Law, but for unpolarized light, you apply a factor of 1/2. Thus, the intensity after the first polarizer, \(I_1\), is given by:\[ I_1 = \frac{I_0}{2} = \frac{1260.0 \, \text{W/m}^2}{2} = 630.0 \, \text{W/m}^2. \]

Step 2: Passing Through Second Polarizer

The light intensity after passing through the second polarizer is determined using Malus's Law. Since the angle between the first and second polarizer is \(\theta_{2} - \theta_{1}\), we calculate it as:\[ I_2 = I_1 \cdot \cos^2(\theta_2 - \theta_1) = 630.0 \, \text{W/m}^2 \cdot \cos^2(55.0^\circ - 19.0^\circ). \]Calculate the angle difference: \(\theta_2 - \theta_1 = 36.0^\circ\). Therefore, the intensity is:\[ I_2 = 630.0 \, \text{W/m}^2 \cdot \cos^2(36.0^\circ) \approx 630.0 \, \text{W/m}^2 \cdot 0.8090 = 509.67 \, \text{W/m}^2. \]

Step 3: Passing Through Third Polarizer

For the third polarizer, use Malus's Law with the angle between the second and third polarizer given by \(\theta_{3} - \theta_{2}\):\[ I_3 = I_2 \cdot \cos^2(\theta_3 - \theta_2) = 509.67 \, \text{W/m}^2 \cdot \cos^2(100.0^\circ - 55.0^\circ). \]Calculate the angle difference: \(\theta_3 - \theta_2 = 45.0^\circ\). So, the intensity after the third polarizer is:\[ I_3 = 509.67 \, \text{W/m}^2 \cdot \cos^2(45.0^\circ) \approx 509.67 \, \text{W/m}^2 \cdot 0.5 = 254.835 \, \text{W/m}^2. \]

Key Concepts

Malus's Law

Malus's Law is essential when studying polarized light. It helps us understand how the intensity of light changes as it passes through polarizers. The law states that when light passes through a polarizing filter, the intensity of the transmitted light is proportional to the square of the cosine of the angle between the light polarization direction and the filter's transmission axis. Mathematically, it is expressed as:\[ I = I_0 \cdot \cos^2(\theta) \]where:

• \(I\) is the intensity after passing through the polarizer.
• \(I_0\) is the initial light intensity.
• \(\theta\) is the angle between the light's polarization direction and the polarizer's transmission axis.

Malus's Law is vital in determining light intensity, especially when dealing with multiple polarizers. With multiple polarizing sheets, as in the original problem, calculating the angle between each sheet's transmission axis guides how the light intensity evolves through each step.

Unpolarized Light

Unpolarized light consists of waves vibrating in multiple planes. It is common in natural light sources like the sun. When unpolarized light encounters a polarizing filter, the filter only allows waves aligned to its transmission axis to pass through. This process reduces the light's intensity to half because half of the light's energy aligns with the transmission axis, while the other half does not. This reduction is a crucial starting point in problems involving multiple polarizers. Through the first polarizer, as shown in the exercise, the light's intensity halves, setting the stage for calculations involving subsequent polarizers according to Malus's Law.

Intensity of Light

The intensity of light describes the power carried by light waves per unit area. It is a measure of how much energy the light transfers to a given area in a specific time, expressed in watts per square meter (W/m²). Unpolarized light's intensity halves upon contact with the first polarizer, as previously discussed. Using Malus's Law, we further predict changes in intensity as light traverses additional polarizers. Each polarizer reduces the intensity based on the cosine of the angle from the prior polarizer's orientation. This understanding allows us to predict the final intensity after passing through multiple polarizers, crucial for optical applications in science and engineering.