Problem 27 - University Physics with Modern Physics

Question

Unpolarized light with intensity \(I_{0}\) is incident on two polarizing filters. The axis of the first filter makes an angle of \(60.0^{\circ}\) with the vertical, and the axis of the second filter is horizontal. What is the intensity of the light after it has passed through the second filter?

Step-by-Step Solution

Answer

The intensity after the second filter is \(\frac{3I_0}{8}\).

Step 1: Determine the Intensity After the First Polarizer

When unpolarized light passes through a polarizer, its intensity is reduced by half. The intensity after the first polarizer, which is oriented at \(60.0^{\circ}\) to the vertical, is given by:\[ I_1 = \frac{I_0}{2} \]

Step 2: Apply Malus's Law for the Second Polarizer

The light that exits the first polarizer is polarized at an angle of \(60.0^{\circ}\) with respect to the vertical. When this polarized light passes through the second polarizer, which is oriented horizontally, we use Malus's Law to determine the intensity. Malus's Law is given by:\[ I_2 = I_1 \cos^2(\theta) \]where \(\theta\) is the angle between the light's polarization direction and the axis of the polarizer. Here, \(\theta = 60.0^{\circ} - 90.0^{\circ} = -30.0^{\circ}\).Therefore, the intensity after the second polarizer is:\[ I_2 = \frac{I_0}{2} \cos^2(30.0^{\circ}) \]

Step 3: Calculate Cosine Square and Final Intensity

The cosine of \(30.0^{\circ}\) is \(\frac{\sqrt{3}}{2}\), so:\[ \cos^2(30.0^{\circ}) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]Substitute this value back into the equation for \(I_2\):\[ I_2 = \frac{I_0}{2} \times \frac{3}{4} = \frac{3I_0}{8} \]

Key Concepts

Malus's Law

Malus's Law is a key principle in understanding how light behaves when it passes through polarizing filters.
This law describes how the intensity of polarized light changes as it passes through a second polarizer.

According to Malus's Law, the intensity of light (\(I\)) after passing through a polarizer is given by:
\[ I = I_0 \cos^2(\theta) \]where:

• \(I_0\) is the initial intensity of the polarized light before it encounters the second polarizer.
• \(\theta\) is the angle between the light's current polarization direction and the axis of the polarizing filter.

This law reveals that as the angle \(\theta\) increases towards 90°, less light is allowed through the filter, resulting in reduced intensity.
When \(\theta\) is 0°, the light passes through with maximum intensity.

Malus's Law is crucial for calculating the intensity of light in optical systems and understanding light transmission through multiple polarizers.

Polarizing Filters

Polarizing filters are optical devices that allow light of a specific polarization to pass through while blocking other polarizations.
They are essential in various applications like photography, sunglasses, and scientific experiments.

Here's how they work:

• Unpolarized light, which has waves vibrating in multiple directions, becomes polarized after passing through a filter.
• The filter's axis determines the direction of polarization that will pass through.
• If another polarizing filter is placed in the path, its orientation relative to the first filter affects the amount of light passing through.

Polarizing filters are often used in pairs to control the intensity and directionality of light.
By adjusting the relative angles of the filters, one can modulate the brightness and color of the observed light.
This is particularly useful in reducing glare and enhancing contrast in visual media.

Intensity of Light

The intensity of light is a measure of the energy carried by the light wave per unit area.
It is crucial in determining how much light is detected by our eyes or instruments after passing through various media.

• Unpolarized light has intensity \(I_0\), which is evenly distributed among all polarization directions.
• When light passes through a polarizer, its intensity is reduced to half of the original, given by \(\frac{I_0}{2}\).
• Intensity changes further when light passes through additional polarizers due to Malus's Law, described by \(I = I_0 \cos^2(\theta)\).

Understanding how light intensity changes helps in designing optical systems and is vital in fields like optical engineering and physics.
This knowledge allows us to predict the behavior of light in complex systems and develop better imaging and lighting solutions.