Problem 57 - College Physics

Question

\(\bullet\) Unpolarized light with intensity \(I_{0}\) is incident on an ideal polarizing filter. The emerging light strikes a second ideal polarizing filter whose axis is at \(41.0^{\circ}\) to that of the first. Determine (a) the intensity of the beam after it has passed through the second polarizer and (b) its state of polarization.

Step-by-Step Solution

Answer

Intensity: \(0.28485 I_0\); Polarized along second filter axis.

Step 1: Understand Polarization through Filters

When unpolarized light with intensity \( I_0 \) passes through the first polarizing filter, its intensity becomes \( I_1 = \frac{I_0}{2} \). This occurs because the first filter only allows light polarized along its axis to pass through, which is half of the total intensity of unpolarized light.

Step 2: Apply Malus's Law

After passing through the first polarizer, the light is polarized and has intensity \( I_1 = \frac{I_0}{2} \). When this polarized light passes through the second filter, which is at a \( 41.0^{\circ} \) angle to the first, the intensity of the light is given by Malus's Law: \[ I_2 = I_1 \cos^2(41.0^{\circ}) \] Substitute \( I_1 = \frac{I_0}{2} \) into the equation to get \[ I_2 = \frac{I_0}{2} \cos^2(41.0^{\circ}) \].

Step 3: Calculate Final Intensity

Calculate \( \cos(41.0^{\circ}) \) to find the emerging intensity:\[ \cos(41.0^{\circ}) \approx 0.7547 \] Thus, \[ I_2 = \frac{I_0}{2} (0.7547)^2 \approx \frac{I_0}{2} \times 0.5697 \approx 0.28485 I_0 \]. This is the intensity of the light after passing through both polarizers.

Step 4: Determine State of Polarization

The light that emerges after passing through the second polarizing filter is linearly polarized with its polarization direction aligned to the axis of the second polarizing filter.

Key Concepts

Polarization

Polarization is a fascinating phenomenon in the field of optics. It refers to the orientation of light waves in a specific direction. When light is unpolarized, its light waves vibrate in all possible directions perpendicular to the direction of propagation.

However, once light passes through a polarizing filter, only the waves aligned with the filter's axis can pass through, leading to polarized light. This transformation means the light now oscillates in a single plane. Different filters can control this effect and can be used in various practical applications such as sunglasses to reduce glare or in photography to enhance contrast and saturation.

It's important to note that some materials can naturally polarize light, while others need mechanical or optical aids, like polarizing filters, to achieve polarization. Understanding this concept is crucial for appreciating how devices manipulate light in technological and scientific applications.

Malus's Law

Malus's Law is a critical principle when studying polarized light and its behavior. This law allows us to determine how much of the polarized light will pass through a second polarizing filter.

Named after Étienne-Louis Malus, Malus's Law states that the intensity of light ( I) passing through a polarizer is proportional to the cosine squared of the angle ( \(\theta\)) between the light's initial polarization direction and the axis of the polarizer. The formula is given by:

\[ I = I_0 \cos^2(\theta) \]

Here, \(I_0\) is the intensity of the polarized light before entering the second filter. Malus's Law helps explain behaviors such as why rotating one polarizing filter in relation to another changes the intensity of light passing through them.

In practical applications, this law is incredibly useful for adjusting light intensity in optical devices and systems. Understanding Malus's Law can provide insights into how light interacts with materials and can be manipulated for various technological purposes.

Polarizing Filters

Polarizing filters are essential tools in controlling the properties of light. They work by only allowing light waves oscillating in a specific direction to pass through. This specific direction is known as the filter's axis.

When unpolarized light hits a polarizing filter, it emerges polarized. The filter essentially "selects" light that vibrates parallel to its axis, allowing only 50% of the incoming light through when it is initially unpolarized.

Further control comes with using multiple polarizing filters. For example, in this exercise, a second filter is placed at an angle to the first. The result is a reduction in intensity according to Malus's Law.

Polarizing filters are widely used beyond theoretical exercises too. In photography, they enhance image quality by reducing reflections and glare, and in optical devices, they regulate light passage, enhancing clarity and contrast.

Overall, these filters provide a powerful means of manipulating light's behavior in both everyday and scientific contexts, helping us manage how we see and use light in numerous applications.