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Chapter 8: Problem 25

Two ideal linear sheet polarizers are arranged with respect to the vertical with their transmission axis at \(10^{\circ}\) and \(60^{\circ},\) respectively. If a linearly polarized beam of light with its electric ficld at \(40^{\circ}\) enters the first polarizer, what fraction of its irradiance will emerge?

Short Answer

Expert verified
Approximately 61.9% of the irradiance emerges.

Step by step solution

01

Determine the initial intensity

The initial intensity of the light beam is denoted by \(I_0\). As the beam is linearly polarized, the electric field of the light initially makes an angle of \(40^{\circ}\) with the vertical.
02

Calculate irradiance after first polarizer

Use Malus's law, which says that the intensity \(I\) after passing through a polarizer is \(I = I_0 \cos^2(\theta)\). Here, \(\theta = 40^{\circ} - 10^{\circ} = 30^{\circ}\). Thus, the intensity after the first polarizer is \(I_1 = I_0 \cos^2(30^{\circ})\). Since \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\), we have \(I_1 = I_0 \left(\frac{\sqrt{3}}{2}\right)^2 = I_0 \frac{3}{4}\).
03

Calculate irradiance after second polarizer

The angle between the transmission axis of the first polarizer and the second polarizer is \(60^{\circ} - 10^{\circ} = 50^{\circ}\). Apply Malus's law again: \(I_2 = I_1 \cos^2(50^{\circ})\). We use \(\cos(50^{\circ}) \approx 0.643\), so \(I_2 = I_0 \frac{3}{4} \times (0.643)^2 \approx I_0 \times 0.619\).
04

Express final fraction of irradiance

The fraction of the original light's irradiance that emerges is the ratio \(\frac{I_2}{I_0} = 0.619\). Thus, approximately 61.9% of the original light's irradiance emerges from both polarizers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Malus's Law
Malus's Law is named after the French physicist 脡tienne-Louis Malus. It provides a crucial way to determine how light intensity changes as it passes through polarizing materials.
This law states that the intensity of light (\( I \)) after passing through a polarizer is given by:\[ I = I_0 \cos^2(\theta) \]
  • \( I_0 \) is the initial intensity of the light.
  • \( \theta \) is the angle between the light's initial polarization direction and the axis of the polarizer.

In practical terms, Malus's Law helps us understand how much of the initial light can pass through a polarizer based on its alignment. This insight is critical when dealing with multiple polarizers, as you have to apply the law successively for each polarizer encountered by the light.
Intensity of Light
The intensity of light refers to the power transferred per unit area along the direction of propagation. It's a clear indicator of how 'strong' or 'weak' light is in a given area.
In the context of polarization, intensity changes as light interacts with polarizers.
Some key points about light intensity include:
  • Measured in watts per square meter (\( W/m^2 \)).
  • Dependent on both the amplitude and frequency of the light wave.

In polarization exercises, we often begin with some initial intensity (\( I_0 \)) and use Malus's Law to find how much remains after light has passed through polarizers. It's all about the angles; if the light is perfectly aligned with the polarizer, maximum light passes through. If they're perpendicular, minimal (or zero) light intensity emerges.
Linear Polarizers
Linear polarizers are devices that only allow light waves oscillating in a particular direction to pass through. They filter out waves vibrating in all other directions.
This direction is called the transmission axis. As the light beam passes through the polarizer, its intensity and quality change, often reduced according to Malus's Law.
  • For a polarizer aligned with the light's polarization, intensity is maximally preserved.
  • If misaligned, the light's properties are significantly altered.

In our example with two polarizers, the change in light's intensity due to their angled positioning demonstrates how effective these devices are at controlling light properties, either for scientific applications or everyday technologies like sunglasses and camera filters.
Irradiance
Irradiance is essentially the intensity of light but with a focus on how it affects a surface. It's the amount of light power hitting an area of one square meter.
This parameter is vital in fields such as energy technology, meteorology, and health sciences. Expressed mathematically as:\[ E = \frac{P}{A} \]where:
  • \( E \) is the irradiance.
  • \( P \) is the light power striking the surface.
  • \( A \) is the area of the surface receiving the light.

When using irradiance in polarization problems, you're often interested in how the initial light power diminishes after passing through polarizers. The fraction of original irradiance remaining after processing through polarizers provides practical insights into energy propagation and light manipulation.

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Most popular questions from this chapter

Light reflected from a glass \(\left(n_{g}=1.65\right)\) plate immersed in ethyl alcohol \(\left(n_{e}=1.36\right)\) is found to be completely linearly polarized. At what angle will the partially polarized beam be transmitted into the plate?

Two sheets of \(H N-38 S\) linear polarizer are in series one behind the other with their transmission axes aligned. The first is illuminated by \(1000 \mathrm{W} / \mathrm{m}^{2}\) of natural light. Determine the approximate emerging irradiance. What is the value of the resulting transmittance of the pair?

An ideal polarizer is rotated at a rate \(\omega\) between a similar pair of stationary crossed polarizers. Show that the emergent flux density will be modulated at four times the rotational frequency. In other words, show that $$I=\frac{I_{1}}{8}(1-\cos 4 \omega t)$$ where \(I_{1}\) is the flux density emerging from the first polarizer and \(I\) is the final flux density.

Given that \(200 \mathrm{W} / \mathrm{m}^{2}\) of randomly polarized light is incident normally on a stack of ideal linear polarizers that are positioned one behind the other with the transmission axis of the first vertical, the second at \(30^{\circ}\). the third at \(60^{\circ}\), and the fourth at \(90^{\circ}\). How much light emerges?

The light from an ordinary flashlight is passed through a linear polarizer with its transmission axis vertical. The resulting beam, having an irradiance of \(200 \mathrm{W} / \mathrm{m}^{2}\), is incident normally on a vertical \(H N-50\) linear polarizer whose transmission axis is tilted at \(30^{\circ}\) above the horizontal. How much light is transmitted?
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