Physics 53 Polarization (4 of 5) Ninety Polarizers

Michel van Biezen

29 Apr 201304:55

EducationalLearning

32 Likes 10 Comments

TLDRIn this lecture, the presenter explores a fascinating problem involving 91 polarizers arranged sequentially with each subsequent polarizer rotated one degree from the previous one, starting with a vertical orientation. The goal is to calculate the intensity of light that passes through the final, perpendicular polarizer. The explanation walks through the process of using Malus's Law, demonstrating how the intensity diminishes with each polarizer due to the cosine squared of the angle between them. The final calculation reveals that a significant portion of light intensity is lost after the first polarizer, with minimal loss through the remaining 90, resulting in a small but notable overall reduction in light intensity.

Takeaways

🌟 The lecture discusses a problem involving a series of 90 polarizers, each rotated by one degree from the vertical orientation of the first polarizer.
🔬 The first polarizer reduces the light intensity to half of its initial value, represented as \( I_1 = \frac{1}{2} I_0 \).
📏 Each subsequent polarizer further reduces the light intensity by the cosine squared of the angle difference from the previous polarizer.
🔄 The process of calculating the light intensity after each polarizer is iterative, applying the formula \( I_{n+1} = I_n \times \cos^2(1^\circ) \).
📉 The intensity after passing through the 90th polarizer is significantly less than the initial due to the cumulative effect of the angle differences.
🧮 The final intensity is calculated by raising the factor \( 0.999695 \) to the 90th power, which represents the cumulative effect of the 90-degree rotation.
📉 The majority of the light intensity loss occurs at the first polarizer, with only a small percentage lost through the subsequent 90 polarizers.
✅ The final formula for the light intensity after all 91 polarizers is \( I_{91} = I_0 \times 0.473 / 2 \).
📊 The script emphasizes the importance of using a calculator for such complex calculations, which would be impractical with a slide rule.
🤓 The problem demonstrates the principles of light polarization and the effects of multiple polarizers on light intensity.
🔚 The script concludes by highlighting that only 48.6% of the initial light intensity remains after passing through 91 polarizers.

Q & A

What is the purpose of the lecture?
-The purpose of the lecture is to discuss an interesting problem involving polarizers and to calculate the intensity of light passing through a series of 91 polarizers, each rotated by one degree relative to the previous one.
What is the initial orientation of the first polarizer in the series?
-The first polarizer is oriented vertically.
How are the subsequent polarizers oriented relative to the first one?
-Each subsequent polarizer is oriented one degree further than the previous one, creating a one-degree difference between each polarizer.
What is the total number of polarizers in the setup?
-There are a total of 91 polarizers, including the first one.
What is the initial intensity of light, denoted as I sub 0, in the problem?
-The initial intensity of light, I sub 0, is not specified in the script, but it is the reference intensity from which all calculations are based.
How is the intensity of light after the first polarizer, I sub 1, related to I sub 0?
-The intensity of light after the first polarizer, I sub 1, is equal to half of I sub 0, which is 1/2 I sub 0.
What mathematical operation is used to calculate the intensity after each subsequent polarizer?
-The intensity after each subsequent polarizer is calculated by multiplying the previous intensity by the cosine squared of the angle difference between the polarizers.
What is the angle difference between each polarizer in the series?
-The angle difference between each polarizer is 1 degree.
What is the final intensity of light after passing through all 91 polarizers, denoted as I sub 91?
-The final intensity of light, I sub 91, is calculated as I sub 0 times 0.99969 raised to the 90th power, and then divided by 2.
What percentage of the initial light intensity is lost after passing through the first polarizer?
-After passing through the first polarizer, 50% of the initial light intensity is lost.
How much additional intensity is lost as the light passes through the remaining 90 polarizers?
-An additional 3% of the initial intensity is lost as the light passes through the remaining 90 polarizers.
What is the overall intensity loss through the entire setup of 91 polarizers?
-The overall intensity loss through the entire setup is approximately 51.4%, with half of that loss occurring at the first polarizer and the remaining 3% through the subsequent 90 polarizers.

Outlines

00:00

🔍 Understanding Polarizers: An Introductory Problem

In this section, the presenter introduces a complex problem related to polarizers. The problem involves 91 polarizers, with the first one being vertical and each subsequent polarizer rotated 1 degree further than the previous one. The goal is to determine the intensity of light that makes it through the last polarizer. The speaker sets up the scenario by explaining the arrangement and orientation of the polarizers, leading to a total of 90 polarizers positioned such that the last one is perpendicular to the first.

📉 Calculating Light Intensity Through Polarizers

The process of calculating light intensity as it passes through multiple polarizers is explained here. Initially, light passes through the first polarizer, reducing its intensity by half, described as I₁ = ½I₀. The intensity of light after passing through each subsequent polarizer is calculated using the formula Iₙ = Iₙ₋₁ × cos²(θ), where θ is the angle difference between the polarizers. The presenter calculates the cosine of 1 degree, which is approximately 0.999695, and uses this value to determine how light intensity changes incrementally through each polarizer.

🔄 Iterative Intensity Calculations

Continuing the calculation, the speaker shows the iterative process of determining the light intensity after passing through each polarizer. By applying the formula I₃ = I₂ × cos²(1°), and subsequently, I₄ = I₃ × cos²(1°), the pattern is established. The formula's repetitive nature reveals a consistent reduction in light intensity. Each iteration involves multiplying by 0.999695, indicating a small but cumulative effect as light passes through each additional polarizer.

🔗 Connecting the Intensity Calculations

Here, the speaker ties the individual calculations together, explaining how each intensity step relates to the initial intensity. The equation for I₃ is reformulated as I₃ = I₁ × (0.999695)², connecting back to the starting intensity, I₁. This pattern continues such that I₄ = I₁ × (0.999695)³, demonstrating a cumulative multiplication effect that ties each stage of light intensity reduction to the first polarizer's output.

🧮 Solving the 90 Polarizer Challenge

The speaker addresses the challenge of calculating light intensity after passing through all 90 polarizers. The intensity equation I₉₁ = I₁ × (0.999695)⁹⁰ is derived, representing the cumulative effect of passing through each polarizer. The importance of modern calculators is highlighted, as they allow for precise computation of such complex equations, a task that would have been daunting with older tools like slide rules.

💡 Final Intensity and Key Observations

The solution is reached by calculating the final intensity, I₉₁, using I₁ = ½I₀, resulting in I₉₁ = 0.486 × I₀. This result illustrates that the majority of light intensity is lost at the first polarizer, with only a small fraction (3%) lost as light passes through the subsequent 90 polarizers. The speaker emphasizes how understanding the principles of polarizers can lead to insights into the nature of light and its interaction with materials, showcasing the importance of the problem-solving approach presented.

Mindmap

Keywords

💡Polarisers

Polarisers are optical filters that allow light waves to pass through them in a specific orientation, blocking light waves that vibrate in other directions. In the video's theme, they are used to demonstrate how light intensity decreases as it passes through multiple polarisers oriented at different angles. The script describes a setup with 90 polarisers, each rotated one degree from the previous one, to illustrate the cumulative effect on light intensity.

💡Intensity of Light

Intensity of light refers to the amount of light energy passing through a unit area. It is a central concept in the video, as the problem presented involves calculating the intensity of light after it has passed through a series of polarisers. The script explains how the initial intensity is halved by the first polariser and then further reduced by subsequent polarisers.

💡Cosine Squared

The term 'cosine squared' is used in the context of calculating the intensity of light after it has passed through each polariser. It is derived from the Malus's law, which states that the intensity of transmitted light through a polariser is proportional to the square of the cosine of the angle between the polariser's transmission axis and the light's plane of vibration. The script uses cosine squared to calculate the decrease in light intensity at each stage.

💡Malus's Law

Malus's Law is a fundamental principle in optics that describes the interaction between polarised light and a polariser. It is implicitly used in the script to explain how the intensity of light is reduced as it passes through each polariser at an angle. The law is mathematically represented as I = I0 * cos²θ, where I0 is the initial intensity, θ is the angle of the polariser, and I is the transmitted intensity.

💡Degrees

In the script, 'degrees' are used to measure the angle of rotation between each successive polariser. The problem setup involves 90 polarisers, each rotated one degree from the vertical orientation of the first polariser. This incremental rotation is crucial for understanding the cumulative effect on light intensity as it passes through the series of polarisers.

💡Transmitted Intensity

Transmitted intensity refers to the intensity of light that has passed through a medium or an optical component, such as a polariser. In the video, the transmitted intensity is calculated at each stage as the light passes through multiple polarisers, with the script demonstrating how this intensity decreases progressively.

💡Optical Filters

Optical filters, such as polarisers, are used to selectively transmit, block, or modify light based on its properties. In the context of the video, polarisers are a type of optical filter that only allows light vibrating in a certain direction to pass through, which is essential for the problem discussed regarding the decrease in light intensity.

💡Plane of Vibration

The plane of vibration refers to the plane in which the electric field of a light wave oscillates. In the script, the plane of vibration is important when discussing how light interacts with polarisers, as only light vibrating in the plane parallel to the polariser's axis will pass through.

💡Cumulative Effect

The cumulative effect in the video refers to the total impact of multiple polarisers on the intensity of light. As light passes through each polariser, its intensity is reduced according to Malus's Law, and the script calculates the overall decrease in intensity after 90 polarisers.

💡Calculators

Calculators are mentioned in the script as a tool for performing the complex calculations required to determine the final intensity of light after passing through 90 polarisers. The mention of calculators highlights the practical aspect of solving the problem, contrasting with the older method of using a slide rule.

💡Slide Rule

A slide rule is an obsolete analog computing instrument used for multiplication, division, and other calculations. In the script, it is mentioned to emphasize the difficulty of performing the calculations required for the problem without modern computational tools like calculators or computers.

Highlights

Introduction to an interesting problem involving polarizers.

Description of a setup with 91 polarizers, each rotated by one degree relative to the previous one.

The first polarizer is vertical, and the setup aims to measure the intensity of light after passing through all 91 polarizers.

Explanation of the initial light intensity, I_1, being half of the initial intensity, I_0.

Calculation of I_2 using the cosine squared of the angle difference between polarizers.

Demonstration of how to calculate the light intensity after passing through each subsequent polarizer.

The process of iteratively calculating the light intensity through multiple polarizers.

The mathematical formula used to calculate the intensity after the 90th polarizer.

The significance of the small angle approximation used in the calculations.

The cumulative effect of the small angle differences on the final light intensity.

The final calculation of I_{91} as I_0 times 0.473.

Discussion on the majority of light intensity loss occurring at the first polarizer.

The minimal loss of light intensity through the remaining 90 polarizers.

The practical application of understanding light polarization and its intensity after passing through multiple polarizers.

The importance of using calculators for complex calculations in physics problems.

The conclusion that only a small percentage of the initial light intensity is lost after passing through 90 polarizers.

The overall process of solving a physics problem involving polarizers and light intensity.

Transcripts

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Physics 53 Polarization (4 of 5) Ninety Polarizers

Takeaways

Q & A

What is the purpose of the lecture?

What is the initial orientation of the first polarizer in the series?

How are the subsequent polarizers oriented relative to the first one?

What is the total number of polarizers in the setup?

What is the initial intensity of light, denoted as I sub 0, in the problem?

How is the intensity of light after the first polarizer, I sub 1, related to I sub 0?

What mathematical operation is used to calculate the intensity after each subsequent polarizer?

What is the angle difference between each polarizer in the series?

What is the final intensity of light after passing through all 91 polarizers, denoted as I sub 91?

What percentage of the initial light intensity is lost after passing through the first polarizer?

How much additional intensity is lost as the light passes through the remaining 90 polarizers?

What is the overall intensity loss through the entire setup of 91 polarizers?