91Ó°ÊÓ

A circular drop of oil lies on a smooth, horizontal surface. The drop is thickest in the center and tapers to zero thickness at the edge. When illuminated from above by blue light \((\lambda=455 \mathrm{nm}), 56\) concentric bright rings are visible, including a bright fringe at the edge of the drop. In addition, there is a bright spot in the center of the drop. When the drop is illuminated from above by red light \((\lambda=637 \mathrm{nm}),\) a bright spot again appears at the center, along with a different number of bright rings. Ignoring the bright spot, how many bright rings appear in red light? Assume that the index of refraction of the oil is the same for both wavelengths. The ability to exhibit interference effects is a fundamental characteristic of any kind of wave. Our understanding of these effects depends on the principle of linear superposition, which we first encountered in Chapter 17\. Only by means of this principle can we understand the constructive and destructive interference of light waves that lie at the heart of every topic in this chapter. Problem 67 serves as a review of the essence of this principle. Problem 68 deals with thin-film interference and reviews the factors that must be considered in such cases.

Step by step solution

01

Introduction to the Problem

The problem involves thin-film interference in an oil drop with different wavelengths of light. We need to find the number of bright rings produced when illuminated with red light, given the information for blue light and the index of refraction is constant.

02

Identify the Principle

The appearance of bright rings is due to constructive interference. For a film, like the oil drop, bright rings occur when the optical path difference (OPD) is a multiple of the wavelength.

03

Formulate the Condition for Bright Rings

The constructive interference condition is given by: \(2nt = m\lambda\), where \(t\) is the thickness, \(n\) is the index of refraction, \(m\) is an integer, and \(\lambda\) is the wavelength of light used.

04

Apply Blue Light Condition

For blue light with \(\lambda = 455 \ nm\), and 56 rings including the one at the edge and one at the center, the largest order of interference can be written as \(m_{blue} = 55\).

05

Apply Red Light Condition with Same Optical Conditions

With the same thickness \(t\), and assuming the same \(n\), determine the largest integer \(m_{red}\) using the red light wavelength \(\lambda = 637 \ nm\). This uses the same thickness and optical condition: \(2nt = m_{red}(637)\).

06

Calculate Maximum Order of Interference for Red Light

We relate the number of bright rings for both wavelengths: \(\frac{m_{blue}\cdot 455}{m_{red}\cdot 637} =1\). Solving for \(m_{red}\), we find that \(m_{red} = \frac{455}{637} \times 55\approx 39.26\). Hence the integer value is \(m_{red} = 39\).

07

Conclude with the Number of Bright Rings in Red Light

Since the order \(m_{red} = 39\) corresponds to 39 bright rings excluding the central spot, thus, there are 39 rings for red light.

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Key Concepts

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

Constructive Interference

When light waves meet under specific conditions, they combine in a process known as constructive interference. This occurs when the waves are in phase, meaning their peaks and troughs align perfectly.

• In thin-film interference, like with the oil drop, this is how we see bright fringes.
• The condition for constructive interference is met when the optical path difference (OPD) equals a whole number multiple of the light's wavelength.

This condition ensures that the waves reinforce each other instead of cancelling out. For bright rings in our oil drop, whenever this interference condition is satisfied, a bright ring appears.
Remember, it is crucial for the waves to travel through the film and bounce back such that their journey adds up to a full wavelength or multiple of it.

Optical Path Difference

Optical Path Difference (OPD) is foundational in understanding thin-film interference.
As light waves travel through different mediums, their speed changes based on the medium's properties.

• The OPD is essentially the extra distance a wave travels in one medium compared to another.
• Mathematically, for our problem, OPD in the film is expressed as: \(2nt\), where \(n\) is the index of refraction and \(t\) is the thickness of the film.

For constructive interference and thus bright rings, the OPD needs to be a multiple of the wavelength \(\lambda\), i.e., \(2nt = m\lambda\), where \(m\) is an integer.
Understanding OPD helps predict how light will interfere and thus explains the patterns we see at different wavelengths.

Index of Refraction

The index of refraction is a critical value that describes how much light slows down when entering a new medium.
Think of it as the medium's fingerprint for light speed. In our exercise:

• The index of refraction is consistent for both light wavelengths (blue and red) in the oil drop.
• It is denoted as \(n\), and higher \(n\) means greater bending and slowing of light.

This consistency allows us to directly compare how different wavelengths behave in the same medium. For example, in calculating the number of bright rings, knowing \(n\) helped determine the optical path changes. Since the index remains the same for both the red and blue light, their interaction with the film depends on their wavelengths, leading to the differing number of rings apparent in either case.