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Chapter 26: Problem 49

Suppose you illuminate two thin slits by monochromatic coherent light in air and find that they produce their first interference minima at \(\pm 35.20^{\circ}\) on either side of the central bright spot. You then immerse these slits in a transparent liquid and illuminate them with the same light. Now you find that the first minima occur at \(\pm 19.46^{\circ}\) instead. What is the index of refraction of this liquid?

Short Answer

Expert verified
The index of refraction of the liquid is approximately 1.73.

Step by step solution

01

Understand the Problem Setup

We have two interference minima of a double-slit experiment with monochromatic light, occurring at specific angles. In air, the first minima occur at \(\pm 35.20^{\circ}\). When the setup is immersed in a liquid, the minima are at \(\pm 19.46^{\circ}\). We need to find the index of refraction of this liquid.
02

Identify the Relevant Equations

The condition for the first interference minimum in a double-slit setup is given by the equation: \(d \sin \theta = \pm \frac{\lambda}{2}\), where \(d\) is the distance between the slits and \(\lambda\) is the wavelength of the light. In a medium, the angle of minimum changes, and the relation becomes \(d \sin \theta' = \pm \frac{\lambda'}{2}\), where \(\lambda' = \frac{\lambda}{n}\) and \(n\) is the index of refraction of the liquid.
03

Setup the Equations for Air and Liquid

For air (index of refraction \(n = 1\)), the equation is: \(d \sin 35.20^{\circ} = \frac{\lambda}{2}\).For the liquid, the equation is:\(d \sin 19.46^{\circ} = \frac{\lambda}{2n}\).
04

Solve for the Index of Refraction

Divide the equation for the liquid by the equation for air to eliminate \(\lambda\) and \(d\): \[ \frac{\sin 19.46^{\circ}}{\sin 35.20^{\circ}} = \frac{1}{n} \]Rearrange to solve for \(n\):\[ n = \frac{\sin 35.20^{\circ}}{\sin 19.46^{\circ}} \].
05

Calculate the Indices

Compute \(\sin 35.20^{\circ}\) and \(\sin 19.46^{\circ}\), then find the index of refraction:The sine values are:\(\sin 35.20^{\circ} \approx 0.5767\)\(\sin 19.46^{\circ} \approx 0.3331\)Thus,\[ n = \frac{0.5767}{0.3331} \approx 1.73 \]

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Key Concepts

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

Double-Slit Experiment
The double-slit experiment is a fascinating physics experiment that demonstrates the wave nature of light and other particles. When light passes through two closely spaced slits, it creates an interference pattern on a screen behind the slits. This pattern consists of alternating bright and dark bands.

In this setup:
  • The slits act as two coherent light sources.
  • The distances between the slits and the screen cause the light waves to overlap.
  • When the waves align perfectly, they create bright spots (constructive interference).
  • When they are out of sync, they create dark spots (destructive interference).
The angle at which these patterns form depends on the wavelength of light and the distance between the slits. In essence, the double-slit experiment beautifully showcases the principle of superposition, which is key to understanding wave behavior.
Interference Minima
Interference minima refer to the positions where destructive interference occurs in a pattern from the double-slit experiment. These are the dark bands in the interference pattern. Their positions are determined by the path difference between the two waves emerging from the slits.

For the first minimum, the condition is given by:
  • Path difference = \(n + \frac{1}{2}\) wavelengths, where \(n\) is an integer (typically starting at zero for the first minimum).
  • The angle \(\theta\) at which the minima occur satisfies \(d \sin \theta = \pm \frac{\lambda}{2}\), where \(d\) is the distance between the slits and \(\lambda\) is the wavelength of the light.
This equation explains why the angle changes when the experiment is conducted in a different medium, as seen in the problem setup with the transparent liquid.
Monochromatic Light
Monochromatic light is light that has a single wavelength and thus a single color. This consistency is essential in the double-slit experiment to create a clear and stable interference pattern.

The characteristics of monochromatic light include:
  • Uniform wavelength: This ensures that the interference patterns produced remain consistent over time.
  • Coherence: The constant phase relationship necessary for creating meaningful patterns.
When using monochromatic light, the interference patterns are easy to observe and calculate because they simplify many variables. This simplification allows for precise experiments, such as when determining the index of refraction of a liquid by observing the angle of interference minima.
Transparent Liquid
A transparent liquid in an optical experiment changes how light behaves due to its refractive properties. Immersing light sources like slits in a liquid affects the angles of interference patterns.

Key effects of a transparent liquid include:
  • Changing of light's speed: Light slows down when entering a liquid, which causes the light's wavelength to shorten within the medium.
  • Inducing refraction: The angle of light shifts according to the medium鈥檚 index of refraction.
Refractive index ( ), a crucial property of transparent liquids, is the ratio of the speed of light in a vacuum to its speed in the medium. It quantifies how much the light will bend. By observing how interference minima shift when a double-slit setup is immersed in a liquid, scientists can deduce the liquid's refractive index.

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Most popular questions from this chapter

Light of wavelength \(631 \mathrm{nm}\) passes through a diffraction grating having 485 lines \(/ \mathrm{mm}\). (a) What is the total number of bright spots (indicating complete constructive interference) that will occur on a large distant screen? Solve this problem without finding the angles. (Hint: What is the largest that \(\sin \theta\) can be? What does this imply for the largest value of \(m ?\) (b) What is the angle of the bright spot farthest from the center?

The lenses of a particular set of binoculars have a coating with index of refraction \(n=1.38,\) and the glass itself has \(n=1.52 .\) If the lenses reflect a wavelength of \(525 \mathrm{nm}\) the most strongly, what is the minimum thickness of the coating?

Two thin parallel slits that are \(0.0116 \mathrm{~mm}\) apart are illuminated by a laser beam of wavelength \(585 \mathrm{nm}\). (a) How many bright fringes are there in the angular range of \(0<\theta<20^{\circ} ?\) (b) How many dark fringes are there in this range?

Coherent light from a sodium-vapor lamp is passed through a filter that blocks everything except for light of a single wavelength. It then falls on two slits separated by \(0.460 \mathrm{~mm}\). In the resulting interference pattern on a screen \(2.20 \mathrm{~m}\) away, adjacent bright fringes are separated by \(2.82 \mathrm{~mm}\). What is the wavelength of the light that falls on the slits?

Two rectangular pieces of plane glass are laid one upon the other on a table. A thin strip of paper is placed between them at one edge, so that a very thin wedge of air is formed. The plates are illuminated at normal incidence by \(546 \mathrm{nm}\) light from a mercury-vapor lamp. Interference fringes are formed, with 15.0 fringes per centimeter. Find the angle of the wedge.
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