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A thin slit illuminated by light of frequency \(f\) produces its first dark band at \(\pm\)38.2\(^\circ\) in air. When the entire apparatus (slit, screen, and space in between) is immersed in an unknown transparent liquid, the slit's first dark bands occur instead at \(\pm\)21.6\(^\circ\). Find the refractive index of the liquid.

Short Answer

Expert verified
The refractive index of the liquid is approximately 1.672.

Step by step solution

01

Understanding the Problem

We are given that the first dark band is caused by diffraction and it occurs at specific angles in air and in an unknown liquid. We need to find the refractive index of this liquid. The angles given are for the first dark band, which is a characteristic of single-slit diffraction.
02

Apply Diffraction Formula

The formula for the angle of the first dark band in single-slit diffraction is given by \( a \sin \theta = m \lambda \), where \(a\) is the slit width, \(\theta\) is the angle of the dark band, \(m\) is the order of minimum, and \(\lambda\) is the wavelength. For the first minimum, \(m = 1\). We use this formula to find relations between the angles in air and in the liquid.
03

Find Wavelength in the Liquid

Given that the angle in air is \(\theta_{air} = 38.2^\circ\), so we have \( a \sin 38.2^\circ = \lambda \). In the liquid, \(\theta_{liquid} = 21.6^\circ\), so using the same formula \( a \sin 21.6^\circ = \lambda' \). The wavelength in the liquid \(\lambda'\) relates to the wavelength in air \(\lambda\) by \( \lambda' = \lambda/n \), where \(n\) is the refractive index of the liquid.
04

Calculate the Refractive Index

From both conditions, we know \( a = \frac{\lambda}{\sin 38.2^\circ} = \frac{\lambda'}{\sin 21.6^\circ} \). Substitute \(\lambda' = \lambda/n\) into the second equation, so \( \frac{\lambda}{n \cdot \sin 21.6^\circ} = \frac{\lambda}{\sin 38.2^\circ} \). Cancel \( \lambda \) from both sides and solve for \( n \): \( n = \frac{\sin 38.2^\circ}{\sin 21.6^\circ} \).
05

Perform the Calculation

Calculate \(n\) using the sine values: \( n = \frac{\sin 38.2^\circ}{\sin 21.6^\circ} \approx \frac{0.6157}{0.3681} \approx 1.672 \).

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Key Concepts

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

Single-Slit Diffraction
Single-slit diffraction is an important phenomenon in wave optics. It occurs when waves, such as light, pass through a narrow opening and then spread out, creating a pattern of light and dark areas on a screen. The pattern results from constructive and destructive interference of the waves.

In a single-slit setup, the dark bands or minima represent positions where waves cancel out due to destructive interference. This results in less light being observed at these angles. The first minimum is typically observed at a specific angle depending on the wavelength of the light and the width of the slit through which the light passes. The angle at which the first dark band occurs is crucial in determining various properties of the light and the medium it passes through.

One can use this knowledge to determine other factors such as the wavelength of light in different mediums by observing how the angles of the dark bands shift when the setup is moved from one medium to another, as seen in the exercise problem.
Diffraction Formula
The diffraction formula for single-slit diffraction helps us find the angles at which these dark bands appear. The formula is given by:

\[ a \sin \theta = m \lambda \]
where:
  • \(a\) is the width of the slit,
  • \(\theta\) is the angle of the dark band,
  • \(m\) is the order of the minimum (for the first dark band, \(m = 1\)), and
  • \(\lambda\) is the wavelength of the light.
This equation allows us to solve for the angle, slit width, or wavelength, depending on the known quantities. In an experimental setup, if the angle and slit width are known, it becomes possible to determine the wavelength of the light.

In different mediums, such as air and a liquid, the light's wavelength changes, which results in a shift in the angles where these minima occur. This principle is employed in the exercise, where the angle shift is used to find the refractive index of a liquid.
Wavelength in Liquid
The wavelength of light changes when it passes from one medium to another. This is due to the refractive index of the medium, which is a measure of how much the speed of light is reduced inside the medium compared to vacuum.

In an unknown liquid, the wavelength is typically shorter than in air, leading to different diffraction patterns. The relationship between the wavelength in air \(\lambda\) and in a liquid \(\lambda'\) is described by the equation:

\[ \lambda' = \frac{\lambda}{n} \]
where \(n\) is the refractive index of the liquid.

By measuring the angles of diffraction in both air and the liquid, and using the known refractive index of air (which is approximately 1), one can solve for the refractive index of the liquid using:

\[ n = \frac{\sin \theta_{air}}{\sin \theta_{liquid}} \]
This calculation provides a practical way to determine the properties of an unknown transparent medium by using its effect on the behavior of light passing through it.

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Most popular questions from this chapter

A slit \(0.240 \mathrm{~mm}\) wide is illuminated by parallel light rays of wavelength \(540 \mathrm{nm} .\) The diffraction pattern is observed on a screen that is \(3.00 \mathrm{~m}\) from the slit. The intensity at the center of the central maximum \(\left(\theta=0^{\circ}\right)\) is \(6.00 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\). (a) What is the distance on the screen from the center of the central maximum to the first minimum? (b) What is the intensity at a point on the screen midway between the center of the central maximum and the first minimum?

Light of wavelength 585 nm falls on a slit 0.0666 mm wide. (a) On a very large and distant screen, how many \(totally\) dark fringes (indicating complete cancellation) will there be, including both sides of the central bright spot? Solve this problem \(without\) calculating all the angles! (\(Hint\): What is the largest that sin \theta can be? What does this tell you is the largest that \(m\) can be?) (b) At what angle will the dark fringe that is most distant from the central bright fringe occur?

Monochromatic light from a distant source is incident on a slit 0.750 mm wide. On a screen 2.00 m away, the distance from the central maximum of the diffraction pattern to the first minimum is measured to be 1.35 mm. Calculate the wavelength of the light.

The maximum resolution of the eye depends on the diameter of the opening of the pupil (a diffraction effect) and the size of the retinal cells. The size of the retinal cells (about 5.0 \(\mu\)m in diameter) limits the size of an object at the near point (25 cm) of the eye to a height of about 50 \(\mu\)m. (To get a reasonable estimate without having to go through complicated calculations, we shall ignore the effect of the fluid in the eye.) (a) Given that the diameter of the human pupil is about 2.0 mm, does the Rayleigh criterion allow us to resolve a 50-\(\mu\)m- tall object at 25 cm from the eye with light of wavelength 550 nm? (b) According to the Rayleigh criterion, what is the shortest object we could resolve at the 25-cm near point with light of wavelength 550 nm? (c) What angle would the object in part (b) subtend at the eye? Express your answer in minutes (60 min = 1\(^\circ\)), and compare it with the experimental value of about 1 min. (d) Which effect is more important in limiting the resolution of our eyes: diffraction or the size of the retinal cells?

(a) What is the wavelength of light that is deviated in the first order through an angle of 13.5\(^\circ\) by a transmission grating having 5000 slits/cm? (b) What is the second-order deviation of this wavelength? Assume normal incidence.

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