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Chapter 26: Problem 102

A person working on the transmission of a car accidentally drops a bolt into a tray of oil. The oil is \(5.00 \mathrm{~cm}\) deep. The bolt appears to be \(3.40\mathrm{~cm}\) beneath the surface of the oil, when viewed from directly above. What is the index of refraction of the oil?

Short Answer

Expert verified
The index of refraction of the oil is approximately 1.47.

Step by step solution

01

Understand the Concept

The observation is based on the apparent depth seen from above a medium due to the refraction of light. The apparent depth is the depth at which the object seems to be, and the real depth is the actual depth of the object.
02

Identify Given Values

The actual depth of the bolt is given as the depth of oil, which is 5.00 cm. The apparent depth is given as 3.40 cm.
03

Use the Refraction Formula

The formula relating apparent depth (_\(d_a\)_) and real depth (_\(d_r\)_) is:\[ n = \frac{d_r}{d_a} \]where \(n\) is the index of refraction of the oil.
04

Plug in the Values

Substitute the given values into the formula:\[ n = \frac{5.00 \text{ cm}}{3.40 \text{ cm}} \]
05

Calculate the Index of Refraction

Calculate the value:\[ n = \frac{5.00}{3.40} \approx 1.47 \]
06

Conclusion

The index of refraction of the oil is approximately 1.47.

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Key Concepts

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

Apparent Depth
When you look at an object submerged in a liquid from above, it often appears closer to you than it actually is. This phenomenon is known as apparent depth. Apparent depth occurs because the light rays bend as they exit the medium, such as water or oil, into the air. This bending causes your eyes to trace the light rays back along a straight line, creating the illusion that the object is at a shallower level.
  • Apparent depth is less than or equal to the real depth.
  • It changes depending on the medium through which the light travels.
Understanding apparent depth is crucial in fields such as photography or underwater navigation, where accurate perception of distance is essential.
Real Depth
Real depth refers to the actual physical distance between the surface of a medium and an object submerged within it. Unlike apparent depth, real depth does not change based on the observer's viewpoint or the medium's properties.

In the context of our exercise, the real depth of the bolt in the oil is 5.00 cm. Despite what our eyes perceive, this is the true measurement of how deep the bolt sits within the oil. This concept is used for correcting visual perceptions in various scientific and practical applications.
Light Refraction
Light refraction is the bending of the path of light as it passes from one medium into another. This change in direction occurs due to a change in the speed of light in different materials. When light moves from air into a denser medium like oil, it slows down and bends towards the normal.
  • Refraction is responsible for various optical phenomena, including apparent depth.
  • It affects how we perceive the location and size of objects in different mediums.
Understanding light refraction helps us comprehend and predict how light behaves when interacting with various materials.
Optics
Optics is a branch of physics that deals with the study of light and its interactions with different materials. It covers various phenomena such as reflection, refraction, and diffraction.

In our case, optics explains why the bolt appears to be at a different depth due to the refraction of light in oil. Optics combines principles from both physics and engineering to design instruments that improve our vision, like glasses and microscopes, by manipulating light flows and angles.
  • Optics helps us understand how we visually perceive the world.
  • It is crucial for developing various technologies, from simple lenses to complex imaging devices.
The study of optics illuminates how we can improve tools we use for seeing and interacting with our environment.
Snell's Law
Snell's Law is an essential formula in optics, describing how light refracts when passing through the border between two different mediums. It is expressed as: \[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]Where:
  • \( n_1 \) and \( n_2 \) are the indices of refraction of the two media.
  • \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively.
In our exercise, Snell's Law facilitates finding the index of refraction by relating apparent and real depths.

By understanding Snell's Law, you gain insight into how angles and refraction indices work together to create everyday visual experiences. This core principle is instrumental to designers and researchers working to develop new optical devices and systems.

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Most popular questions from this chapter

An object has an angular size of 0.0150 rad when placed at the near point \((21.0 \mathrm{~cm})\) of an eye. When the eye views this object using a magnifying glass, the largest possible angular size of the image is 0.0380 rad. What is the focal length of the magnifying glass?

(a) For astronomical telescopes that have large angular magnifications, which lens has the greater focal length, the objective or the eyepiece? (b) How is the length \(L\) of the telescope related to the focal length \(f_{0}\) of the objective and the focal length \(f_{c}\) of the eyepiece? (c) Three astronomical telescopes have different lengths \(L\), such that \(L_{\mathrm{A}}

A camper is trying to start a fire by focusing sunlight onto a piece of paper. The diameter of the sun is \(1.39 \times 10^{9} \mathrm{~m},\) and its mean distance from the earth is \(1.50 \times 10^{11} \mathrm{~m} .\) The camper is using a converging lens whose focal length is \(10.0 \mathrm{~cm}\) (a) What is the area of the sun's image on the paper? (b) If \(0.530 \mathrm{~W}\) of sunlight pass through the lens, what is the intensity of the sunlight at the paper?

Violet light and red light travel through air and strike a block of plastic at the same angle of incidence. The angle of refraction is \(30.400^{\circ}\) for the violet light and \(31.200^{\circ}\) for the red light. The index of refraction for violet light in plastic is greater than that for red light by \(0.0400 .\) Delaying any rounding off of calculations until the very end, find the index of refraction for violet light in plastic.

An object is placed to the left of a lens, and a real image is formed to the right of the lens. The image is inverted relative to the object and is onehalf the size of the object. (a) What kind of lens, converging or diverging, is used to produce this image? (b) How is the height \(h_{\mathrm{i}}\) of the image related to the height \(h_{0}\) of the object? Don't forget to take into account the fact that the image is inverted relative to the object. (c) What is the ratio \(d_{i} / d_{0}\) of the image distance to the object distance? For the situation described in the Concept Questions, the distance between the object and the image is \(90.0 \mathrm{~cm}\). (a) How far from the lens is the object? (b) What is the focal length of the lens?
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