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Chapter 26: Problem 58

Light is refracted as it travels from a point A in medium 1 to a point \(\mathrm{B}\) in medium 2 . If the index of refraction is 1.33 in medium 1 and 1.51 in medium 2 , how long does it take light to go from \(A\) to \(B\), assuming it travels \(331 \mathrm{cm}\) in medium 1 and \(151 \mathrm{cm}\) in medium \(2 ?\)

Short Answer

Expert verified
Calculate total time by summing times computed using speed = c/n for each medium.

Step by step solution

01

Understand the Problem

We need to find the total time taken by light to travel from point A to point B, passing through two different media. The indices of refraction and the distances in each medium are given.
02

Recall the Formula for Speed of Light in a Medium

Use the formula for the speed of light in a medium, which is given by \( v = \frac{c}{n} \), where \( c \) is the speed of light in vacuum \( (3 \times 10^8 \text{ m/s}) \) and \( n \) is the refractive index of the medium.
03

Calculate Speed of Light in Medium 1

Plug in the given refractive index for medium 1: \( n_1 = 1.33 \). Therefore, the speed of light in medium 1 is \( v_1 = \frac{3 \times 10^8}{1.33} \text{ m/s} \).
04

Calculate Speed of Light in Medium 2

Using the refractive index for medium 2: \( n_2 = 1.51 \), the speed of light in medium 2 is \( v_2 = \frac{3 \times 10^8}{1.51} \text{ m/s} \).
05

Calculate Time Taken in Medium 1

The time taken to travel a distance in a medium is given by \( t = \frac{d}{v} \). For medium 1: \( d_1 = 331 \text{ cm} = 3.31 \text{ m} \) and \( v_1 = \frac{3 \times 10^8}{1.33} \), so \( t_1 = \frac{3.31}{v_1} \).
06

Calculate Time Taken in Medium 2

Similarly for medium 2: \( d_2 = 151 \text{ cm} = 1.51 \text{ m} \) and \( v_2 = \frac{3 \times 10^8}{1.51} \), so \( t_2 = \frac{1.51}{v_2} \).
07

Calculate Total Time

The total time taken \( t_{total} \) is the sum of \( t_1 \) and \( t_2 \). So, \( t_{total} = t_1 + t_2 \).
08

Substitute and Compute

Substitute the values from the previous steps into \( t_1 \) and \( t_2 \), compute each time, and then add them to find \( t_{total} \). \[ t_1 = \frac{3.31}{\frac{3 \times 10^8}{1.33}} \] \[ t_2 = \frac{1.51}{\frac{3 \times 10^8}{1.51}} \] Calculate these and sum them up for \( t_{total} \).
09

Conclusion

The total time taken for light to travel from A to B is approximately calculated from the above steps.

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Key Concepts

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

Speed of Light
The speed of light is a fundamental concept in the study of physics. Light travels incredibly fast, achieving its maximum speed in a vacuum, which is roughly \(3 \times 10^8 \text{ m/s}\). This speed is significantly higher than anything encountered in our everyday experiences. However, as light enters a different medium, such as water or glass, its speed changes.
  • This change in speed is influenced by the medium's refractive index.
  • The greater the refractive index, the slower light travels through that medium.
Understanding how light slows down is essential for calculations in optics, which can affect everything from calculating travel time through different substances to designing lenses.
Refractive Index
The refractive index, represented by the symbol \(n\), is a measure that describes how much a medium can bend, or refract, light. Every material has its unique refractive index, with vacuum having the lowest, at exactly 1. This property directly influences the speed of light in any material.
  • If we know the refractive index, we can determine how much slower light travels in comparison to a vacuum.
  • The formula \( v = \frac{c}{n} \) allows us to calculate the speed of light, \(v\), within the medium where \(c\) is the speed of light in a vacuum.
For example, water has a refractive index of about 1.33, which means light travels slower through water compared to air. Similarly, in the given exercise, medium 1 with its index of 1.33 and medium 2 with 1.51, allow us to determine precisely the effect on light's speed.
Time Calculation
Calculating the time taken for light to travel through different media is crucial in understanding real-world applications like lenses, glasses, and even time estimation in underwater activities. The formula \( t = \frac{d}{v} \) is used to determine the time, \(t\), it takes light to travel a certain distance, \(d\), where \(v\) is the speed of light in the medium.
  • To find the time taken in different media, we first need to compute the speed of light in each, using the medium's refractive index.
  • By substituting the known values of distance and speed, we can solve for the time spent in each section.
In our exercise, after determining the speeds for each medium using their refractive indices, we computed the time it took for light to travel the given distances. Summing these times gives us the total time for light's journey from point A to point B.

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Most popular questions from this chapter

An object with a height of \(2.54 \mathrm{cm}\) is placed \(36.3 \mathrm{mm}\) to the left of a lens with a focal length of \(35.0 \mathrm{mm}\). (a) Where is the image located? (b) What is the height of the image?

An object is located to the left of a convex lens whose focal length is \(34 \mathrm{cm}\). The magnification produced by the lens is \(m=3.0 .\) (a) To increase the magnification to \(4.0,\) should the object be moved closer to the lens or farther away? Explain. (b) Calculate the distance through which the object should be moved.

Standing \(2.0 \mathrm{m}\) in front of a small vertical mirror you see the reflection of your belt buckle, which is \(0.70 \mathrm{m}\) below your eyes. If you remain \(2.0 \mathrm{m}\) from the mirror but climb onto a a stool, how high must the stool be to allow you to see your knees in the mirror? Assume that your knees are \(1.2 \mathrm{m}\) below your eyes.

During a daytime football game you notice that a player's reflective helmet forms an image of the Sun \(4.8 \mathrm{cm}\) behind the surface of the helmet. What is the radius of curvature of the helmet, assuming it to be roughly spherical?

Two thin lenses, with focal lengths \(f_{1}\) and \(f_{2}\), are placed in contact. What is the effective focal length of the double lens?
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