Ch 17: Superposition

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 17: Superposition

Problem 64

Chapter 17, Problem 64

Microwaves with a frequency of 10.5 GHz are aimed downward into a flat-bottomed beaker that contains sunflower oil. A microwave detector above the beaker finds that there are strong reflections when the oil depth is 2.76 cm and 3.68 cm but at no depths in between. What is the index of refraction of sunflower oil at microwave frequencies?

Verified step by step guidance

Step 1: Understand the problem. The strong reflections occur due to constructive interference of microwaves reflected from the top and bottom surfaces of the sunflower oil. This happens when the path difference between the reflected waves is an integer multiple of the wavelength in the oil.

Step 2: Write the condition for constructive interference. The path difference is twice the depth of the oil (since the wave travels down and back up). For constructive interference, this path difference must equal an integer multiple of the wavelength in the oil:

2d = mλ_o

, where

d

is the depth,

m

is an integer, and

λ_o

is the wavelength in the oil.

Step 3: Relate the wavelength in the oil to the wavelength in vacuum. The wavelength in the oil is given by

λ_o = λ/n

, where

λ

is the wavelength in vacuum and

n

is the index of refraction of the oil. The wavelength in vacuum can be calculated using the frequency of the microwaves:

λ = c/f

, where

c

is the speed of light and

f

is the frequency.

Step 4: Use the given depths to find the difference in constructive interference orders. The depths 2.76 cm and 3.68 cm correspond to consecutive constructive interference orders, so the difference in path lengths is equal to one wavelength in the oil:

2(3.68 - 2.76) = λ_o

. Solve for

λ_o

Step 5: Calculate the index of refraction. Substitute

λ_o

and

λ = c/f

into the equation

λ_o = λ/n

to solve for

n

, the index of refraction of sunflower oil.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Microwave Frequency and Wavelength

Microwaves are a form of electromagnetic radiation with frequencies ranging from 300 MHz to 300 GHz. The frequency of 10.5 GHz corresponds to a wavelength of approximately 2.86 cm, calculated using the speed of light. Understanding the relationship between frequency and wavelength is crucial for analyzing how microwaves interact with materials, such as sunflower oil.

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. For microwaves, the index of refraction can be determined by analyzing reflection and transmission at the boundaries of different media, which is essential for solving the given problem.

Standing Waves and Resonance

Standing waves occur when two waves of the same frequency interfere with each other, creating nodes and antinodes. In the context of the beaker, strong reflections at specific depths indicate the formation of standing waves, which correspond to resonant conditions. These depths relate to the wavelength of the microwaves and are critical for determining the index of refraction of sunflower oil.

Recommended video:

Guided course

07:58

Intro to Transverse Standing Waves

Related Practice

Textbook Question

The three identical loudspeakers in FIGURE P17.71 play a 170 Hz tone in a room where the speed of sound is 340 m/s. You are standing 4.0 m in front of the middle speaker. At this point, the amplitude of the wave from each speaker is a. How far must speaker 2 be moved to the left to produce a maximum amplitude at the point where you are standing?

views

Textbook Question

Two loudspeakers emit sound waves of the same frequency along the x-axis. The amplitude of each wave is a. The sound intensity is minimum when speaker 2 is 10 cm behind speaker 1. The intensity increases as speaker 2 is moved forward and first reaches maximum, with amplitude 2a, when it is 30 cm in front of speaker 1. What is The amplitude of the sound (as a multiple of a) if the speakers are placed side by side?

views

Textbook Question

An old mining tunnel disappears into a hillside. You would like to know how long the tunnel is, but it's too dangerous to go inside. Recalling your recent physics class, you decide to try setting up standing-wave resonances inside the tunnel. Using your subsonic amplifier and loudspeaker, you find resonances at 4.5 Hz and 6.3 Hz, and at no frequencies between these. It's rather chilly inside the tunnel, so you estimate the sound speed to be 335 m/s . Based on your measurements, how far is it to the end of the tunnel?

Textbook Question

A 1.0-m-tall vertical tube is filled with 20°C water. A tuning fork vibrating at 580 Hz is held just over the top of the tube as the water is slowly drained from the bottom. At what water heights, measured from the bottom of the tube, will there be a standing wave in the tube above the water?

views

Textbook Question

Engineers are testing a new thin-film coating whose index of refraction is less than that of glass. They deposit a 560-nm-thick layer on glass, then shine lasers on it. A red laser with a wavelength of 640 nm has no reflection at all, but a violet laser with a wavelength of 400 nm has a maximum reflection. How the coating behaves at other wavelengths is unknown. What is the coating’s index of refraction?

Textbook Question

Scientists are testing a transparent material whose index of refraction for visible light varies with wavelength as n = 30.0 nm^1/2/λ^1/2 , where λ is in nm. If a 295-nm-thick coating is placed on glass (n=1.50), for what visible wavelengths will the reflected light have maximum constructive interference?

views