/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 An electromagnetic standing wave... [FREE SOLUTION] | 91影视

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Chapter 32: Problem 35

An electromagnetic standing wave in a certain material has frequency \(1.20 \times 10 ^ { 10 } \mathrm { Hz }\) and speed of propagation \(2.10 \times\) \(10 ^ { 8 } \mathrm { m } / \mathrm { s } .\) (a) What is the distance between a nodal plane of \(\vec { \boldsymbol { B } }\) and the closest antinodal plane of \(\vec { \boldsymbol { B } } ?\) (b) What is the distance between an antinodal plane of \(\vec { \boldsymbol { E } }\) and the closest antinodal plane of \(\vec { \boldsymbol { B } }\) ? (c) What is the distance between a nodal plane of \(\vec { \boldsymbol { E } }\) and the closest nodal plane of \(\vec { \boldsymbol { B } }\) ?

Short Answer

Expert verified
(a) 0.004375 m; (b) 0.00875 m; (c) 0.00875 m.

Step by step solution

01

Calculate the Wavelength

The wavelength \( \lambda \) of the electromagnetic wave can be found using the formula:\[\lambda = \frac{v}{f}\]where \(v = 2.10 \times 10^8 \ \text{m/s}\) is the speed of propagation and \(f = 1.20 \times 10^{10} \ \text{Hz}\) is the frequency. Plugging in the values:\[\lambda = \frac{2.10 \times 10^8 \ \text{m/s}}{1.20 \times 10^{10} \ \text{Hz}} = 0.0175 \ \text{m}\]
02

Distance from Node to Antinode of \(\vec{\boldsymbol{B}}\)

For electromagnetic waves, the distance between a nodal plane and the closest antinodal plane is \( \frac{\lambda}{4} \). Since the wavelength \( \lambda \) is 0.0175 m, the distance is:\[\text{Distance} = \frac{0.0175 \ \text{m}}{4} = 0.004375 \ \text{m}\]
03

Distance from Antinode of \(\vec{\boldsymbol{E}}\) to Antinode of \(\vec{\boldsymbol{B}}\)

In electromagnetic standing waves, the electric field \(\vec{\boldsymbol{E}}\) and the magnetic field \(\vec{\boldsymbol{B}}\) are \(\frac{\lambda}{2}\) out of phase. Hence, the distance between antinodal planes of \(\vec{\boldsymbol{E}}\) and \(\vec{\boldsymbol{B}}\) is \( \frac{\lambda}{2} \):\[\text{Distance} = \frac{0.0175 \ \text{m}}{2} = 0.00875 \ \text{m}\]
04

Distance from Node of \(\vec{\boldsymbol{E}}\) to Node of \(\vec{\boldsymbol{B}}\)

Because the \(\vec{\boldsymbol{E}}\) and \(\vec{\boldsymbol{B}}\) fields are \(\frac{\lambda}{2}\) out of phase in a nodal arrangement as well, the distance between nodal planes is the same \(\frac{\lambda}{2} \):\[\text{Distance} = \frac{0.0175 \ \text{m}}{2} = 0.00875 \ \text{m}\]

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Key Concepts

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

Standing Waves
Standing waves are a fascinating phenomenon seen in various types of waves, including electromagnetic ones. In a standing wave, certain points, known as nodes, remain stationary, while other points, called antinodes, reach maximum displacement.
These waves form due to the interference between two waves traveling in opposite directions with the same frequency and amplitude.
In the context of electromagnetic standing waves, both the electric and magnetic fields produce nodes and antinodes. The nodes for the magnetic field (\(\vec{B}\)) are typically where no movement or change occurs, while antinodes are the areas of maximum magnetic activity.
For the electric field (\(\vec{E}\)), nodes and antinodes can be found in a similar manner, often displaced relative to the magnetic field's nodes and antinodes. Understanding this positioning is key to grasping how standing waves function and propagate within various media.
Wave Propagation
Wave propagation refers to the way waves travel through a medium. For electromagnetic waves, propagation involves the transmission of electric and magnetic fields through space.
These waves travel by alternating electric and magnetic fields, which recreate and sustain each other, allowing the wave to keep moving forward.
Key characteristics influencing wave propagation include:
  • Speed: Determined by the medium and the wave type.
  • Frequency: The rate at which wave crests pass a point.
  • Medium: Physical material through which the wave travels, impacting speed and behavior.
In our step-by-step solution, the given speed of the wave is \(2.10 \times 10^8\) m/s, impacting how quickly the electromagnetic wave progresses through the material.
Understanding wave propagation is crucial as it affects how we perceive and use electromagnetic waves in technology and communication.
Wavelength
Wavelength is a fundamental concept in wave physics, defined as the distance between consecutive crests (high points) or troughs (low points) of a wave.
In electromagnetic waves, the wavelength (\(\lambda\)) is related to both the speed (\(v\)) and the frequency (\(f\)) of the wave by the formula: \[\lambda = \frac{v}{f}\]This formula was used in the step-by-step solution to find the electromagnetic wave鈥檚 wavelength, giving us \(0.0175\) meters after calculation.
The wavelength provides essential information about the wave, such as how long each complete wave cycle takes in terms of distance. Wavelengths vary from long radio waves to microscopic gamma rays, each with different applications and behaviors.
Electric and Magnetic Fields
Electric and magnetic fields are core components of electromagnetic waves. These fields oscillate perpendicularly to one another and to the direction of wave propagation.

The interaction between these fields allows for the continuous propagation of electromagnetic waves, even in a vacuum.
In standing waves, \(\vec{E}\) and \(\vec{B}\) are \(\frac{\lambda}{2}\) out of phase, meaning one field's antinode corresponds to the other's node. This spacing determines regions of maximum and zero field strength in the standing wave.
Understanding the behavior of these fields is vital for comprehending how electromagnetic waves carry energy and information. Practical applications range from radio transmissions to medical imaging techniques.

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Most popular questions from this chapter

Radio station WCCO in Minneapolis broadcasts at a frequency of 830\(\mathrm { kHz }\) . At a point some distance from the transmitter, the magnetic-field amplitude of the electromagnetic wave from WCCO is \(4.82 \times 10 ^ { - 111 } \mathrm { T.Calculate }\) (a) the wavelength; (b) the wave number; (c) the angular frequency; (d) the electric-field amplitude.

A sinusoidal electromagnetic wave from a radio station passes perpendicularly through an open window that has area 0.500\(\mathrm { m } ^ { 2 } .\) At the window, the electric field of the wave has rms value 0.0200\(\mathrm { V } / \mathrm { m } .\) How much energy does this wave carry through the window during a 30.0 -s commercial?

A small helium-neon laser emits red visible light with a power of 4.60\(\mathrm { mW }\) in a beam that has a diameter of 2.50\(\mathrm { mm }\) . (a) What are the amplitudes of the electric and magnetic fields of the light? (b) What are the average energy densities associated with the electric field are the magnetic field? (c) What is the total energy contained in a \(1.00 - \mathrm { m }\) length of the beam?

An electromagnetic wave with frequency \(5.70 \times 10 ^ { 14 } \mathrm { Hz }\) propagates with a speed of \(2.17 \times 10 ^ { 8 } \mathrm { m } / \mathrm { s }\) in a certain piece of glass. Find (a) the wavelength of the wave in the glass; (b) the wavelength of a wave of the same frequency propagating in air; (c) the index of refraction \(n\) of the glass for an electromagnetic wave with this frequency; (d) the dielectric constant for glass at this frequency, assuming that the relative permeability is unity.

An electromagnetic wave with frequency 65.0 Hz travels in an insulating magnetic material that has dielectric constant 3.64 and relative permeability 5.18 at this frequency. The electric field has amplitude \(7.20 \times 10 ^ { - 3 } \mathrm { V } / \mathrm { m }\) . (a) What is the speed of propagation of the wave? (b) What is the wavelength of the wave? (c) What is the amplitude of the magnetic field?
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