/*! 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 11 An electromagnetic wave has an e... [FREE SOLUTION] | 91影视

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Chapter 32: Problem 11

An electromagnetic wave has an electric field given by \(\vec { \boldsymbol { E } } ( y , t ) = \left( 3.10 \times 10 ^ { 5 } \mathrm { V } / \mathrm { m } \right) \hat { \boldsymbol { k } } \cos \left[ k y - \left( 12.65 \times 10 ^ { 12 } \mathrm { rad } / \mathrm { s } \right) t \right]\) (a) In which direction is the wave traveling? (b) What is the wave-length of the wave? (c) Write the vector equation for \(\vec { \boldsymbol { B } } ( y , t )\)

Short Answer

Expert verified
(a) +y direction (b) 位 unknown without k (c) \( \vec{B} = 1.03 \times 10^{-3} \hat{i} \cos[ky - (12.65 \times 10^{12})t] \)

Step by step solution

01

Identify Wave Direction

The electric field, \( \vec{\boldsymbol{E}}(y, t) \), given in the exercise has a cosine argument of the form \( ky - \omega t \). This indicates that the wave is traveling in the +\( y \)-direction since the argument follows the general form \( kx - \omega t \) for a wave propagating in the positive direction of the x, y, or z coordinate, depending on the variable used.
02

Determine Wavelength from Wave Number

The wave number \( k \) is related to the wavelength \( \lambda \) by the formula \( k = \frac{2\pi}{\lambda} \). Since we have the function in terms of \( ky \), we need to first calculate \( k \). Right now, \( k \) is not directly given, but the parameter \( \omega = 12.65 \times 10^{12} \text{ rad/s} \) is related to other wave properties. With this, determine \( \lambda = \frac{2\pi}{k} \) once \( k \) is found from further context (such contexts involve also being provided or actually knowing \( k \)), we generally assume an ideal environment.
03

Find Expression for Magnetic Field

For an electromagnetic wave, the magnetic field \( \vec{\boldsymbol{B}}(y, t) \) can be found using the relation \( B = \frac{E}{c} \), where \( c \) is the speed of light \( c = 3.00 \times 10^{8} \text{ m/s} \). The magnetic field will have the same sinusoidal form and will be perpendicular to both \( \vec{\boldsymbol{E}} \) and the direction of propagation. Thus, the equation for \( \vec{\boldsymbol{B}}(y, t) \) becomes:\[\vec{\boldsymbol{B}}(y, t) = \left( \frac{3.10 \times 10^{5}}{3.00 \times 10^{8}} \right) \hat{\boldsymbol{i}} \cos\left[ ky - (12.65 \times 10^{12}) t \right]\]

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Key Concepts

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

Electric Field
An electric field in an electromagnetic wave represents the force felt by a charged particle in space. This field is essential for transmitting energy through the wave. In our exercise, the electric field is denoted as \( \vec { \boldsymbol { E } } ( y , t ) = \left( 3.10 \times 10 ^ { 5 } \mathrm { V } / \mathrm { m } \right) \hat { \boldsymbol { k } } \cos \left[ k y - \left( 12.65 \times 10 ^ { 12 } \mathrm { rad } / \mathrm { s } \right) t \right]\).
The equation captures several core elements:
  • The strength or amplitude of the field, here given as \( 3.10 \times 10^{5} \mathrm{ V/m} \).
  • The direction of the field, noted by the unit vector \( \hat { \boldsymbol { k } } \), demonstrating that the field is oriented along one of the axis perpendicular to wave travel.
  • Its variation over space (\( y \)) and time (\( t \)), described by the cosine function, indicative of its oscillatory nature.
This oscillation is what constitutes the wave-like behavior of electromagnetic fields, enabling information transfer across distances.
Magnetic Field
A magnetic field complements the electric field in an electromagnetic wave, forming a duo responsible for wave propagation. The magnetic field is always perpendicular to the electric field and the direction of wave travel.
To find the magnetic field \( \vec{\boldsymbol{B}}(y, t) \), use the relationship \( B = \frac{E}{c} \), where \( c = 3.00 \times 10^{8} \text{ m/s} \) is the speed of light. For our particular setup, the expression becomes:
\[\vec{\boldsymbol{B}}(y, t) = \left( \frac{3.10 \times 10^{5}}{3.00 \times 10^{8}} \right) \hat{\boldsymbol{i}} \cos\left[ ky - (12.65 \times 10^{12}) t \right]\]
Key points include:
  • The reduction in amplitude compared to the electric field due to division by \( c \).
  • The unit vector \( \hat{\boldsymbol{i}} \) shows orientation, ensuring the magnetic field remains orthogonal to both the electric field and propagation.
Wave Direction
The direction in which a wave travels is vital to understanding its behavior and interaction with the environment. In general, the direction is indicated by the variations in the wave's equation. In this exercise, the electric field function is of the form \( ky - \omega t \). This format is significant because:
  • It follows the generic propagation equation \( kx - \omega t \) for waves.
  • The presence of \( y \) in \( ky \) designates that this specific wave is propagating in the positive \( y \)-direction as the variable corresponds to the direction of travel.
This positive \( y \)-direction reveals how the wave's energy moves through space, crucial for practical applications like antenna alignment.
Wavelength
The wavelength of a wave is a critical parameter that determines its spatial extent. It refers to the distance over which the wave's shape repeats. Wavelength \( \lambda \) is inversely related to the wave number \( k \) by the equation \( k = \frac{2\pi}{\lambda} \).
Although \( k \) is not directly provided here, once known, it will guide us to:
  • Understanding the scale of the wave's repeating pattern.
  • Calculating the exact spatial distance per cycle.
From the broader electromagnetic wave perspective, knowing the wavelength helps in identifying the wave type (e.g., visible light, radio waves) and its potential applications.
Wave Number
The wave number \( k \) is a fundamental metric for waves. It characterizes the wave's spatial frequency, indicating how many oscillations exist within a unit distance.
The relationship between the wave number and wavelength is given by \( k = \frac{2\pi}{\lambda} \). Hence:
  • A higher wave number corresponds to a shorter wavelength, meaning more cycles per unit distance.
  • This measure assists in determining wave characteristics and properties.
Because \( k \) isn't provided directly, solving for it involves further context or given values, central for complete wave analysis and accurate prediction of wave interactions and behaviors.

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Most popular questions from this chapter

A satellite 575\(\mathrm { km }\) above the earth's surface transmits sinusoidal electromagnetic waves of frequency 92.4 MHz uniformly in all directions, with a power of 25.0\(\mathrm { kW }\) . (a) What is the intensity of these waves as they reach a receiver at the surface of the earth directly below the satellite? (b) What are the amplitudes of the electric and magnetic fields at the receiver? (c) If the receiver has a totally absorbing panel measuring 15.0\(\mathrm { cm }\) by 40.0\(\mathrm { cm }\) oriented with its plane perpendicular to the direction the waves travel, what average force do these waves exert on the panel? Is this force large enough to cause significant effects?

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 cylindrical conductor with a circular cross section has a radius \(a\) and a resistivity \(\rho\) and carries a constant current \(I\) (a) What are the magnitude and direction of the electric-field vector \(E\) at a point just inside the wire at a distance \(a\) from the axis? (b) What are the magnitude and direction of the magneticfield vector \(\vec { \boldsymbol { B } }\) at the same point? \(( \mathrm { c } )\) What are the magnitude and direction of the Poynting vector \(\vec { \boldsymbol { S } }\) at the same point? (The direction of \(\vec { \boldsymbol { S } }\) is the direction in which electromagnetic energy flows into or out of the conductor.) (d) Use the result in part (c) to find the rate of flow of energy into the volume occupied by a length \(l\) of the conductor. (Hint: Integrate \(\vec { S }\) over the surface of this volume.) Compare your result to the rate of generation of thermal energy in the same volume. Discuss why the energy dissipated in a current carrying conductor, due to its resistance, can be thought of as entering through the cylindrical sides of the conductor.

Testing a Space Radio Transmitter. You are a NASA mission specialist on your first flight aboard the space shuttle. Thanks to your extensive training in physics, you have been assigned to evaluate the performance of a new radio transmitter on board the International Space Station (ISS). Perched on the shuttle's movable arm, you aim a sensitive detector at the ISS, which is 2.5\(\mathrm { km }\) away. You find that the electric-field amplitude of the radio waves coming from the ISS transmitter is 0.090\(\mathrm { V } / \mathrm { m }\) and that the frequency of the waves is 244\(\mathrm { MH } z\) . Find the following: (a) the intensity of the radio wave at your location; (b) the magnetic-field amplitude of the wave at your location; (c) the total power output of the ISS radio transmitter. (d) What assumptions, if any, did you make in your calculations?

The electric field of a sinusoidal electromagnetic wave obeys the equation \(E = ( 375 \mathrm { V } / \mathrm { m } ) \cos \left[ \left( 1.99 \times 10 ^ { 7 } \mathrm { rad } / \mathrm { m } \right) x + \right.\) \(\left( 5.97 \times 10 ^ { 15 } \mathrm { rad } / \mathrm { s } \right) t ] . ( \mathrm { a } )\) What are the amplitudes of the electric and magnetic fields of this wave? (b) What are the frequency, wavelength, and period of the wave? Is this light visible to humans? (c) What is the speed of the wave?
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