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Chapter 23: Problem 14

Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a) \(\vec{E}\) in the \(+x\) direction, \(\vec{B}\) in the \(+y\) direction (b) \(\vec{E}\) in the \(-y\) direction, \(\vec{B}\) in the \(+x\) direction (c) \(\vec{E}\) in the \(+z\) direction, \(\vec{B}\) in the \(-x\) direction (d) \(\vec{E}\) in the \(+y\) direction, \(\vec{B}\) in the \(-z\) direction

Short Answer

Expert verified
(a) +z direction, (b) -z direction, (c) -y direction, (d) +x direction.

Step by step solution

01

Understanding the Right-Hand Rule for Waves

The direction of propagation of an electromagnetic wave is determined using the right-hand rule. If you point your index finger in the direction of the electric field \(\vec{E}\), and your middle finger in the direction of the magnetic field \(\vec{B}\), your thumb points in the direction of wave propagation \(\vec{k}\). This rule applies to each of the scenarios below.
02

Apply the Right-Hand Rule to Scenario (a)

For scenario (a), \(\vec{E}\) is in the \(+x\) direction and \(\vec{B}\) is in the \(+y\) direction. Point your index finger along \(+x\) and middle finger along \(+y\): your thumb will point in the \(+z\) direction. Hence, the wave propagates in the \(+z\) direction.
03

Apply the Right-Hand Rule to Scenario (b)

For scenario (b), \(\vec{E}\) is in the \(-y\) direction and \(\vec{B}\) is in the \(+x\) direction. Point your index finger along \(-y\) and middle finger along \(+x\): your thumb will point in the \(-z\) direction. Thus, the wave moves in the \(-z\) direction.
04

Apply the Right-Hand Rule to Scenario (c)

For scenario (c), \(\vec{E}\) is in the \(+z\) direction and \(\vec{B}\) is in the \(-x\) direction. Point your index finger along \(+z\) and middle finger along \(-x\): your thumb will point in the \(-y\) direction. Therefore, the wave propagates in the \(-y\) direction.
05

Apply the Right-Hand Rule to Scenario (d)

For scenario (d), \(\vec{E}\) is in the \(+y\) direction and \(\vec{B}\) is in the \(-z\) direction. Point your index finger along \(+y\) and middle finger along \(-z\): your thumb will point in the \(+x\) direction. Consequently, the wave propagates in the \(+x\) direction.

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Key Concepts

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

Right-Hand Rule
The right-hand rule is a simple, yet powerful tool used in physics to determine the direction of wave propagation, especially in electromagnetism. When dealing with electromagnetic waves, we usually have two vectors to consider: the electric field vector (\(\vec{E}\)) and the magnetic field vector (\(\vec{B}\)). To use the right-hand rule, follow these steps:
  • Point your index finger in the direction of the electric field (\(\vec{E}\)).
  • Point your middle finger in the direction of the magnetic field (\(\vec{B}\)).
  • Your thumb will naturally point in the direction of wave propagation (\(\vec{k}\)).
This systematic approach vividly illustrates how electromagnetic waves propagate through space, ensuring we correctly interpret the interactions between electric and magnetic fields.
Electric Fields
Electric fields (\(\vec{E}\)) are a fundamental concept in electromagnetism, representing the force experienced by a charge in space. These fields originate from electric charges, positive or negative, and can exert forces on other charges within their vicinity. The strength and direction of an electric field at any given point are defined as the force per unit charge experienced by a positive test charge placed at that point. As electric fields interact with magnetic fields in wave propagation, understanding them is crucial for grasping the behavior of electromagnetic waves.
Magnetic Fields
Magnetic fields (\(\vec{B}\)) arise from the movement of electric charges and are central to electromagnetism. They describe the influence a magnetic force exerts over an area, affecting charges in motion. When combined with electric fields in electromagnetic waves, they play a pivotal role in wave dynamics. A changing magnetic field generates an electric field, and vice versa鈥攄emonstrating the intertwined nature of these fields in electromagnetic phenomena. Understanding their direction and magnitude provides insight into how waves behave and propagate.
Wave Propagation
Wave propagation is the way electromagnetic waves move through space. It is governed by the interaction between electric and magnetic fields, which continuously regenerate each other in a cycle. This self-perpetuating motion allows waves to travel over long distances without needing a medium. Electromagnetic wave propagation direction, as dictated by the right-hand rule, ensures consistent and predictable outcomes. The propagation ability is key in numerous applications, from radio transmissions to optical communication.
Physics Education
Physics education aims to demystify concepts like electromagnetic waves by breaking down complex principles into digestible parts. Teaching strategies involve hands-on activities, like using the right-hand rule, to connect theory with practice. Educational tools play a crucial role in enhancing understanding, fostering curiosity, and inspiring students to explore further. This approach not only builds foundational knowledge but also cultivates problem-solving skills, preparing students for real-world applications.

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Most popular questions from this chapter

NASA is doing research on the concept of solar sailing. A solar sailing craft uses a large, low-mass sail and the energy and momentum of sunlight for propulsion. (a) Should the sail be absorptive or reflective? Why? (b) The total power output of the sun is \(3.9 \times 10^{26} \mathrm{~W} .\) How large a sail is necessary to propel a \(10,000 \mathrm{~kg}\) spacecraft against the gravitational force of the sun? Express your result in square kilometers. (c) Explain why your answer to part (b) is independent of the distance from the sun.

Scientists are working on a new technique to kill cancer cells by zapping them with ultrahighenergy (in the range of \(10^{12} \mathrm{~W}\) ) pulses of electromagnetic waves that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk \(5.0 \mu \mathrm{m}\) in diameter, with the pulse lasting for \(4.0 \mathrm{~ns}\) with an average power of \(2.0 \times 10^{12} \mathrm{~W}\). We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in \(\mathrm{W} / \mathrm{m}^{2}\) ) delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?

Plane-polarized light passes through two polarizers whose axes are oriented at \(35.0^{\circ}\) to each other. If the intensity of the original beam is reduced to \(15.0 \%,\) what was the polarization direction of the original beam, relative to the first polarizer?

There have been many studies of the effects on humans of electromagnetic waves of various frequencies. Using these studies, the International Commission on NonIonizing Radiation Protection (ICNIRP) produced guidelines for limiting exposure to electromagnetic fields, with the goal of protecting people against known adverse health effects. At frequencies of \(1 \mathrm{~Hz}\) to \(25 \mathrm{~Hz}\), the maximum exposure level of electric-field amplitude, \(E_{\max },\) for the general public is \(14 \mathrm{kV} / \mathrm{m}\). (Different guidelines were created for people who have occupational exposure to radiation.) At frequencies of \(25 \mathrm{~Hz}\) to \(3 \mathrm{kHz},\) the corresponding \(E_{\max }\) is \(\frac{350}{f} \mathrm{kV} / \mathrm{m},\) where \(f\) is the frequency in \(\mathrm{kHz}\). Doubling the frequency of a wave in the range of \(25 \mathrm{~Hz}\) to \(3 \mathrm{kHz}\) represents what change in the maximum allowed electromagneticwave intensity? A. A factor of 2 B. A factor of \(1 / \sqrt{2}\) C. A factor of \(\frac{1}{2}\) D. A factor of \(\frac{1}{4}\)

We can reasonably model a \(75 \mathrm{~W}\) incandescent lightbulb as a sphere \(6.0 \mathrm{~cm}\) in diameter. Typically, only about \(5 \%\) of the energy goes to visible light; the rest goes largely to nonvisible infrared radiation. (a) What is the visible light intensity (in \(\mathrm{W} / \mathrm{m}^{2}\) ) at the surface of the bulb? (b) What are the amplitudes of the electric and magnetic fields at this surface, for a sinusoidal wave with this intensity?
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