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Chapter 32: Problem 2

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; (b) -z; (c) +y; (d) +x.

Step by step solution

01

Understanding Electromagnetic Wave Propagation

In electromagnetic wave propagation, the direction of propagation of the wave is given by the cross product of the electric field \(\vec{E}\) and the magnetic field \(\vec{B}\). The direction of this propagation is given by the right-hand rule, meaning that if you point your index finger in the direction of \(\vec{E}\) and your middle finger in the direction of \(\vec{B}\), then your thumb will point in the direction of wave propagation.
02

Case (a): Fields in +x and +y Directions

Given \(\vec{E}\) in the +\(x\)-direction and \(\vec{B}\) in the +\(y\)-direction. Using the right-hand rule, point your index finger in the +\(x\)-direction and your middle finger in the +\(y\)-direction. Your thumb points in the +\(z\)-direction, indicating the wave propagates in the +\(z\)-direction.
03

Case (b): Fields in -y and +x Directions

Here \(\vec{E}\) is in the -\(y\)-direction and \(\vec{B}\) in the +\(x\)-direction. Point your index finger in the -\(y\)-direction and your middle finger in the +\(x\)-direction. Your thumb points in the -\(z\)-direction, so the wave propagates in the -\(z\)-direction.
04

Case (c): Fields in +z and -x Directions

With \(\vec{E}\) in the +\(z\)-direction and \(\vec{B}\) in the -\(x\)-direction, point your index finger in the +\(z\)-direction and your middle finger in the -\(x\)-direction. Your thumb points in the +\(y\)-direction, indicating that the wave propagates in the +\(y\)-direction.
05

Case (d): Fields in +y and -z Directions

Here \(\vec{E}\) is in the +\(y\)-direction and \(\vec{B}\) is in the -\(z\)-direction. Using the right-hand rule, point your index finger in the +\(y\)-direction and your middle finger in the -\(z\)-direction. Your thumb will point in the +\(x\)-direction, meaning 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 handy mnemonic that helps you determine the direction of electromagnetic wave propagation. It uses the orientation of your fingers to describe vectors in relation to one another. Imagine you're holding your right hand out:
  • Extend your index finger in the direction of the electric field \(\vec{E}\).
  • Extend your middle finger perpendicular to your index finger, pointing in the direction of the magnetic field \(\vec{B}\).
  • Your thumb, extended at a right angle to both fingers, will then indicate the direction of wave propagation.
This is a quick and effective way to visualize and calculate the direction of an electromagnetic wave using vector quantities. It helps you comprehend how these two orthogonal components, the electric and magnetic fields, interact to produce a wave in space.
Electric Field Orientation
An electric field is a vector field that represents the force exerted on a charged particle. Its direction can be described as the path along which a positive charge would accelerate.
In the context of electromagnetic waves, this field determines the orientation of one component of the wave. It oscillates perpendicular to the magnetic field component.
  • In Case (a), the electric field \(\vec{E}\) is oriented in the +\(x\)-direction.
  • In Case (b), it points in the -\(y\)-direction.
  • In Case (c), \(\vec{E}\) is aligned in the +\(z\)-direction.
  • In Case (d), it's directed in the +\(y\)-direction.
These orientations are fundamental in evaluating the electromagnetic wave's overall direction using the right-hand rule.
Magnetic Field Orientation
Magnetic field orientation is another critical component in electromagnetic wave propagation. The magnetic field, much like its electric counterpart, is a vector field that affects charged particles. It oscillates perpendicular to both the electric field and the wave propagation direction.
  • In Case (a), the magnetic field \(\vec{B}\) points in the +\(y\)-direction.
  • In Case (b), it is oriented in the +\(x\)-direction.
  • In Case (c), \(\vec{B}\) emerges in the -\(x\)-direction.
  • In Case (d), it is arranged in the -\(z\)-direction.
These orientations guide the formation of electromagnetic waves when used in conjunction with the electric field direction and the right-hand rule. Understanding the relationship between these vector positions is crucial for identifying the wave's path.

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Most popular questions from this chapter

A circular wire loop has a radius of \(7.50 \mathrm{~cm}\). A sinusoidal electromagnetic plane wave traveling in air passes through the loop, with the direction of the magnetic field of the wave perpendicular to the plane of the loop. The intensity of the wave at the location of the loop is \(0.0275 \mathrm{~W} / \mathrm{m}^{2}\), and the wavelength of the wave is \(6.90 \mathrm{~m}\). What is the maximum emf induced in the loop?

A small helium-neon laser emits red visible light with a power of 5.80 mW in a beam of diameter 2.50 mm. (a) What are the amplitudes of the electric and magnetic fields of this light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a 1.00-m length of the beam?

Electromagnetic waves propagate much differently in conductors than they do in dielectrics or in vacuum. If the resistivity of the conductor is sufficiently low (that is, if it is a sufficiently good conductor), the oscillating electric field of the wave gives rise to an oscillating conduction current that is much larger than the displacement current. In this case, the wave equation for an electric field \(\vec{E} (x, t) = E_y(x, t)\hat{\jmath}\) en propagating in the +\(x\)-direction within a conductor is $${\partial^2E_y(x, t)\over \partial x^2} = {\mu \over \rho} {\partial Ey(x, t)\over \partial t}$$ where \(\mu\) is the permeability of the conductor and \(\rho\) is its resistivity. (a) A solution to this wave equation is \(E_y(x, t) = E_{max} e^{-k_C x} cos(k_Cx - \omega t)\), where \(k_C = \sqrt{(\omega \mu/2\rho}\). Verify this by substituting E_y(x, t) into the above wave equation. (b) The exponential term shows that the electric field decreases in amplitude as it propagates. Explain why this happens. (\(Hint\): The field does work to move charges within the conductor. The current of these moving charges causes \(i^2R\) heating within the conductor, raising its temperature. Where does the energy to do this come from?) (c) Show that the electric-field amplitude decreases by a factor of 1/\(e\) in a distance \(1/k_C = \sqrt{2\rho/\omega\mu}\), and calculate this distance for a radio wave with frequency \(f\) = 1.0 MHz in copper (resistivity 1.72 \(\times\) 10\(^{-8 } \Omega \bullet m\); permeability \(\mu = \mu_0\)). Since this distance is so short, electromagnetic waves of this frequency can hardly propagate at all into copper. Instead, they are reflected at the surface of the metal. This is why radio waves cannot penetrate through copper or other metals, and why radio reception is poor inside a metal structure.

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} V/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?

The electron in a hydrogen atom can be considered to be in a circular orbit with a radius of 0.0529 nm and a kinetic energy of 13.6 eV. If the electron behaved classically, how much energy would it radiate per second (see Challenge Problem 32.51)? What does this tell you about the use of classical physics in describing the atom?
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