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Consider each of the following electric- and magneticfield orientations. In each case, what is the direction of propagation of the wave? (a) \(\vec{E} = E\hat{\imath}\), \(\vec{B} = -B\hat{\jmath}\); (b) \(\vec{E} = E\hat{\jmath}\), \(\vec{B} = B\hat{\imath}\); (c) \(\vec{E} = -E\hat{k}\) , \(\vec{B} = -B\hat{\imath}\); (d) \(vec{E} = E\hat{\imath}\), \(\vec{B} = -B\hat{k}\).

Short Answer

Expert verified
(a) -\hat{k}, (b) -\hat{k}, (c) \hat{\jmath}, (d) \hat{\jmath}.

Step by step solution

01

Understand the Wave Propagation

For an electromagnetic wave in free space, the electric field \(\vec{E}\), the magnetic field \(\vec{B}\), and the direction of propagation \(\vec{k}\) are all perpendicular to each other. This relationship can be expressed as \(\vec{k} = \vec{E} \times \vec{B}\), where \(\times\) denotes the cross product.
02

Determine Direction for Part (a)

Given \(\vec{E} = E\hat{\imath}\) and \(\vec{B} = -B\hat{\jmath}\), compute \(\vec{k}\) using the cross product. \(\vec{k} = (E\hat{\imath}) \times (-B\hat{\jmath}) = -EB(\hat{\imath} \times \hat{\jmath}) = -EB\hat{k}\). Thus, the wave propagates in the \(-\hat{k}\) direction.
03

Determine Direction for Part (b)

Given \(\vec{E} = E\hat{\jmath}\) and \(\vec{B} = B\hat{\imath}\), compute \(\vec{k}\). \(\vec{k} = (E\hat{\jmath}) \times (B\hat{\imath}) = EB(\hat{\jmath} \times \hat{\imath}) = -EB\hat{k}\). Thus, the wave propagates in the \(-\hat{k}\) direction.
04

Determine Direction for Part (c)

Given \(\vec{E} = -E\hat{k}\) and \(\vec{B} = -B\hat{\imath}\), compute \(\vec{k}\). \(\vec{k} = (-E\hat{k}) \times (-B\hat{\imath}) = EB(\hat{k} \times \hat{\imath}) = EB\hat{\jmath}\). Thus, the wave propagates in the \(\hat{\jmath}\) direction.
05

Determine Direction for Part (d)

Given \(\vec{E} = E\hat{\imath}\) and \(\vec{B} = -B\hat{k}\), compute \(\vec{k}\). \(\vec{k} = (E\hat{\imath}) \times (-B\hat{k}) = -EB(\hat{\imath} \times \hat{k}) = EB\hat{\jmath}\). Thus, the wave propagates in the \(\hat{\jmath}\) direction.

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Key Concepts

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

Wave Propagation Direction
Understanding the direction of wave propagation is crucial in electromagnetism. When dealing with electromagnetic waves, such as light, they propagate through space in a specific direction. This direction is denoted by the vector \( \vec{k} \). In simpler terms, \( \vec{k} \) shows where the wave is headed.

In the electromagnetic spectrum, both the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) are always perpendicular to each other and to the wave propagation direction. This means if you know the orientations of \( \vec{E} \) and \( \vec{B} \), you can determine the direction of \( \vec{k} \).

To find \( \vec{k} \), you use the formula \( \vec{k} = \vec{E} \times \vec{B} \). This equation is a mathematical representation of the direction of the wave by using the cross product. By calculating \( \vec{k} \), you determine the wave's travel path.
Electric Field Orientation
The electric field \( \vec{E} \) is a vector field that represents the electric force per unit charge at different points in space. In an electromagnetic wave, the direction of \( \vec{E} \) is one of the key factors influencing the wave's propagation.

Electric fields are typically oriented along one axis in a three-dimensional coordinate system, such as \( \hat{i} \), \( \hat{j} \), or \( \hat{k} \). The chosen axis depends on the situation or the problem at hand.

Orientation of \( \vec{E} \) is significant because it is directly perpendicular to where the wave will travel and also perpendicular to \( \vec{B} \), the magnetic field. In exercises involving electromagnetic waves, identifying \( \vec{E} \) is the first step in figuring out the direction of \( \vec{k} \), which ultimately determines the propagation direction.
Magnetic Field Orientation
Just like the electric field, the magnetic field \( \vec{B} \) is a vector. It represents the magnetic influence on moving charges, current loops, and magnetized materials. In electromagnetic waves, \( \vec{B} \) interacts with the electric field to dictate the wave’s direction.

Magnetic fields in these scenarios are also oriented along an axis in the three-dimensional system, such as \( \hat{i} \), \( \hat{j} \), or \( \hat{k} \). A crucial point to remember is \( \vec{B} \) is perpendicular to both the electric field \( \vec{E} \) and the wave propagation direction \( \vec{k} \).

This perpendicular relationship forms the right-handed coordinate system, where \( \vec{E} \), \( \vec{B} \), and \( \vec{k} \) interact according to the right-hand rule. Thus, knowing the orientation of \( \vec{B} \) is vital in determining \( \vec{k} \).
Cross Product in Physics
The cross product, denoted by \( \times \), is a fundamental operation in vector algebra used in physics to find a vector perpendicular to two given vectors. In the context of electromagnetic waves, it is employed to calculate the direction of wave propagation \( \vec{k} \).

In mathematical terms, the cross product of two vectors \( \vec{A} \) and \( \vec{B} \) is another vector \( \vec{C} \), which is orthogonal to both \( \vec{A} \) and \( \vec{B} \). The magnitude of \( \vec{C} \) is given by \( |\vec{C}| = |\vec{A}| |\vec{B}| \sin(\theta) \), where \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \). The direction of \( \vec{C} \) follows the right-hand rule.

In electromagnetic problems, \( \vec{k} = \vec{E} \times \vec{B} \), helps us find the wave's direction. Using the cross product ensures that \( \vec{k} \) is perpendicular to both the electric and magnetic fields, adhering to the fundamental properties of wave propagation in free space.

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Most popular questions from this chapter

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?

A sinusoidal electromagnetic wave having a magnetic field of amplitude 1.25 \(\mu\)T and a wavelength of 432 nm is traveling in the +\(x\)-direction through empty space. (a) What is the frequency of this wave? (b) What is the amplitude of the associated electric field? (c) Write the equations for the electric and magnetic fields as functions of \(x\) and t in the form of Eqs. (32.17).

The microwaves in a certain microwave oven have a wavelength of 12.2 cm. (a) How wide must this oven be so that it will contain five antinodal planes of the electric field along its width in the standing-wave pattern? (b) What is the frequency of these microwaves? (c) Suppose a manufacturing error occurred and the oven was made 5.0 cm longer than specified in part (a). In this case, what would have to be the frequency of the microwaves for there still to be five antinodal planes of the electric field along the width of the oven?

An intense light source radiates uniformly in all directions. At a distance of 5.0 m from the source, the radiation pressure on a perfectly absorbing surface is 9.0 \(\times\) 10\(^{-6}\) Pa. What is the total average power output of the source?

An electromagnetic wave of wavelength 435 nm is traveling in vacuum in the -\(z\)-direction. The electric field has amplitude 2.70 \(\times\) 10\(^{-3}\) V/m and is parallel to the \(x\)-axis. What are (a) the frequency and (b) the magnetic-field amplitude? (c) Write the vector equations for \(\vec{E} (z, t)\) and \(\vec{B} (z, t)\).

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