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Chapter 12: Problem 44

Electromagnetic waves are supposed to have their electric and magnetic fields perpendicular to each other. (Throughout this problem, assume we're talking about waves traveling through a vacuum, and that there is only a single sine wave traveling in a single direction, not a superposition of sine waves passing through each other.) Suppose someone claims they can make an electromagnetic wave in which the electric and magnetic fields lie in the same plane. Prove that this is impossible based on Maxwell's equations.

Short Answer

Expert verified
Electromagnetic waves cannot have electric and magnetic fields in the same plane based on Maxwell's equations.

Step by step solution

01

Understanding the Problem

We need to prove, using Maxwell's equations, that electromagnetic waves cannot have electric and magnetic fields lying in the same plane.
02

Identifying Maxwell's Equations Relevant to the Problem

The two Maxwell's equations relevant here are Faraday's Law of Induction and the Amp猫re-Maxwell Law. Faraday's Law is \( abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \), and the Amp猫re-Maxwell Law is \( abla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \).
03

Analyzing Faraday's Law

Faraday's Law of Induction \( abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \) implies that a changing magnetic field \( \mathbf{B} \) will induce a circulating electric field \( \mathbf{E} \). For \( \mathbf{E} \) and \( \mathbf{B} \) to be perpendicular, \( abla \times \mathbf{E} \) must not lie in the same direction as \( \mathbf{E} \).
04

Analyzing the Amp猫re-Maxwell Law

The Amp猫re-Maxwell Law \( abla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \) suggests that a changing electric field \( \mathbf{E} \) will induce a circulating magnetic field \( \mathbf{B} \). Again, perpendicularly implies that \( abla \times \mathbf{B} \) must not lie in the same direction as \( \mathbf{B} \).
05

Conclusion from Analysis

Both Faraday's Law and the Amp猫re-Maxwell Law indicate that if the electric and magnetic fields were in the same plane, they could not appropriately induce each other as required in a propagating electromagnetic wave.

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Key Concepts

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

Maxwell's Equations
Maxwell's Equations are a set of four fundamental equations that describe classical electromagnetism. These equations were formulated by James Clerk Maxwell in the 19th century. They explain how electric and magnetic fields interact, shine light on the nature of electromagnetic waves, and lay the foundation for various technologies like radios, televisions, and even the internet.

  • Gauss's Law: It states the electric field \(\mathbf{E}\) flux through a closed surface is proportional to the charge enclosed within the surface.
  • Gauss's Law for Magnetism: This law conveys that there are no 'magnetic charges'; magnetic field lines are closed loops.
  • Faraday's Law of Induction: It shows how a time-varying magnetic field creates an electric field.
  • Amp猫re-Maxwell Law: It describes how a time-varying electric field and electric currents generate a magnetic field.
The interplay between these four laws describes how changes in electric and magnetic fields propagate through space as waves, specifically electromagnetic waves like light waves. They are foundational in both theoretical physics and technological applications.
Faraday's Law
Faraday's Law of Induction is a crucial part of electromagnetism, expressing how electric fields can be generated by changing magnetic fields. The law is mathematically expressed as:

\[ abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
This equation tells us that a time-varying magnetic field \( \mathbf{B} \) induces an electric field \( \mathbf{E} \) that circulates around the change.

- **Circular Induction:** When the magnetic field changes, it sets up loops of electric field lines around the variation. This is why loops or circles often represent electrical phenomena in various applications, like transformers or electric generators.- **Behavior of Fields:** For the fields to collaborate effectively in electromagnetic wave propagation, they must not lie in the same direction. The electric field induced by a changing magnetic field can鈥檛 be in the same plane as the magnetic field.
Faraday's Law is utilized in many practical devices, such as electric generators, which transform mechanical energy into electrical energy.
Amp猫re-Maxwell Law
The Amp猫re-Maxwell Law extends Amp猫re's original formula by including the effect of changing electric fields. This comprehensive take is vital for describing the behavior of electromagnetic waves. The equation is represented as follows:

\[ abla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]
- **Current and the Magnetic Field:** The original Amp猫re's Law described how electric currents produce magnetic fields. Maxwell enhanced this by introducing the displacement current \( \frac{\partial \mathbf{E}}{\partial t} \) term, acknowledging the role of changing electric fields in making magnetic fields.- **Field Interaction:** When electric fields vary over time, a corresponding magnetic field arises around this change. For electromagnetic waves, it's crucial that these fields perpetuate each other through 90-degree angles, and not within the same plane.
Maxwell's addition was significant for explaining how electromagnetic waves, including light, can propagate through a vacuum even without a material medium. This understanding revolutionized the field of electromagnetism and supported the later development of wireless communications.

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Most popular questions from this chapter

Suppose a charged particle is moving through a region of space in which there is an electric field perpendicular to its velocity vector, and also a magnetic field perpendicular to both the particle's velocity vector and the electric field. Show that there will be one particular velocity at which the particle can be moving that results in a total force of zero on it. Relate this velocity to the magnitudes of the electric and magnetic fields. (Such an arrangement, called a velocity filter, is one way of determining the speed of an unknown particle.)

A positively charged particle is released from rest at the origin at \(t=0\), in a region of vacuum through which an electromagnetic wave is passing. The particle accelerates in response to the wave. In this region of space, the wave varies as \(\mathbf{E}=\hat{\mathbf{x}} \tilde{E} \sin \omega t, \mathbf{B}=\hat{\mathbf{y}} \tilde{B} \sin \omega t\), and we assume that the particle has a relatively large value of \(m / q\), so that its response to the wave is sluggish, and it never ends up moving at any speed comparable to the speed of light. Therefore we don't have to worry about the spatial variation of the wave; we can just imagine that these are uniform fields imposed by some external mechanism on this region of space. (a) Find the particle's coordinates as functions of time.(answer check available at lightandmatter.com) (b) Show that the motion is confined to \(-z_{\max } \leq z \leq z_{\max }\), where \(z_{\max }=1.101\left(q^{2} \tilde{E} \tilde{B} / m^{2} \omega^{3}\right)\).

(a) A line charge, with charge per unit length \(\lambda\), moves at velocity \(v\) along its own length. How much charge passes a given point in time \(d t\) ? What is the resulting current? Nwans\\{hwans:linechargecurrent \(\\}\) (b) Show that the units of your answer in part a work out correctly. This constitutes a physical model of an electric current, and it would be a physically realistic model of a beam of particles moving in a vacuum, such as the electron beam in a television tube. It is not a physically realistic model of the motion of the electrons in a current-carrying wire, or of the ions in your nervous system; the motion of the charge carriers in these systems is much more complicated and chaotic, and there are charges of both signs, so that the total charge is zero. But even when the model is physically unrealistic, it still gives the right answers when you use it to compute magnetic effects. This is a remarkable fact, which we will not prove. The interested reader is referred to E.M. Purcell, Electricity and Magnetism, McGraw Hill, \(1963 .\)

On page 713, I proved that \(\mathbf{H}_{\|, 1}=\mathbf{H}_{\|, 2}\) at the boundary between two substances if there is no free current and the fields are static. In fact, each of Maxwell's four equations implies a constraint with a similar structure. Some are constraints on the field components parallel to the boundary, while others are constraints on the perpendicular parts. Since some of the fields referred to in Maxwell's equations are the electric and magnetic fields \(\mathbf{E}\) and \(\mathbf{B}\), while others are the auxiliary fields \(\mathbf{D}\) and \(\mathbf{H}\), some of the constraints deal with \(\mathbf{E}\) and \(\mathbf{B}\), others with \(\mathbf{D}\) and \(\mathbf{H}\). Find the other three constraints.

A charged particle is in a region of space in which there is a uniform magnetic field \(\mathbf{B}=B \hat{\mathbf{z}}\). There is no electric field, and no other forces act on the particle. In each case, describe the future motion of the particle, given its initial velocity. a. \(\mathbf{v}_{\mathrm{o}}=0\) b. \(\mathbf{v}_{\mathrm{o}}=(1 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{z}}\) c. \(\mathbf{v}_{\mathrm{o}}=(1 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{y}}\)
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