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Chapter 13: Problem 3

Assume from electrodynamics the following equations which are valid in free space. (They are called Maxwell's equations.) $$\nabla \cdot \mathbf{E}=0 \quad \nabla \cdot \mathbf{B}=0$$ $$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \quad \nabla \times \mathbf{B}=\frac{1}{c^{2}} \frac{\partial \mathbf{E}}{\partial t}$$ where \(\mathbf{E}\) and \(\mathbf{B}\) are the electric and magnetic fields, and \(c\) is the speed of light in a vacuum. From them show that any component of \(\mathbf{E}\) or \(\mathbf{B}\) satisfies the wave equation (1.4) with \(v=c\)

Short Answer

Expert verified
... satisfies the wave equation.

Step by step solution

01

Write Down Maxwell’s Equations

First, note the given Maxwell's equations in free space: 1. abla abla abla abla abla abla abla abla abla abla abla
02

Take the Curl of Faraday’s Law

Start by taking the curl of the third Maxwell equation: abla abla abla abla abla abla abla abla
03

Substitute Ampere’s Law

Use the fourth Maxwell equation abla abla abla
04

Apply the Vector Identity

Recall and apply the vector identity:
05

Simplify the Equation

Since .... .. ...
06

Repeat for the Magnetic Field

Repeat the same steps starting

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Key Concepts

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

Electromagnetic Waves
Electromagnetic waves are waves composed of electric and magnetic fields that oscillate perpendicular to each other and to the direction of the wave's propagation. These waves travel through the vacuum of space at the speed of light, denoted as \(c\). Maxwell's equations describe how electromagnetic waves propagate and interact.
Maxwell's equations predict that changes in electric fields produce magnetic fields and vice versa. This interdependency forms the basis of electromagnetic wave generation. Hence, an accelerating charge can generate electromagnetic waves.
Visible light, radio waves, X-rays, and microwaves are all examples of electromagnetic waves, differing only in their frequencies and wavelengths.
Wave Equation
The wave equation is a fundamental mathematical expression that describes the propagation of various wave phenomena. Generically, it is represented as: \[ abla^2 \textbf{E} - \frac{1}{c^2} \frac{\text{d}^2 \textbf{E}}{\text{d} t^2} = 0 \] In the context of Maxwell’s equations, it applies to both the electric field \( \textbf{E} \) and magnetic field \( \textbf{B} \). For an electric field component \( \textbf{E}_i \), this equation shows how variations in space and time evolve.
To derive this:
  • Start from Maxwell's equations in free space.
  • Take the curl of Faraday's Law to get the wave equation for \mathbf{E}. \
  • Use the curl of Ampere’s Law to relate changes in \( \textbf{B} \) to \( \textbf{E} \).
The process reveals that both \( \textbf{E} \) and \( \textbf{B} \) satisfy the wave equation, showing that they propagate as waves with speed \( c \).
Electric and Magnetic Fields
Electric and magnetic fields encompass the core components of electromagnetic waves. The field vectors \( \textbf{E} \) and \( \textbf{B} \) describe the magnitude and direction of electric and magnetic influences at any point in space.
Electric Fields (\textbf{E}):
  • Created by electric charges.
  • Represent the force exerted by a charge.
  • Example: The static field created around a point charge.
Magnetic Fields (\textbf{B}):
  • Generated by moving charges (currents).
  • Represent the force exerted on a moving charge.
  • Example: The field around a current-carrying wire.
Maxwell's equations unify these fields, showing that the time-varying electric field induces a magnetic field and vice versa, resulting in the propagation of electromagnetic waves. They are always perpendicular to each other in these waves, creating a transverse wave.

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Most popular questions from this chapter

Find the interior temperature in a hemisphere if the curved surface is held at \(u=\) \(\cos \theta\) and the equatorial plane at \(u=1\).

Find the temperature distribution in a rectangular plate \(10 \mathrm{cm}\) by \(30 \mathrm{cm}\) if two adjacent sides are held at \(100^{\circ}\) and the other two sides at \(0^{\circ}\).

Show that the gravitational potential \(V=-G m / r\) satisfies Laplace's equation, that is, show that \(\nabla^{2}(1 / r)=0\) where \(r^{2}=x^{2}+y^{2}+z^{2}, r \neq 0\).

Find the electrostatic potential outside a conducting sphere of radius \(a\) placed in an originally uniform electric field, and maintained at zero potential. Hint: Let the original field \(\mathbf{E}\) be in the negative \(z\) direction so that \(\mathbf{E}=-E_{0} \mathbf{k} .\) Then since \(\mathbf{E}=-\nabla \Phi,\) where \(\Phi\) is the potential, we have \(\Phi=E_{0} z=E_{0} r \cos \theta\) (Verify this!) for the original potential. You then want a solution of Laplace's equation \(\nabla^{2} u=0\) which is zero at \(r=a\) and becomes \(u \sim \Phi\) for large \(r\) (that is, far away from the sphere). Select the solutions of Laplace's equation in spherical coordinates which have the right \(\theta\) and \(\phi\) dependence (there are just two such solutions) and find the combination which reduces to zero for \(r=a\).

Find the energy eigenvalues and eigenfunctions for the hydrogen atom. The potential energy is \(V(r)=-e^{2} / r\) in Gaussian units, where \(e\) is the charge of the electron and \(r\) is in spherical coordinates. since \(V\) is a function of \(r\) only, you know from Problem 18 that the eigenfunctions are \(R(r)\) times the spherical harmonics \(Y_{l}^{m}(\theta, \phi),\) so you only have to find \(R(r) .\) Substitute \(V(r)\) into the \(R\) equation in Problem 18 and make the following simplifications: Let \(x=2 r / \alpha, y=r R ;\) show that then $$r=\alpha x / 2, \quad R(r)=\frac{2}{\alpha x} y(x), \quad \frac{d}{d r}=\frac{2}{\alpha} \frac{d}{d x}, \quad \frac{d}{d r}\left(r^{2} \frac{d R}{d r}\right)=\frac{2}{\alpha} x y^{\prime \prime}$$ Let \(\alpha^{2}=-2 M E / \hbar^{2}\) (note that for a bound state, \(E\) is negative, so \(\alpha^{2}\) is positive) and \(\lambda=M e^{2} \alpha / \hbar^{2},\) to get the first equation in Problem 22.26 of Chapter \(12 .\) Do this problem to find \(y(x),\) and the result that \(\lambda\) is an integer, say \(n\). [Caution: not the same \(n \text { as in equation }(22.26)] .\) Hence find the possible values of \(\alpha\) (these are the radii of the Bohr orbits), and the energy eigenvalues. You should have found \(\alpha\) proportional to \(n ;\) let \(\alpha=n a,\) where \(a\) is the value of \(\alpha\) when \(n=1,\) that is, the radius of the first Bohr orbit. Write the solutions \(R(r)\) by substituting back \(y=r R\) and \(x=2 r /(n a),\) and find \(E_{n}\) from \(\alpha\).
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