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Chapter 8: Problem 5

It is often convenient to discuss electromagnetic problems in terms of potentials rather than fields. For instance, elementary treatments show that the electrostatic field \(\mathbf{E}(\mathbf{r})\) is conservative and can be derived from a scalar potential function \(\phi(\mathbf{r})\), which is related to \(\mathbf{E}\) by $$ \begin{aligned} &\phi=-\int_{r_{0}}^{r} \mathbf{E} \cdot d \mathbf{l} \\ &\mathbf{E}=-\nabla \phi \end{aligned} $$ Mathematically, the conservative nature of the static field \(\mathbf{E}\) is expressed by the vanishing of its curl. Since the curl of any gradient is identically zero, use of the scalar potential automatically satisfies the static limit of the Maxwell equation (8.2.2); the other constraint on \(\phi\) is Gauss' law (8.2.1). Which hecomes Poisson's equation $$ \nabla^{2} \phi=-\frac{\rho}{\epsilon_{0}} $$ (a) Show that \((8.2 .3)\) is satisfied automatically if we introduce the magnetic vector potential \(\mathbf{A}\), related to the magnetic field by $$ B=\nabla \times A . $$ (b) Show that in the general (nonstatic) case, the electric field is given in terms of the scalar and vector potentials by $$ \mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t} $$ (c) Complete the prescription of \(\mathbf{A}\) by defining its divergence by the Lorents condition $$ \boldsymbol{\nabla} \cdot \mathbf{A}=-\frac{1}{c^{2}} \frac{\partial \phi}{\partial t} $$ and show that the two potentials obey the symmetrical inhomogeneous wave equations $$ \begin{aligned} &\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=-\frac{\rho}{\epsilon_{0}} \\ &\nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=-\mu_{0} \mathbf{J} . \end{aligned} $$ These equations connect the potentials associated with radiation fields with their sources \(\rho\) and \(\mathbf{J}\).

Short Answer

Expert verified
Question: Show that the static limit of Maxwell equation is satisfied automatically with the introduction of the magnetic vector potential. Also, find the general expression for the electric field in terms of scalar and vector potentials, and define the divergence of the vector potential by the Lorents condition. Lastly, derive the symmetrical inhomogeneous wave equations. Answer: The static limit of Maxwell equation is satisfied automatically since the curl of the magnetic field, \(\nabla \times \mathbf{B}\), becomes \(\nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}\) when using the magnetic vector potential \(\mathbf{A}\). The general expression for the electric field in terms of scalar and vector potentials is given by \(\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}\). Using the Lorentz condition, the divergence of the vector potential is defined as \(\nabla \cdot \mathbf{A} = -\frac{1}{c^{2}} \frac{\partial \phi}{\partial t}\). The resulting symmetrical inhomogeneous wave equations are: $$ \begin{aligned} &\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}} = -\frac{\rho}{\epsilon_{0}} \\ &\nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}} = -\mu_{0} \mathbf{J} . \end{aligned} $$

Step by step solution

01

(a) Static limit of Maxwell equation

According to the question, the magnetic field \(\mathbf{B}\) is related to the magnetic vector potential \(\mathbf{A}\) by the equation \(\mathbf{B} = \nabla \times \mathbf{A}\). To show that the static limit of Maxwell equation \((8.2.3)\) is satisfied automatically, we need to calculate the curl of this equation: \(\nabla \times \mathbf{B}\). Since \(\nabla \times \mathbf{B} = \nabla \times (\nabla \times \mathbf{A})\), we can use the identity \(\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}\). Notice that when \(\mathbf{A}\) is the magnetic vector potential, \(\nabla \cdot \mathbf{A}\) is a scalar since it is the multiplication of a vector and a scalar. Thus, the previous identity reduces to the equation \(\nabla \times \mathbf{B} = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}\). This equation satisfies the static limit of Maxwell equation \((8.2.3)\) automatically.
02

(b) Electric field in terms of scalar and vector potential

Recall that the Maxwell-Faraday equation is given by \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\), which display the nonstatic nature of the magnetic field. Since \(\mathbf{B} = \nabla \times \mathbf{A}\), we can write the Maxwell-Faraday equation as \(\nabla \times \mathbf{E} = -\frac{\partial (\nabla \times \mathbf{A})}{\partial t}\). Now, considering the time derivative of the magnetic field \(\mathbf{B}\) with respect to time, we can say that \(\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \frac{\partial \mathbf{A}}{\partial t}\). So, the Maxwell-Faraday equation becomes \(\nabla \times \mathbf{E} = -\nabla \times \frac{\partial \mathbf{A}}{\partial t}\), or equivalently \(\nabla \times (\mathbf{E} + \frac{\partial \mathbf{A}}{\partial t}) = 0\). Since the curl of any gradient is identically zero, we can define \(\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}\).
03

(c) Lorents condition and symmetrical inhomogeneous wave equations

Let's first define the divergence of the vector potential using the Lorents condition: \(\nabla \cdot \mathbf{A} = -\frac{1}{c^{2}} \frac{\partial \phi}{\partial t}\). Then, let's calculate \(\nabla^{2}\phi - \frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}\) and \(\nabla^{2} \mathbf{A} - \frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}\). For \(\nabla^{2}\phi - \frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}\), we can use Gauss' law which is given by \(\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_{0}}\), or equivalently, \(\nabla \cdot (-\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}) = \frac{\rho}{\epsilon_{0}}\). Notice that the right-hand-side is independent of time, which means that \(\nabla \cdot \frac{\partial \mathbf{A}}{\partial t} = 0\). So, the Gauss' law reduces to the equation \(\nabla^{2}\phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}} = -\frac{\rho}{\epsilon_{0}}\). To find the other equation, we can use the Ampère-Maxwell equation which is given by \(\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}\). Since \(\mathbf{H} = \frac{1}{\mu_{0}}\mathbf{B}\) and \(\mathbf{D} = \epsilon_{0} \mathbf{E}\), we can write the Ampère-Maxwell equation as \(\nabla \times (\frac{1}{\mu_{0}}\nabla \times \mathbf{A}) = \epsilon_{0} \frac{\partial}{\partial t}(-\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}) + \mathbf{J}\). With some calculations and using the Lorents condition, we can find the equation \(\nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}} = -\mu_{0} \mathbf{J}\). Thus the two potentials obey the symmetrical inhomogeneous wave equations: $$ \begin{aligned} &\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}} = -\frac{\rho}{\epsilon_{0}} \\ &\nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}} = -\mu_{0} \mathbf{J} . \end{aligned} $$

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Most popular questions from this chapter

(a) For waves varying sinusoidally with time as \(e^{j w t}\), show that the conductivity can be eliminated from (8.5.7) and (8.5.8) by substituting for the relative permittivity the complex quantity $$ \vec{k}_{e}=\kappa_{q}-j \frac{g}{\omega t_{0}} . $$ Then all electromagnetic properties of the medium are contained in only two constants, \(\vec{x}_{e}\) and \(\kappa_{m}\). (b) When currents flow nonuniformly in space, it is possible that a net charge density Pfres builds up at certain locations, Show that the complex permittivity formalism of part (a) not only eliminates the \(\mathbf{J}_{\text {freo }}\) term in Maxwell's equation \((8.2 .20)\) but also eliminates the piros term in \((8.2 .17)\). (c) As an alternative to the formalism of part ( \(a\) ), show that the relative permittivity can be disregarded, i.e., set equal to unity, by introducing the complex conductivity $$ \ddot{g}=g+j \cot \theta\left(\kappa_{c}-1\right) \text {. } $$ In this case, the properties of the medium are specified by the two constants \(g{g}\) and \(\kappa_{w}\) -

Substitute (8.9.3) in (8.9.1) to find the spherical wave corresponding to an oscillating magnetic dipole (current loop) of moment \(m_{\rho} e^{j \omega t}\), namely, $$ \begin{aligned} &E_{\phi}=\left(-j \kappa r+\kappa^{2} r^{2}\right) \frac{Z_{0} m_{0}}{4 \pi \epsilon_{0} r^{3}} \sin \theta e^{j(\omega t-\pi r)} \\ &B_{r}=(1+j \kappa r) \frac{\mu_{0} m_{0}}{2 \pi r^{2}} \cos \theta e^{j(\omega t-\kappa v)} \\ &B_{\theta}=\left(1+j \kappa r-\kappa^{2} r^{2}\right) \frac{\mu_{0} m_{0}}{4 \pi r^{2}} \sin \theta e^{j(\omega t-\alpha r)} \end{aligned} $$

Develop Poynting's theorem for the general material medium of relative permittivity \(\kappa_{\&}\) and permeability \(\kappa_{m}\) introduced in Prob. \(8.2 .2\); i.e., substitute the Maxwell curl equations \((8.2 .18)\) and \((8.2 .20)\) in the expansion of \(\nabla \cdot(\mathbf{E} \times \mathbf{H})\) to obtain $$ \oint_{S}(\mathbf{E} \times \mathbf{H}) \cdot d \mathbf{S}+\int_{V}\left(\mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t}+\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}\right) d v+\int_{V} \mathbf{E} \cdot \mathbf{J} d v=0 $$ from which it follows that the Poynting vector is $$ s=\mathbf{E} \times \mathbf{H} $$ and the energy density is $$ \begin{aligned} \left(W_{1}\right)_{\text {leld }} &=\frac{1}{2} \mathbf{E} \cdot \mathbf{D}+\frac{1}{2} \mathbf{H} \cdot \mathbf{B} \\ &=\frac{1}{2} \kappa_{\ell} \epsilon_{0} E^{2}+\frac{1}{2 \kappa_{\mathrm{m}} \mu_{0}} B^{2} \end{aligned} $$ What restrictions on \(\kappa_{\&}\) and \(\kappa_{m}\) are necessary to obtain \((8.4 .24)\) ?

Consider an ionized gas of uniform electron density \(n\). Regard the positive ions as a smeared-out continuous fluid which renders the gas macroscopically neutral and through which the electrons can move without friction. Now assume that, by some external means, each electron is shifted in the \(x\) direction by the displacement \(\xi=\xi(x)\), a function of its initial, unperturbed location \(x\). (a) Show from Gauss' law (8.2.1) that the resulting electric field is \(E_{x}=n e \xi / e_{0} .(b)\) Show that each electron experiences a linear (Hooke's law) restoring force such that when the external forces are removed, it oscillates about the equilibrium position \(\xi=0\) with simple harmonic motion at the angular frequency $$ \omega_{p}=\left(\frac{n e^{2}}{\epsilon_{0} m}\right)^{1 / 2} $$ which is known as the electron plasma frequency. † For fuller discussion, see J. A. Ratcliffe, "The Magneto-ionic Theory and Its Applications to the Ionosphere," Cambridge University Press, New York, 1959 ; M. A. Heald and C. B. Wharton, "Plasma Diagnostics with Microwaves," John Wiley \& Sons Inc., New York, 1965; I. P. Shkarofsky, T. W. Johnson, and M. P. Bachynski, "Particle Kinetics of Plasma," Addison-Wesley Publishing Company, Reading, Mass., \(1965 .\)

For normal incidence and nonmagnetic materials show that the power coefficients \((8.6 .38)\) and \((8.6 .39)\) reduce to $$ \begin{aligned} &R_{p}=\left(\frac{n-1}{n+1}\right)^{x} \\ &T_{p}=\frac{4 n}{(n+1)^{2}} \end{aligned} $$ where \(n=c_{1} / c_{2}=Z_{01} / Z_{02}\) is the relative refractive index. Account for the difference in sign between the amplitude reflection coefficients \((8.6 .28)\) and \((8.6 .36)\) at normal incidence (see footnote, page 102). Compare with equations (1.9.6), (4.2.15), and (5.6.13).
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