Chapter 8

(a)\( Generalize the boundary conditions \)(8.6 .5)\( to \)(8.6 .8)\( to include the case where a surface charge density \)\sigma=\Delta q_{\text {tree }} / \Delta S\( and a surface current of magnitude \)K=\Delta I_{\text {fres }} / \Delta l\( exist on the boundary surface, establishing the conditions $$ \text { fi } \begin{aligned} & \cdot\left(x_{22} \mathbf{E}_{2}-x_{41} \mathbf{E}_{1}\right)=\frac{\sigma}{\epsilon_{0}} \\ \text { fi } \times\left(\frac{\mathbf{B}_{2}}{k_{m 2}}-\frac{\mathbf{B}_{1}}{k_{m 1}}\right) &=\mu_{0} \mathbf{K} . \end{aligned} $$ (b) Show that the boundary conditions remain valid when the boundary is not plane and when the respective media are not homogeneous. (c) What are the boundary conditions on the \)\mathbf{D}\( and \)\mathbf{H}$ fields?

Consider an electric dipole consisting of a charge \(-e\) oscillating sinusoidally in position about a stationary charge \(+e\). Show that the instantaneous total power radiated can be written in the form $$ \frac{d W}{d t}=\frac{e^{2}[a]^{2}}{6 \pi \epsilon_{0} c^{3}} $$ where \([a]=a_{0} e^{i(\omega t-k n)}\) is the instantaneous (retarded) acceleration of the moving charge Since this result does not depend upon the oscillator frequency, and since by Fourier analysis, an arbitrary motion can be described by superposing many sinusoidal motions of proper frequency, amplitude, and phase, this rate-of-radiation formula has general validity for any accelerated charge (in the nonrelativistic limit \(\diamond \ll c\) ).

(a) Show that the skin depth \(\delta\) can be put in the form $$ \delta=\left(\frac{\lambda_{0}}{\pi Z_{\mu} \pi_{m} g}\right)^{1 / 2} $$ where \(Z_{0}\) is the free-space wave impedance \((8.3,10)\) and \(\lambda_{0}=2 \pi c / \omega\) is the vacuum wavelength. (b) Evaluate s for copper, for waves having wavelengths in vacuum of \(5,000 \mathrm{~km}(60-\mathrm{Hz}\) power line); \(100 \mathrm{~m}\) ( \(\sim\) AM broadcast band); \(1 \mathrm{~m}\) ( \(\sim\) television and FM broadcast); \(3 \mathrm{~cm}\) ( \(\sim\) radar); \(500 \mathrm{~nm}\) ( \(\sim\) visible light). How does the size of the skin depth affect the technology of these various applications" (c) The electrical conductivity of sea water is about \(4 \mathrm{mhos} / \mathrm{m}\). How would you communicate by radio with a submarine \(100 \mathrm{~m}\) below the surface?

Consider an E-field line of force, i.e., a continuous line everywhere parallel to the local direction of \(\mathbf{E}\), deflected at the boundary between two uniform media. Show that the exit line of force lies in the plane determined by the entrance line and the normal to the boundary surface and that the angles of incidence \(\theta_{1}\) and exit \(\theta_{2}\), measured with respect to the normal, are related by the Snell's law equation $$ \frac{1}{\kappa_{\theta 1}} \tan \theta_{1}=\frac{1}{K_{\theta 2}} \tan \theta_{2} \text {. } $$ What are the corresponding equations for \(\mathbf{B}, \mathbf{D}\), and \(\mathbf{H}\) ?

Use the results of Prob. 8.2.3 to compute the Poynting vector for a coaxial transmission line. Integrate it over the annular area between conductors and show that the power carried down the line by the wave is $$ P=i^{2} Z_{0}=\frac{v^{2}}{Z_{0}}, $$ where \(i\) and \(v\) are the instantaneous current and voltage and \(Z_{0}\) is the characteristic impedance \((8.1 .9)\), that is, just the result one would expect from elementary circuit analysis.

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, $$ \begin{aligned} &E_{\phi}=\left(-j \kappa r+\kappa^{2} r^{2}\right) \frac{Z_{0} m_{0}}{4 \pi \epsilon_{0} r^{3}} \sin \theta e^{j(\omega t-\pi r)} \\ &B_{r}=(1+j \kappa r) \frac{\mu_{0} m_{0}}{2 \pi r^{2}} \cos \theta e^{j(\omega t-\kappa v)} \\ &B_{\theta}=\left(1+j \kappa r-\kappa^{2} r^{2}\right) \frac{\mu_{0} m_{0}}{4 \pi r^{2}} \sin \theta e^{j(\omega t-\alpha r)} \end{aligned} $$

The text following (8.2.10) refers to low-frequency (or dc) laboratory measurements of \(\epsilon_{0}\) and \(\mu_{0}\). How could you determine these constants? What logical chain of definitions and calibrations would be needed?

A long straight wire of radius a carries a current \(I\) and has resistance \(R_{1}\) per unit length. Compute \(\mathbf{E}\) and \(\mathbf{B}\) at its surface and show that the rate of energy flow into the wire via the Poynting flux is \(I^{2} R_{1}\) per unit length.

Consider \(\mathbf{E}\) and \(\mathbf{B}\) wave fields whose only dependence on \(z\) and \(t\) is included in the factor \(e^{i\left(\omega t-x_{1} \theta\right)}\). Further assume TE waves such that \(E_{z}=0\). Write out Maxwell's curl equations \((82.2)\) and \((8.2 .4)\) in cartesian components and show \((a)\) that all four transverse field components can be obtained from \(B_{t}\) by first-order partial differentiation and \((b)\) that \(B_{*}\) must be a solution of the Helmholtz equation \((8.7 .16)\). Thus the scalar function \(\phi\) of the text may be interpreted as proportional to \(B_{z}\) for TE waves or proportional to \(E_{s}\) for TM waves.

Consider a general tu o-conductor transmission line for which the conductors have a (round-trip) series resistance per unit length \(R_{1}\) and the medium between conductors has a leakage conductance per unit length \(G_{1}\). Show that the valtage and current waves then obey the telegrapher's equalion $$ \frac{\partial^{s} v}{\partial s^{2}}=L_{1} C_{3} \frac{\partial^{s} v}{\partial r^{2}}+\left(R_{1} C_{1}+G_{1} L_{1}\right) \frac{\partial v}{\partial t}+R_{1} G_{1} v $$ What can you discover about the solutions of this equation? Show that the characteristic impedance for monochromatic waves is $$ \breve{Z}_{0}=\left(\frac{R_{1}+j \omega L_{1}}{G_{1}+j \omega C_{1}}\right)^{1 / 2} $$