12.1.1^*

In Sec. 10.1, the electric field in an MQS system was divided into a particular part E_p satisfying Faraday's law, and an irrotational part E_h. The latter was adjusted to make the sum satisfy appropriate boundary conditions. Show that in terms of and A, as defined in this section, E_p = - A/ t and E_h = -, where these potentials satisfy (12.1.8) and (12.1.10) with the time derivatives neglected.

12.1.2

In Sec. 3.3, dimensional arguments were used to show that the quasistatic limits were valid in a system having a typical length L and time if L/c . Use similar arguments to show that the second term on the right in either (12.1.8) or (12.1.10) is negligible when this condition prevails. Note that the resulting equations are those for MQS (8.1.5) and EQS (4.2.2) systems.

Electrodynamic Fields of Source Singularities

12.2.1

An electric dipole has q(t) = 0 for t < 0 and t > T. When 0 < t < T, q(t) = Q[1 - cos (2 t/T)]/2. Use sketches similar to those of Figs. 12.2.5 and 12.2.6 to show the field distributions when t < T and T < t.

12.2.2^*

Use the "interchange of variables" property of Maxwell's equations to show that the sinusoidal steady state far fields of a magnetic dipole, (12.2.35) and (12.2.36), follow directly from (12.2.23), (12.2.24), and (12.2.32).

12.2.3^*

A magnetic dipole has a moment m(t) having the time dependence shown in Fig. 12.2.5a where dq(t) m(t). Show that the fields are then much as shown in Fig. 12.2.6 with E H, H -E, and .

Antenna Radiation Fields in the Sinusoidal Steady State

12.4.1

An "end-fed" antenna consists of a wire stretching between z = 0, where it is driven by the current I_o cos ( t - _o), and z = l. At z = l, it is terminated in such a resistance that the current distribution over its length is a wave traveling with the velocity of light in the z direction; i(z, t) = Re [I_o exp [j( t - kz + _o)]] where k /c.

(a) Determine the radiation pattern, ( ).

(b) For a one-wavelength antenna (kl = 2 ), use a plot of | ( )| to show that the lobes of the radiation pattern tend to be in the direction of the traveling wave.

12.4.2^*

An antenna is modeled by a distribution of incremental magnetic dipoles, as shown in Fig. P12.4.2. Define (z) as a dipole moment per unit length so that for an incremental dipole located at z^' , (z^' )dz^'. Given , show that equation GIF #12.146

where

Figure P12.4.2

12.4.3

A linear distribution of magnetic dipoles, described in general in Prob. 12.4.2, is excited so that (z^' ) = - _o exp (j_o) sin (z - l)/sin l, 0 z l where is a given parameter (not necessarily /c). Determine _o( ).

12.4.4^*

For the three-element array shown in Fig. P12.4.4, the spacing is /4.

(a)	Show that the array factor is
(b)	Show that for an array of in-phase short dipoles, the "broadside" radiation intensity pattern is
(c)	Show that for an array of short dipoles differing progressively by 90 degrees so that ₁ - _o = /2 and ₂ - _o = , the end-fire radiation pattern is

Figure P12.4.4

12.4.5

Collinear elements have the half-wave spacing and configuration shown in Fig. P12.5.5.

(a)	Determine the array factor _a ( ).
(b)	What is the radiation pattern if the elements are "short" dipoles driven in phase?
(c)	What is the gain G( ) for the array of part (b)?

Figure P12.5.5

Complex Poynting's Theorem and Radiation Resistance

12.5.1^*

A center-fed wire antenna has a length of 3 /2. Show that its radiation resistance in free space is 104 . (The definite integral can be evaluated numerically.)

12.5.2

The spherical coil of Example 8.5.1 is used as a magnetic dipole antenna. Its diameter is much less than a wavelength, and its equivalent circuit is an inductance L in parallel with a radiation resistance R_rad. In terms of the radius R, number of turns N, and frequency , what are L and R_rad?

Waves

12.6.1^*

In the plane y = 0, K_z = 0 and the surface charge density is given as the traveling wave _s = Re _o exp [j( t - k_x x)] = Re [_o exp (-jk_x x) exp (j t)], where _o, , and k_x are given real numbers.

(a)	Show that the current density is
(b)	Show that the fields are where upper and lower signs, respectively, refer to the regions where 0 < y and 0 > y.
(c)	Sketch the field distributions at a given instant in time for imaginary and real.

12.6.2

In the plane y = 0, the surface current density is a standing wave, K = Re [i_z K_o sin (k_x x) exp (j t)], and there is no surface charge density.

(a)	Determine E and H.
(b)	Sketch these fields at a given instant in time for real and imaginary.
(c)	Show that these fields can be decomposed into waves traveling in the x directions with the phase velocities /k_x.

12.6.3^*

In the planes y = d/2, shown in Fig. P12.6.3, there are surface current densities K_z = Re exp [j( t - k_x x)], where = ^a at y = d/2 and = ^b at y = -d/2. The surface charge density is zero in each plane.

(a)	Show that
(b)	Show that if ^b = -^a exp (-j d), the fields cancel in region (b) where (y < -d/2), so that the combined radiation is unidirectional.
(c)	Show that under this condition, the field in the region y > d/2 is
(d)	With the structure used to impose the surface currents such that k_x is fixed, show that to maximize the wave radiated in region (a), the frequency should be and that under this condition, the direction of the radiated wave is k = k_x i_x + [(2n + 1) /2d]i_y where n = 0, 1, 2, .

12.6.4

Surface charges in the planes y = d/2 shown in Fig. P12.7.3 have the densities _s = Re exp [j( t - k_x x)] where = ^a at y = d/2 and = ^b at y = -d/2.

related to produce field cancellation in region (b)?

(a) How should ^a and ^b be

(b) Under this condition, what is H_z in region (a)?

(c) What frequencies give a maximum H_z in region (a), and what is the direction of propagation under this condition?

Figure P12.7.3

Conductors

12.7.1^*

An antenna consists of a ground plane with a 3 /4 vertical element in which a "quarter-wave stub" is used to make the current in the top half-wavelength in phase with that in the bottom quarter-wavelength. In each section, the current has the sinusoidal distribution shown in Fig. P12.7.1. Show that the radiation intensity factor is equation GIF #12.157

Figure P12.7.1 touching GIF #P12.7.2

Figure P12.7.2

12.7.2

A vertical half-wave antenna with a horizontal perfectly conducting ground plane is shown in Fig. P12.7.2. What is its radiation resistance?

12.7.3

Plane parallel perfectly conducting plates in the planes x = a/2 form the walls of a waveguide, as shown in Fig. 12.7.3. Waves in the free-space region between are excited by a sheet of surface charge density _s = Re _o cos ( x/a) exp (j t) and K_z = 0.

solutions that meet both the continuity conditions at the sheet and the boundary conditions on the perfectly conducting plates.) Are they TE or TM?

(a) Find the fields in regions 0 < y and 0 > y. (Guess

(b) What are the distributions of _s and K on the perfectly conducting plates?

(c) What is the "dispersion equation" relating to ?

(d) Sketch E and H for imaginary and real.

12.7.4

Consider the configuration of Prob. 12.7.3, but with _s = 0 and K_z = Re _o cos ( x/a) exp (j t) in the plane y = 0. Complete parts (a)-(d) of Prob. 12.7.3.