Chapter 39, Radiation Iii Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

The radiation patterns of the electric dipole and of the half-wave antenna
(a) Show that in the far field of an electric dipole $E_{\mathrm{rms}}=6.71 P^{1 / 2} \sin \theta / r$.
(b) Show that for the half-wave antenna
$$
E_{\mathrm{rms}}=7.02 \frac{P^{1 / 2} \cos \{(\pi / 2) \cos \theta\}}{r \sin \theta}
$$

Manish Jain

Numerade Educator

01:33

Problem 2

Calculate $E$ at a distance of 1 kilometer in the equatorial plane of a half-wave radio antenna radiating 1 kilowatt of power. Set $\lambda \ll 1$ kilometer.

Ajay Singhal

Numerade Educator

02:17

Problem 3

An antenna is normally situated near a conductor (the earth, an airborne vehicle, a satellite, etc.). Energy radiated toward the conductor is reflected, and the total field is thus the vector sum of the direct wave plus the reflected wave. It is convenient to consider that the latter is generated, not by reflection, but by an image of the antenna located behind the surface of the conductor.
(a) Show that the current in the image of a horizontal half-wave antenna and the current in the antenna flow in opposite directions.
(b) Show that the current in the image of a vertical half-wave antenna and that in the antenna flow in the same direction.
Both rules apply to oblique half-wave antennas.
(c) We have shown that the radiation resistance of a half-wave antenna is $73.1$ ohms. Find the radiation resistance of a quarter-wave antenna perpendicular to a conducting plane.

Khoobchandra Agrawal

Numerade Educator

06:24

Problem 4

A linear array consists of parallel half-wave antennas lying in a plane. Say there are $N$ antennas, uniformly separated by a distance $D$ and excited in phase.
(a) Show that, in the plane perpendicular to the antennas,
$$
E \propto \frac{\sin \{(N D / 2 \chi) \cos \phi\}}{\sin \{(D / 2 \lambda) \cos \phi\}}
$$
where $\phi$ is the angle between the direction of observation and the plane of the array. The best approach is to sum the individual $\boldsymbol{E}$ phasors graphically in the complex plane.
(b) Find the angular positions of the minima and maxima of $E$. Differentiation yields only the maxima.
(c) Show that, for a given spacing $D$, the main lobe at $\phi=\pi / 2$ becomes narrower as $N$ increases.
(d) Draw a polar diagram of $E$ as a function of $\theta$ between 0 and $360^{\circ}$ for an array of 30 parallel half-wave antennas that are in phase and spaced by $\lambda / 4$.
(e) Now plot the same function, using Cartesian coordinates, between 0 and $180^{\circ}$ with a log scale for the $E$-axis.
(f) Explain why the main lobe is twice as wide as the two neighboring lobes.
(g) Show that its half-width (the angle between the maximum and the first minimum on one side or the other) is approximately equal to $\lambda / l$, where $l$ is the length of the arrav

Mayukh Banik

Numerade Educator

08:15

Problem 5

By definition, the directivity of an antenna is equal to the ratio of the Poynting vector at the maximum of the radiation pattern to the Poynting vector averaged over a spherical surface surrounding the antenna:
$$
D=\frac{\mathscr{S}_{\max }}{P /\left(4 \pi r^{2}\right)}=\frac{\mathscr{S}_{\max }}{\frac{1}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \mathscr{S} \sin \theta d \theta d \phi}
$$
(a) Show that, the directivity of an electric or magnetic dipole is $1.5$.
(b) Show that the directivity of a half-wave antenna is $1.64$.

Mandeep Mangat

Numerade Educator

03:04

Problem 6

Show that, for a given diameter and for a given mass of copper, the ratio $\omega L / R$ for a magnetic dipole antenna is independent of the number of turns, $L$ and $R$ being, respectively, the inductance and the resistance of the dipole.

Increasing the number of turns increases the resistance, the inductance, and the stray capacitance of the coil. The impedance is thus, in fact, a complicated function of the number of turns.

Amit Srivastava

Numerade Educator

06:51

Problem 7

You have two receiving antennas. One is an electric dipole of length $l$, and the other is a single-turn magnetic dipole of diameter $l$.
(a) Calculate the ratio of the induced voltages far away from an electric dipole and close by, in the equatorial plane.
(b) Repeat the calculation for the field of a magnetic dipole.

Katie Mcalpine

Numerade Educator

02:49

Problem 8

The azimuthal and centripetal accelerations in the magnetic dipole; synchrotron radiation
We found that the radiation fields of electric and magnetic dipoles result from the accelerations of the electric charges.
Show that the centripetal acceleration is negligible in the oscillating magnetic dipole. Assume a copper torus of major radius $R_{1}$ and minor radius $R_{2}$ and a current $I_{m} \cos \omega t$. Assume also that the current is uniformly distributed over a thickness equal to the skin depth.
Set $R_{1}=200$ millimeters, $R_{2}=10$ millimeters, $I_{m}=1$ ampere, $\rho=1.3 \times$ $10^{10}$ coulombs/meter $^{3}$ of conduction electrons, and $f=1$ megahertz.

Yaqub Khan

Numerade Educator

02:00

Problem 9

Figure $39-9$ shows two parallel electric dipoles, one of which acts as a transmitting antenna and the other as a receiving antenna. The distance $r$ is much larger than the lengths of the dipoles. How does the signal at the receiving antenna vary with the distance $r \gg \lambda$ and with the angle $\theta$ ?

Manish Jain

Numerade Educator

01:16

Problem 10

A loop receiving antenna of inductance $L$ feeds a load resistance $R$. The resistance of the loop is negligible compared to $R$.

Show that there is maximum power transfer to the load when $R=\omega L$.

Surendra Kumar

Numerade Educator

01:13

Problem 11

Compare the responses of electric and of magnetic dipole antennas used as receivers in seawater. Assume a frequency of 20 kilohertz, a typical frequency for communicating with submarines. Assume that the loop antenna has a single turn, that its diameter is equal to the length $l$ of the electric dipole, and that $l<\lambda$ in seawater.

Averell Hause

Carnegie Mellon University

03:43

Problem 12

Two identical magnetic dipoles are perpendicular and have a common diameter.
(a) Show that the radiation pattern is a circle in the plane perpendicular to the common diameter if one dipole leads the other by $\pi / 2$ radians.
(b) Explain the nature of the resulting field.
(c) How would you connect these antennas to a common source?
Such a pair of crossed coils forms an omnidirectional transmitting or receiving antenna.

Mayukh Banik

Numerade Educator