ELECTROMAGNETIC WAVES

Introduction.
In 1864 the English physicist, James Clerk Maxwell (1831-1879), presented to the Royal Society his famous paper "On a Dynamical Theory of the Electromagnetic Field," in which he set forth his four, now famous, equations that describe all electromagnetic radiation. Building on the work of Coulomb, Ampere, Faraday, and their contemporaries who set forth the intimate relationship between electricity and magnetism, that an electric charge and a magnetic pole can exert a force on each other, provided that they are in relative motion, Maxwell translated these relationships into a mathematical form. Using as a model the concept of the field that Michael Faraday (1791-1867) had introduced and the lines-of-force he used to describe them, Maxwell converted the descriptions of electric and magnetic phenomena into mathematical terms. Faraday's great discovery of electromagnetic induction, which he expressed clearly in terms of his lines-of-force model of the electric and magnetic fields, became the fundamental postulate of Maxwell's theory. The work of Faraday, who is regarded as one of greatest experimentalists, was complemented by the work of Maxwell, who ranks among the greatest theoretical physicists. Even though Faraday expressed his discoveries in non-mathematical form, his concepts are eminently mathematical. As Maxwell writes in the preface to his paper,

"As I proceeded with the study of Faraday, I perceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. For instance, Faraday, in his mind's eye, saw lines of force traversing all space where the mathematicians saw centers of force attracting at a distance; Faraday saw a medium where they saw nothing but distance; Faraday sought the seat of the phenomena in real actions going on in the medium, [whereas] they were satisfied that they had found it in a power of action at a distance impressed on the electric fluid."

Maxwell began by developing a clearer picture of how the lines of force functioned in transmitting electric and magnetic forces through the aether. Using the lines of force model, he converted the descriptions of electric and magnetic phenomena into mathematical symbols. Maxwell wrote, "When I had translated what I considered to be Faraday's ideas into a mathematical form, I found that in general the results of the two methods coincided, so that the same phenomena were accounted for, …."

The set of field equations that Maxwell obtained by the translation into mathematical notation was expressed in the language of the calculus; not just in the calculus of functions of one independent variable, but the calculus of functions of more than one independent variable, called partial differentiation and multiple integration. Maxwell did not have the concise notation of the later calculus of vectors that is called vector analysis. The form of vector analysis found in present day American texts was developed by one of the outstanding mathematical physicist of the nineteenth century, the American, John Willard Gibbs (1839-1903). In lecturing to his students at Yale, Gibbs thought that a simpler mathematical language was needed for the theoretical aspects of such subjects as electromagnetics and thermodynamics. His familiarly with the work of the famous Irish theoretical physicist, William Rowan Hamilton (1805-1865), on quaternions and of the German mathematician Herman Grassmann (1809-1877) on generalized multiplication enabled him to pick out those aspects of their work that seemed to apply best to the needs of theoretical physics. Apparently Gibbs did not think that his development of vector analysis was original enough to be worthy of publication; his notes on the subject were circulated among only his students and those interested in the subject. It was twenty years after the development of his original notes, that he allowed them to be published in book form. In 1901, J. B. Wilson published the first book on the subject. Maxwell field equations today are usually expressed most concisely in the notation of the vector calculus. The following four equations in the notation of vector analysis are Maxwell's four equations of electromagnetic fields.
∇ · D = ρ, (1)
∇ · B = 0, (2)
∇ × E = -∂B/∂t, (3)
∇ × H = J + ∂D/∂t, (4)
where D = εE and B = μH. These four equations (1) through (4) are called Maxwell's electromagnetic equations, and they express concisely the experimental knowledge of electromagnetic fields as contained in the laws of Coulomb, Ampere, Faraday, and Maxwell.

These equations can be summarized in words as follows:
Equation (1) says that an electric field is produced by electric charge; this equation is a generalized form of Coulomb's Law of Electrostatics Forces called Gauss' Law for electricity.
Equation (2) says that magnetic field lines are continuous; they do not begin or end, as do electric field lines do. This equation is Gauss' law for magnetism. It essentially says that there are no isolated magnetic poles (monopoles) that are the sources of magnetic fields.
Equation (3) says that an electric field is produced by a changing magnetic field; this equation is a generalization of Faraday's Law of Electromagnetic Induction.
Equation (4) says that a magnetic field is produced by an electric current or by a changing electric field; the equation is a generalization of Ampere's Law.

These equations can also be stated in an integral form as applied to free space, that is, to a space where there are no dielectric or magnetic materials. The four equations are:
∫_SD · dA = q, (5)
where q = q₁ + q₂ + q₃ + ... is the total net charge inside the closed surface.
∫_SB · dA = 0, (6)
∫_λE · λ = -(∂φ/∂t), (7)
where φ is magnetic field flux lines.
∫_λH · λ = I + ε₀(∂ψ/∂t), (8)
where ψ is electric field flux lines.

Let us discuss these integral equations (5) through (8) in detail.
Equation (5) is Gauss's Law for Electricity which states that the total electric flux (lines of force) through any closed surface equals the net charge inside that surface. This relates the electric field to the total charge distribution, where the electric field lines orginate on positive charges and they terminate on negative charges.
Equation (6) is Gauss's Law for Magnetism which states that the net magnetic flux through a closed surface is zero. That is, the number of magnetic field lines that enter into a closed volume must equal the number that leave the volume. This implies that magnetic field lines cannot begin or end at any point. If they did, this would mean that there are isolated magnetic monopoles at these points. The fact that isolated magnetic monopoles have not been experimentally detected in the physical world may be taken as a confirmation of this law.
Equation (7) is Faraday's Law of Induction which states that the line integral of a electric field intensity around any closed path (which equals the emf) equals the time rate of change of the magnetic flux through any surface area bounded by that path. One consequence of Faraday's Law is that a current will be induced in conducting loop placed in a time-varying magnetic field.
Equation (8) is Maxwell's generalization of Ampere's Law, which describes the relationship between magnetic fields and changing electric currents and changing electric fields. It states that the line integral of the magnetic field intensity around any closed path is determined by the sum of the net conduction current through that path and the time rate of change of the electric flux through any surface bounded by that path.

Note the symmetry of Maxwell's equations. Equations (5) and (6) are symmetric, apart from the absence of a magnetic monopole term in equation (6). And equation (7) and (8) are symmetric, apart from the electric conductance current term in equation (8), in that the line integrals of E and H around a closed path are related to the time rate of change of magnetic and electric flux, respecively.

The Electromagnetic Wave Equations.
Let us obtain the equations of electromagnetic waves. In empty space, where q = 0 and I = 0, these equations have a wavelike solution, where the wave velocity is equal to the measured speed of light. This result led Maxwell to predict that light waves are, in fact, a form of electromagnetic radiation.

To obtain these wave equations, we must assume that the electromagnetic waves are plane waves, that is, the waves travel in one direction. Also we must assume that E and B at any point P depend upon only the coordinates of the direction of travel and time, and not upon any other coordinates of the point P. Separate equations for these vectors E and B must be obtained that contain only one vector. Let us obtain, for example, the wave equation for the vector E in empty space.

Now, for empty space, in the equations (1) and (4), ρ and J must be set to zero, so that Maxwell's equations take the following symmetrical form.
∇ · D = 0, (9)
∇ · B = 0, (10)
∇ × E = -∂B/∂t, (11)
∇ × H = ∂D/∂t, (12)
From these equations (9) through (12) we can obtain the wave equation for E.
First, using equation (11), let take the curl of the curl of E, that is,
∇ × (∇ × E) = -(∇ × ∂B/∂t).
Expanding the triple vector product on the left side of this equation, we get
∇(∇ · E) - ∇ · ∇E = -(∇ × ∂B/∂t). (13)
From Maxwell's first equation, equation (9), we find that
∇ · D = ∇ · (ε₀E) = ε₀∇ · E = 0.
Thus ∇ · E = 0, and substituting this into equation (13), we get
∇ · ∇E = (∇ × ∂B/∂t). (14)
From Maxwell's last equation, equation (12), and since
D = εE and B = μH, we get
∇ × H = ∇ × (B/μ₀) = (1/μ₀)(∇ × B) = ∂D/∂t = ε₀(∂E/∂t),
or
∇ × B = μ₀ε₀(∂E/∂t).
Taking the time derivative of this last equation, we get
∇ × (∂B/∂t) = μ₀ε₀ (∂²E/∂t²).
Substituting this back equation (14), we get
∇ · ∇E = μ₀ε₀ (∂²E/∂t²). (15)
This is the wave equation for E, as can be seen by comparing this equation (15), writing it in ordinary calculus notation for a wave traveling in the direction of the positive x-coordinate,
(∂²E/∂x²) = μ₀ε₀ (∂²E/∂t²), (16)
with the general three-dimensional wave equation, which has the form
(∂²f/∂x²) = (1/v²) (∂²f/∂t²),
where f is the wave function amplitude and v is the speed of the wave, assuming that the wave is traveling in the direction of the positive x-coordinate.
Letting v = c, then
(1/c²) = μ₀ε₀,
and solving for c, we get
c = 1/√(μ₀ε₀). (17)
Thus we conclnde that the electric field E is propagated in infinite empty space with a speed c.

Similar wave equations may be derived for D, B, and H, and the speed of propagation will be the same as for E. The wave equation for H is
∇ · ∇H = μ₀ε₀ (∂²H/∂t²),
or, written in ordinary calculus notation for a wave traveling in the direction of the positive x-coordinate,
(∂²H/∂x²) = μ₀ε₀ (∂²H/∂t²). (18)

Now let us compute the value of c.
From the definition of the ampere, using Ampere's cuarrent force law,
μ₀ = 4π × 10^-7 nt/amp².
From measurements, using Coulomb's law of force between two known charges, the experimental value of ε₀ can be obtained. This value is very nearly
ε₀ = 1/(4π × 9 × 10⁹) coul²/nt·m².
Multiplying these values, we get
μ₀ε₀ = (4π × 10^-7 nt/amp²) [1/(4π × 9 × 10⁹) coul²/nt·m²] = (1/9) × 10^-16 (sec/m)².
And substituting this value into equation (17), we get
c = 3 × 10⁸ m/sec, (19)
which is, within experimental error, the speed of light. Maxwell's equations leads to the conclusion that electric and magnetic fields may be propagated thorugh empty space as waves whose speed is the same as the speed of light.

The simplest solutions of the partial differential equations (16) and (18) are plane sinusoldal waves, in which the field amplitudes E and H vary with x and t according the equations:
E = E_m cos (kx - ωt), (20)
H = H_m cos (kx - ωt), (21)
where E_m and H_m are the maximum values of the field amplitudes E and H. The constant k is equal to 2π/λ, where λ is the wavelength of the wave, and ω is the angular frequency that is equal to 2πf, where f is the frequency of the wave in the number of cycles per second. The ratio ω/k is equal to the speed c of the wave, since
(ω/k) = (2πf)/(2π/λ)= λf = c.

We shall now show that these electromagnetic waves are transverse, not longitudinal, just as light waves. Consider E to be propagated as a plane wave in the x-direction, so that E and its components are not functions of y and z. From Maxwell's first equation (9) for free space,
∇ · E = 0, we get
∂E_x/∂x + ∂E_y/∂y + ∂E_z/∂z = 0.
For this plane wave ∂E_y/∂y and ∂E_z/∂z are zero, hence ∂E_x/∂x is also zero. This means that E_x does not vary with x, as would a longitudinal wave in the x-direction. Here E_y and E_z may depend on x, but they are components of E perpendicular to the direction of propagation. A similar argument follows for H from the equation ∇ · H = 0. Hence, plane electromagnetic waves are transverse waves.
These similarity between the predicted properties of electromagnetic waves and those of light led Maxwell to postulate that light is electromagnetic waves, Hertz about fifteen years after Maxwell published his theory, first produced electromagnetic waves experimentally, and the waves that he produced experimentally have been found to have all the properties predicted by Maxwell.

The Relation between E and H in a plane wave.
Consider again the electromagnetic plane ray traveling in the positive x-direction, which has been shown above to be a transverse waves. Without any loss of generality, let us take the positive y-axis to be the direction of the vector E, so that E_x = E_z = 0 and E_y = E. Since E_x and E_z are zero, and E_y can vary only with x and t for plane waves in the x-direction, then ∂B_x/∂t = ∂B_y = 0. This means that the only component of B that varies with time, and therefore is being propagated as part of the wave, is B_z. Therefore, if the electric vector is in the y-direction, then the magnetic vector is in the z-direction. The vectors E and B (or H) are mutually perpendicular and E × B, or B × E, is in the direction of the propagation of the wave.

Expressing Maxwell's third equation (11) ∇ × E = -∂B/∂t in calculus notation, we have
∂E_z/∂y - ∂E_y/∂z = -∂B_x/∂t, (11A)
∂E_x/∂z - ∂E_z/∂x = -∂B_y/∂t, (11B)
∂E_y/∂x - ∂E_x/∂y = -∂B_z/∂t. (11C)
Under the above conditions of the wave traveling in the x-direction, with the direction of the E vector in the y-direction, these partial differential equations reduce to
∂E_y/∂x = -∂B_z/∂t, (22)

Similarly, expressing Maxwell's third equation (12) in calculus notation, we have
∂H_z/∂y - ∂H_y/∂z = ∂D_x/∂t, (12A)
∂H_x/∂z - ∂H_z/∂x = ∂D_y/∂t, (12B)
∂H_y/∂x - ∂H_x/∂y = ∂D_z/∂t, (12C)
Again, under the above conditions of the wave traveling in the x-direction, with the direction of the E (or D) vector in the y-direction and B (or H) vector in the z-direction, these partial differential equations reduce to
-∂H_z/∂x = ∂D_y/∂t, (23)
For plane waves traveling in the positive x-direction with the speed c, both E_y and H_z must be functions of f(x - ct). Therefore,
∂f/∂t = -c(∂f/∂x), so that,
∂E_y/∂t = c(∂E_y/∂x), and ∂H_z/∂t = -c(∂H_z/∂x),
and the equations (22) and (23) become
∂E_y/∂x = -∂B_z/∂t = -μ₀(∂H_z/∂x) = -cμ₀(∂H_z/∂x), and
-∂H_z/∂x = ∂D_y/∂t = ε₀(∂E_y/∂t) = cε₀(∂E_y/∂x).
Integrating these equations, we get
E_y = μ₀cH_z, and H_z = ε₀cE_y.
Squarting these equations, we get
E_y² = μ₀²c² H_z², and H_z² = ε₀²c² E_y².
Since c = 1/√(μ₀ε₀) and squaring, we get
c² = 1/(μ₀ε₀).
Substituting into the above equations and simplfying, we get
ε₀E_y² = μ₀H_z², and μ₀H_z² = ε₀E_y².
And since E_y = E and H_z = H, we get a single equation
ε₀E² = μ₀H².

Production of Electromagnetic Waves.
In 1856, Maxwell showed theoretically that electromagnetic waves, which propagate with the velocity of light (3 × 10⁸ m/sec), should be emitted from an oscillating electric circuit. He assumed the existence of a displacement current, ε₀dE/dt, which, like any conduction current, has associated with it a magnetic field. Faraday had shown that a changing magnetic field dB/dt gives rise to an electric current and an electric field. Thus Maxwell's theory showed essentially that a changing electric field should give rise to a changing magnetic field and a changing magnetic field should give rise to a changing electric field. This electromagnetic wave consisting of changing magnetic and electric fields at right angles to each other, is propagated with the velocity of light.

The evidence of these electromagnetic waves was first experimentally shown by Henry in 1842; but, because of his lack of publication, the credit is given to Heinrich Hertz, who demonstrated them about 1888. Oscillations were set up in capacitor-inductor circuit when the spark from an induction coil discharged the capacitor. At some short distance away, the electromagnetic radiation from the high-frequency oscillationsof the capacitor-inductor circuit was picked up. The receiver was a single turn of wire with a small gap in it, and the electrical oscillations caused a spark to pass across the gap.

The oscillations producing the electromagnetic radiation in the Hertzian oscillator were quickly damped, and, though modifications were made in the circuit, it was with some difficulty that dot-and-dashed code messages could be transmitted with this apparatus. Marconi at the turn of the century was able to transmit a short message across the English channel and across the Alantic. However, it was not until the advent of triode vacuum tube and other multigrid tubes that sustained oscillations could be set up so that speech and music could be transmitted.

In order to get sustained oscillations in a circuit, some of the output electrical energy must be fed back into the input portion of the electrical circuit. Armstrong originally introduced the feedback circuit for producting sustained oscillations. Energy is fed by the connection from the plate to the grid. When properly set up, oscillations taking place in the capacitor-inductor circuit, called the tank circuit, are maintained by the feedback from the tube. The frequency of the oscillations is nearly equal to that of the natural resonant frequency of the tank circuit; that is,
f = 1/(2π√LC),
where L is the inductance of tank circuit and C is the capacitance of the tank circuit. By adjusting C, the frequency of tank circuit can be altered. Oscillators may be designed for audio frequencies up to 20,000 cycles/sec and for radio frequencies up to millions cycles per second, that is, megacycles per second.

In order that electromagnetic waves may be used for voice communication is necessary to impress the audio signal upon the wave. One way commonly used is amplitude modulation (AM). This method alters the amplitude of the high frequency waves (called the carrier) by the low frequency oscillations, produced by a microphone or telephone. A simple AM transmitter consists of an RF (Radio Frequency) oscillator that generates a continuous voltage fluctuations across a coil. The RF voltage is then fed to the grid of the triode. A microphone circuit is also inductively coupled to the grid circuit of the triode, so that any voltage fluctuations in the mircrophone circuit will give rise to corresponding voltage fluctuations in the grid circuit. These audio-signals are combined with the RF signal in the grid circuit, thereby changing or modulating the amplitude of the RF signal. The resulting voltage fluctuations are transmitted to the antenna system and radiated out into space. This method of amplitude modulation is called grid modulation.

These amplitude modulated electromagnetic or radio waves are received by an electronic circuit called an AM receiver. A simple AM reciver consists of a receiving antenna that intercepts a portion of the AM electromagnetic waves. As the waves move by, the electric field induces a RF voltage in the antenna system, which is then fed by a transmission line to the resonant circuit that selects the desired carrier frequency or station. This selected RF signal is fed thence to a RF amplifier circuit that increases the voltage of the RF signal. This wave is an exact replica of the transmitted wave. The resulting amplified RF voltage is next fed into a rectifier or detector tube (a diode) wherein the lower half of the RF signal is eliminated. The signal is then fed into a filter circuit that removes the high frequency RF signal. The final audio output is fed into the speaker system.

In transmitting speech and music by means of amplitude modulated waves, the radio wave needs a band of frequencies centered upon the frequency of the carrier. Suppose that f_c is the frequency of the carrier wave and f_a is the frequency of audio signal. It can be shown that the amplitude modulated wave consists of three frequencies, that is, f_c, (f_c + f_a), (f_c - f_a). The frequencies from f_c to (f_c + f_a) is called the upper side band and the frequencies from (f_c - f_a) to f_c is called the lower side band; the band width of the radio wave includes these two side bands and is from the frequency (f_c - f_a) to (f_c + f_a). This phenomena is similar to that of beats in sound.

There is also another way of transmitting audio-signals by radio waves, which is called frequency modulation (FM). In this method of transmitting audio signals the amplitude of the carrier wave remains constant and is not modulated, but its frequency is modulated by the impressed audio-frquencies. Frequency modulated radio waves have band widths similar to those of the amplitude modulated waves.