ELECTROMAGNETIC OSCILLATIONS

Introduction.

The simplest form of electromagnetic oscillation is an inductor connected across an initially charged capacitor. Let us derive an expression for the frequency of these free oscillations.

Electric Free Oscillations.

Consider a LC circuit, in which a capacitor with a capacitance C, initially carrying a charge of magnitude Qm on each plate, is connected with a switch across an inductor with an inductance of L. For simplicity we assume that the circuit contains no resistance. If a wire is connected across the capacitor which had no self-inductance, the charges on capacitor is neutralized. With the inductor present, any change in the current through the inductor and therefore any change in the magnetic flux through the coil, will give rise to a self-induced emf whose direction is always such as to maintain a constant current.

When the switch is closed, at that instant the capacitor starts to discharge. The inductor sets up a back emf, opposing the discharge. When the charge q on either plate has reached zero, a current will still exist in the circuit, and this current continues to exist as the effect of the induced emf, which now opposes the decrease of current. But as the current continues and charges accumulate on the capacitor plates in the reverse sense. The current decreases as the charge arrives as first charge arriving on the capacitor plates repel other charges arriving later. The current falls to zero, and now the capacitor is again fully charged, but with opposite polarity. At this point the electric oscillation has completed exactly one-half of an oscillation cycle. After this the process is repeated, but in the opposite sense. The capacitor again loses its charge, a current is created, now in the opposite direction, until the capacitor again reaches its initial state. Then one cycle of the oscillation has been completed.

The oscillations in electric charge and electric current continues. No energy is dissipated since we assumed no resistance. Indeed, the electric oscillations consists of continuous alternation of energy stored in the electric field of the charged capacitor and energy stored in the magnetic field of the current-carrying inductor. The two circuit elements play different roles:
1. The capacitor stores electric energy in the electric field when charged, but it tends to lose its charge and be restored to its equilibrium state of electric neutrality.
2. The inductor store magnetic energy in the magnetic field when carrying a current, and it displays an electrical inertia in that its induced emf tends to maintain the charges in motion.
These results are all qualitative consideration. Now let us consider the electromagnetic oscillations analytically.
Applying Kirchhoff's second rule (energy conservation) to the LC circuit loop, we get
∑E = ∑V, [1]
or
-Ldi /dt = q/C, [2]
where -Ldi/dt is the inductor's emf and q/C is the potential difference across the capacitor, and q is the charge on either plate of the capacitor at any instant. By definition instantaneous current is i = dq/dt; substituting this in into this equation [2], we get
-L²(d²q/dt²) = q/C,
or rearranging it as a differential equation, we get
d²q/dt² + (1/LC)q = 0, [3]
d²q/dt² + (ω₀)²q = 0, [4]
where (ω₀)² = 1/LC by definition. The subscript 0 signifies that there is no energy lost in this oscillator circuit since there is zero resistance. The equation [3] is a second order differential equation, whose solution is
q = Q_m sin ω₀t, [5]
as can be seen by substituting it into equation [3]. Q_m is the maximum of charge on one plate of the capacitor; it is also the initial charge on one plate of the capacitor, that is q = Q_m at t = 0. Equation [5] says that the charge varies sinusoidally with time and the angular frequency ω₀ of the free oscillation given by
(ω₀)²q = 1/LC. [6]
The frequency f = ω₀/2 and it is the reciprocal of the period T is given by
f = 1/T = 1/2π√(LC).
The instantaneously current i = dq/dt also oscillates sinusoidally, as shown by taking the derivative of equation [5], that is,
i = -ω₀Q_m sin ω₀t = -I_m sin ω₀t, [7]
where I_m = ω₀Q_m is the maximum value of the current i.

Comparing equations [5] and [7] we see that the charge (varies as the cosine) and the current (varying as the sine) are 90 out of phase. That is, when the capacitor is fully charged, the energy resides entirely in the capacitor's electric field, the current through the inductor and magnetic field associated with are zero, and conversely.

The total energy of U of LC circuit remains constant; it consists of the capacitor's energy
U_C = q²/2C and the inductor's energy U_L = Li²/2. Hence,
U = U_C + U_L = q²/2C + Li²/2 = Q_m² (cos ω₀t)²/2C + L(I_m sin ω₀t)²/2. [8]
Since by equation [7] I_m = ω₀Q_m and from equation [6] ω₀² = 1/LC, then
LI_m² = L₀²Q_m² = Q_m²/C.
and equation [8] becomes
U = Q_m²/2C (cos² ω₀t + sin² ω₀t) = Q_m²/2C. [9]
Although the energies of the capacitor and inductor vary sinusoidally with time, there sum with time is a constant.

As equation [6] shows, the frequency of the free oscillation of an LC circuit depends upon the inductance L and capacitance C of the circuit. The frequency increases as the magnitude of L and C decrease. Thus if one is to construct an electromagnetic oscillator of very high frequency, the capacitance C and the inductance L must not only be very small, but also the actual physical dimensions of the capacitor and inductor. First the inductance is reduced by replacing the coil with a single conducting wire; then the capacitance is also reduced by reducing the area of the capacitor plate. Then there is a single conducting loop broken by a gap at one point. This is indeed an oscillator, and if the size of the loop is the order of 1 meter and the gap is perhaps 1 cm, it oscillates at a frequency of tens of megahertz, a frequency lying in the radio frequency range of the electromagnetic spectrum. Oscillators of this type were used in the historic experiment of Heinrich Hertz (1857-1894), who first demonstrated the existence of electromagnetic waves in 1887. Hertz used two such oscillators, both resonant at same frequency. The oscillations were observed by the spark at the gaps. The electromagnetic waves were detected by observing that as the second oscillator is moved relatively far from the first oscillator. The sparks across its gap persisted; the effect could not attributed to any direct action of the electric field and magnetic field of the first oscillator on the second.

Another method of constructing a high-frequency oscillator is by connecting a short straight wire between the two plates of the capacitor. And the inductance is further reduced by connecting an additional short straight wire between the plates. And the inductance is further reduced by connecting additional wires. Indeed, the wires between the plates of the capacitor can be replaced by a surface between the two plates, so forming a hallow, closed right cylinder of conducting material. This does not look like an LC oscillator circuit, but in fact it is simple form of a microwave oscillator. For dimensions of a few centimeters, the free oscillation occur at microwave frequencies in the order of tens of kilomegahertz (or electromagnetic waves whose wavelength is a few centimeters). The oscillating electric field is confined entirely within the closed cylinder, and so is the oscillating magnetic field. Here it becomes most useful to describe the electric oscillations, not in terms of currents and voltages across various pairs of points, but in terms of electric and magnetic fields and configurations within the closed cylinder.

Production of Electromagnetic Waves.

Electromagnetic waves arise as consequence of two effects:
1. a changing magnetic field produces an electric field and
2. a changing electric field produces a magnetic field.
Therefore, it is clear that neither a stationary charges nor steady currents can produce electromagnetic waves. Whenever the current in a wire changes with time, the wire emits electromagnetic radiation. The fundamental mechanism responsible for this radiation is the acceleration of a charged particles. Whenever a charged particle undergoes acceleration, it must radiate energy. An alternating voltage applied to the wires of an antenna forces an electric charge in the antenna to oscillate. This is the common technique for accelerating charged particles and is the source of the radio waves emitted by the antenna of a radio station. Consider the simple antenna system. Two metal rods of an antenna are connected to an a-c generator, which causes charges to oscillate between the two rods. The output voltage of the a-c generator is sinusoidal.

At t = 0, the upper rod is given a maximum positive charge and the bottom rod an equal negative charge. An electric field is produced near the antenna at this instant. As the charges oscillate, the rods becomes less charged, the field near the rods decrease in strength, and the downward-directed maximum electric field produced at t = 0 moves away from the rod. When the charges are neutralized, the electric field has dropped to zero. This occurs at a time equal to one quarter of the period of oscillation. Continuing, the upper rod soon obtains a maximum negative charge and the lower rod becomes positive, resulting in an electric field directed upward. This occurs after a time equal to one half the period of oscillation. The oscillation continue. Note that the electric filed near the antenna oscillates in phase with the charge distribution. That is, the field points down when the upper rod is positive and up when the upper rod is negative. Furthermore, the magnitude of the field at any instant depends on the amount of charge on the rods at that instant. As the charges continue to oscillate (and accelerate) between the rods, the electric field set up by the charges moves away from the antenna at the speed of light. One cycle of charge oscillation produces one full wavelength in the electric field pattern.

Now let us consider the production of electromagnetic waves by a half-wave antenna. In this arrangement, two conducting rods, each one quarter of a wavelength long, are connected to a source of alternating emf (such as LC oscillator). The oscillator forces charges to accelerate back and forth between the two rods. Consider the field configuration at some instant when the current is upward, The electric field lines resemble those of an electric dipole; that is, two equal and opposite charges separated by a distance 2a. Since these charges are continuously oscillating between the two rods, the antenna can approximate by an oscillating electric dipole. The magnetic field lines form concentric circles about the antenna and are perpendicular to the electric field at all points. The magnetic field is zero at all points along the axis of the antenna. Furthermore, the electric field E and the magnetic field B are 90 out of phase in time, that is, E is at some point reaches it maximum value when B is zero and vice versa. This because when the charges at the end rods are at a maximum, the current is zero.

Electromagnetic waves also induces a current in a receiving antenna. The response of a dipole receiving antenna at a given position will be a maximum when its axis is parallel to the electric field at that point and zero when its axis is perpendicular to the electric field.

Radio Transmitting and Receiving.

In order that electromagnetic waves to be used for voice communication, it is necessary to impress an audio-signal on the wave. One method commonly used is amplitude modulation (alternating the amplitude) of the high frequency oscillation by combining them with the low-frequency oscillations, such as are produced by a microphone or telephone.

Consider the circuit of a simple transmitter using the principle of amplitude modulation. A RF oscillator produces a continuous voltage fluctuations across the coil of a parallel LC circuit. The RF voltage is then fed into the grid of a triode. A microphone circuit is also inductively coupled to the grid circuit of this triode, so that any voltage fluctuations in the microphone circuit will give rise to corresponding voltage fluctuations in the grid circuit. These audio signals are combined with the RF signals in the grid circuit, thereby changing or modulating the amplitude of RF wave. The resulting voltage fluctuations are fed to the antenna system and are radiated out into space as electromagnetic wave.