Alternating-Current Circuits and Electromagnetic Waves (Chap. 21)
Whenever a charged particle accelerates
... it radiates energy. An alternating voltage applied to the wires of an antenna forces electric charges in the antenna to oscillate. This common technique for accelerating charged particles is the source of the radio waves emitted by the broadcast antenna of a radio station.
Maxwell's Predictions
1. Electric field lines originate on positive charges and terminate on negative charges. 2. Magnetic field lines always form closed loops; they don't begin or end anywhere. 3. A varying magnetic field induces an emf and hence an electric field. This fact is a statement of Faraday's law (Chapter 20). 4. Magnetic fields are generated by moving charges (or currents), as summarized in Ampère's law (Chapter 19).
electromagnetic waves traveling through free space have the following properties:
1. Electromagnetic waves travel at the speed of light. 2. Electromagnetic waves are transverse waves because the electric and magnetic fields are perpendicular to the direction of propagation of the wave and to each other. 3. The ratio of the electric field to the magnetic field in an electromagnetic wave equals the speed of light. 4. Electromagnetic waves carry both energy and momentum, which can be delivered to a surface.
AC circuit
An AC circuit consists of combinations of circuit elements and an AC generator or an AC source, which provides the alternating current K.I.M: P=i²R Because the heating effect of a current is proportional to the square of the current, it makes no difference whether the sign associated with the current is positive or negative. The heating effect produced by an alternating current with a maximum value of Imax is not the same as that produced by a direct current of the same value, however. The reason is that the alternating current has this maximum value for only an instant of time during a cycle. The important quantity in an AC circuit is a special kind of average value of current, called the rms current: the direct current that dissipates the same amount of energy in a resistor that is dissipated by the actual alternating current. ^find the rms current, we first square the current, then find its average value, and finally take the square root of this average value. Hence, the rms current is the square root of the average (mean) of the square of the current average value of i2 is 1 2Imax 2 1 Therefore, the rms current Irms is related to the maximum value of the alternating current Imax by Irms =Imax /squre root of 2 = 0.707Imax ^^ NOTE: AC ammeters and voltmeters are designed to read rms values.
Electric field and cancer treatment
Cancer cells multiply far more frequently than most normal cells, spreading throughout the body, using its resources and interfering with normal functioning. Most therapies damage both cancerous and healthy cells, so finding methods that target cancer cells is important in developing better treatments for the disease. Because cancer cells multiply so rapidly, it's natural to consider treatments that prevent or disrupt cell division. Treatments such as chemotherapy interfere with the cell division cycle, but can also damage healthy cells. It has recently been found that alternating electric fields produced by alternating currents in the range of 100 kHz can disrupt the cell division cycle, either by slowing the division or by causing a dividing cell to disintegrate. Healthy cells that divide at only a very slow rate are less vulnerable than the rapidly dividing cancer cells, so such therapy holds out promise for certain types of cancer. The alternating electric fields are thought to affect the process of mitosis, which is the dividing of the cell nucleus into two sets of identical chromosomes. Near the end of the first phase of mitosis, called the prophase, the mitotic spindle forms, a structure of fine filaments that guides the two replicated sets of chromosomes into separate daughter cells. The mitotic spindle is made up of a polymerization of dimers of tubulin, a protein with a large electric dipole moment. The alternating electric field exerts forces on these dipoles, disrupting their proper functioning. Electric field therapy is especially promising for the treatment of brain tumors because healthy brain cells don't divide and therefore would be unharmed by the alternating electric fields. Research on such therapies is ongoing
When you walk through the doorway of a courthouse metal detector, you are really walking through a coil of many turns. How might the metal detector work?
EXPLANATION The metal detector is essentially a resonant circuit. The portal you step through is an inductor (a large loop of conducting wire) that is part of the circuit. The frequency of the circuit is tuned to the resonant frequency of the circuit when there is no metal in the inductor. When you walk through with metal in your pocket, you change the effective inductance of the resonance circuit, resulting in a change in the current in the circuit. This change in current is detected, and an electronic circuit causes a sound to be emitted as an alarm
average power delivered by a generator in an AC circuit
Pav = Irms ∆Vrms cos(omega) ; where the quantity cos(omega) is called the power factor the power delivered by an AC source to any circuit depends on the phase difference between the source voltage and the resulting current. APPLICATION: For example, factories often use devices such as large motors in machines, generators, and transformers that have a large inductive load due to all the windings. To deliver greater power to such devices without using excessively high voltages, factory technicians introduce capacitance in the circuits to shift the phase
RLC circuits
The instantaneous voltages across the three elements, have the following phase relations to the instantaneous current: 1. The instantaneous voltage DvR across the resistor is in phase with the instantaneous current. (See Active Fig. 21.9b.) 2. The instantaneous voltage DvL across the inductor leads the current by 90°. (See Active Fig. 21.9c.) 3. The instantaneous voltage DvC across the capacitor lags the current by 90°. (See Active Fig. 21.9d.) The net instantaneous voltage Dv supplied by the AC source equals the sum of the instantaneous voltages across the separate elements: Dv 5 DvR 1 DvC 1 DvL. This doesn't mean, however, that the voltages measured with an AC voltmeter across R, C, and L sum to the measured source voltage( just the instananeous voltage)! In fact, the measured voltages don't sum to the measured source voltage because the voltages across R, C, and L all have different phases. RLC circuit series use phasor diagrams CONCLUSION: the voltages across the resistor, capacitor, and inductor are not in phase, so one cannot simply add them to get the voltage across the combination of element or to get the source voltage Ohm's law, ∆V = IR, with R replaced by the impedance in ohms. Indeed, it can be regarded as a generalized form of Ohm's law applied to a series AC circuit. Both the impedance and therefore the current in an AC circuit depend on the resistance, the inductance, the capacitance, and the frequency (because the reactances are frequency dependent
in a LC circuit
We assume the capacitor has an initial charge of Qmax and the switch is closed at t 5 0. When the capacitor is fully charged, the total energy in the circuit is stored in the electric field of the capacitor and is equal to Q2max/2C. At this time, the current is zero, so no energy is stored in the inductor. As the capacitor begins to discharge, the energy stored in its electric field decreases. At the same time, the current increases and energy equal to LI2/2 is now stored in the magnetic field of the inductor. Thus, energy is transferred from the electric field of the capacitor to the magnetic field of the inductor. When the capacitor is fully discharged, it stores no energy. At this time, the current reaches its maximum value and all the energy is stored in the inductor. The process then repeats in the reverse direction. The energy continues to transfer between the inductor and the capacitor, corresponding to oscillations in the current and charge.
transformer
a device that can change a small AC voltage to a larger one or vice versa. AC transformer consists of two coils of wire wound around a core of soft iron, as shown in Figure 21.15. The coil on the left, which is connected to the input AC voltage source and has N1 turns, is called the primary winding, or the primary. The coil on the right, which is connected to a resistor R and consists of N2 turns, is the secondary. The purposes of the common iron core are to increase the magnetic flux and to provide a medium in which nearly all the flux through one coil passes through the other. When an input AC voltage ∆V1 is applied to the primary, the induced voltage across it is given by: ∆V1= -N1 (∆magentic flux/ ∆t) ^^ If we assume that no flux leaks from the iron core, then the flux through each turn of the primary equals the flux through each turn of the secondary. Hence, the voltage across the secondary coil is: ∆V2= -N2 (∆magentic flux/ ∆t) Since the (∆magentic flux/ ∆t) is common to both equations and can be algebraically eliminated, giving: ∆V2 = (N2/ N1) ∆V1 ^^ this equations implies that When N2 is greater than N1, ∆V2 exceeds ∆V1 and the transformer is referred to as a step-up transformer. When N2 is less than N1, making ∆V2 less than ∆V1, we have a step-down transformer. It may seem that a transformer is a device in which it is possible to get something for nothing. For example, a step-up transformer can change an input voltage from, say, 10 V to 100 V. This means that each coulomb of charge leaving the secondary has 100 J of energy, whereas each coulomb of charge entering the primary has only 10 J of energy. That is not the case, however, because the power input to the primary equals the power output at the secondary: I1 ∆V1= I2 ∆V2 ^^^ THEREFORE: Although the voltage at the secondary may be, say, ten times greater than the voltage at the primary, the current in the secondary will be smaller than the primary's current by a factor of ten. APPLICATION: When electric power is transmitted over large distances, it's economical to use a high voltage and a low current because the power lost via resistive heating in the transmission lines varies as I 2R. If a utility company can reduce the current by a factor of ten, for example, the power loss is reduced by a factor of one hundred. In practice, the voltage is stepped up to around 230 000 V at the generating station, then stepped down to around 20 000 V at a distribution station, and finally stepped down to 120 V at the customer's utility pole.
back emfs
always oppose the change in the current
Ohm's Law for AC circuit
consisting of a resistor connected to an AC generator. A resistor impedes the current in an AC circuit, just as it does in a DC circuit. Ohm's law is therefore valid for an AC circuit ∆VR,rms = IrmsR ; where The rms voltage across a resistor is equal to the rms current in the circuit times the resistance
Electromagnetic Waves
electromagnetic waves, which are composed of fluctuating electric and magnetic fields. Electromagnetic waves in the form of visible light enable us to view the world around us; infrared waves warm our environment; radio- frequency waves carry our television and radio programs, as well as information about processes in the core of our galaxy; and X-rays allow us to perceive structures hidden inside our bodies and study properties of distant, collapsed stars. Light is key to our understanding of the universe
Resonance frequency
f= 1/ 2pi (square root of L time C) ; where L is inductor and C is capacitor The tuning circuit of a radio is an important application of a series resonance circuit. The radio is tuned to a particular station (which transmits a specific radiofrequency signal) by varying a capacitor, which changes the resonance frequency of the tuning circuit. When this resonance frequency matches that of the incoming radio wave, the current in the tuning circuit increases
the idea of Faraday's Law
foundation for transformers -By Faraday's law, a voltage is generated across the secondary only when there is a change in the number of flux lines passing through the secondary. The input current in the primary must therefore change with time, which is what happens when an alternating current is used. When the input at the primary is a direct current, however, a voltage output occurs at the secondary only at the instant a switch in the primary circuit is opened or closed. Once the current in the primary reaches a steady value, the output voltage at the secondary is zero IN OTHER WORDS: A varying magnetic field induces an emf and hence an electric field.
electromagnetic wave ...
is a transvere wave because: the E S and B S fields are perpendicular to each other and (2) both fields are perpendicular to the direction of motion of the wave. This second property is characteristic of transverse waves. . ^^The electric and magnetic fields are sinusoidal and perpendicular to each other. Both fields are perpendicular to the direction of wave propagation Electromagnetic waves travel with the speed of light. In fact, it can be shown that the speed of an electromagnetic wave is related to the permeability and permittivity of the medium through which it travels. Maxwell found this relationship for free space to be c =1/ square root of the permeability times permittivity of the medium.
output of an AC generator
is sinusoidal and varies with time according to: ∆v =∆Vmax sin 2pi ft
Doppler effect for electromagnetic waves
like in sound waves exhibit the Doppler effect when the observer, the source, or both are moving relative to the medium of propagation. Recall that in the Doppler effect, the observed frequency of the wave is larger or smaller than the frequency emitted by the source of the wave. A Doppler effect also occurs for electromagnetic waves, but it differs from the Doppler effect for sound waves in two ways. First, in the Doppler effect for sound waves, motion relative to the medium is most important because sound waves require a medium in which to propagate. In contrast, the medium of propagation plays no role in the Doppler effect for electromagnetic waves because the waves require no medium in which to propagate. Second, the speed of sound that appears in the equation for the Doppler effect for sound depends on the reference frame in which it is measured. In contrast, as we see in Chapter 26, the speed of electromagnetic waves has the same value in all coordinate systems that are either at rest or moving at constant velocity with respect to one another. The single equation that describes the Doppler effect for electromagnetic waves is given by the approximate expression: fo=fs (1 +/- (u/c) where fO is the observed frequency, fS is the frequency emitted by the source, u is the relative speed of the observer and source, and c is the speed of light in a vacuum. Note that Equation is valid only if u is much smaller than c. Further, it can also be used for sound as long as the relative velocity of the source and observer is much less than the velocity of sound. The positive sign in the equation must be used when the source and observer are moving toward each other, whereas the negative sign must be used when they are moving away from each other. Thus, we anticipate an increase in the observed frequency if the source and observer are approaching each other and a decrease if the source and observer recede from each other. APPLICATION: Astronomers have made important discoveries using Doppler observations on light reaching Earth from distant galaxies. Such measurements have shown that the more distant a galaxy is from Earth, the more its light is shifted toward the red end of the spectrum. This cosmological red shift is evidence that the Universe is expanding. The stretching and expanding of space, like a rubber sheet being pulled in all directions, is consistent with Einstein's theory of general relativity. A given star or galaxy, however, can have a peculiar motion toward or away from Earth. For example, Doppler effect measurements made with the Hubble Space Telescope have shown that a galaxy labeled M87 is rotating, with one edge moving toward us and the other moving away. Its measured speed of rotation was used to identify a supermassive black hole located at its center.
Inductors in AC circuit
the effective resistance of the coil in an AC circuit is measured by a quantity called the inductive reactance, XL= 2pifL When f is in hertz and L is in henries, the unit of XL is the ohm. The inductive reactance increases with increasing frequency and increasing inductance. ***Contrast these facts with capacitors, where increasing frequency or capacitance decreases the capacitive reactance ^^^ note that the inductive reactance depends on the inductance L, which is reasonable because the back emf is large for large values of L. Second, note that the inductive reactance depends on the frequency f. This dependence, too, is reasonable because the back emf depends on ∆I/∆t, a quantity that is large when the current changes rapidly, as it would for high frequencies. With inductive reactance defined in this way, we can write an equation of the same form as Ohm's law for the voltage across the coil or inductor: ∆VL,rms = IrmsXL NOTE: When a sinusoidal voltage is applied across an inductor, the voltage reaches its maximum value one-quarter of an oscillation period before the current reaches its maximum value. In this situation we say that the voltage across an inductor always leads the current by 90°.
Capacitors in AC circuit
where the voltage across a capacitor always lags the current by 90° The analogy between capacitive reactance and resistance means that we can write an equation of the same form as Ohm's law to describe AC circuits containing capacitors. This equation relates the rms voltage and rms current in the circuit to the capacitive reactance: ∆VC,rms = IrmsXC capacitive reactance XC, defined as XC =1/2pifC