Chapter 21: Alternating Current Circuits and EM Waves Resistors in an AC Circuits Homework assignment : 22,25,32,42,63  AC circuits An AC circuit consists of combinations of circuit elements and an AC generator or an AC source, which provides the alternating current. The alternating current can be expressed by : The current and the voltage reach their maximum at the same time; they are said to be in phase. The power, which is energy dissipated in a resistor per unit time is:

 Rms current and voltage Resistors in an AC Circuits rms current voltage difference at a resitor rms voltage

 Capacitors in an AC circuit Capacitors in an AC Circuits Phase difference between i C and  v C The voltage across a capacitor always lags the current by 90 o. When the direction of the current is reversed, the amount of the charge at the capacitor starts to decrease. At t=0, there is no charge at the capacitor and the current is free to flow at i=I max. Then the current starts to decrease.

 Capacitors in an AC circuit Capacitors in an AC Circuits Reactance and  V C,rms vs. I rms Reactance: impeding effect of a capacitor capacitive reactance X C in  f in Hz C in F

 Inductors in an AC circuit Inductors in an AC Circuits The changing current output of the generator produces a back emf that impedes the current in the circuit. The magnitude of the back emf is: Phase difference between i L and  v L The effective resistance of coil in an AC circuit is measured by the inductive reactance X L : inductive reactance The voltage across an inductor always leads the current by 90 o. a:  i L /  t maximum  v L max. b:  i L /  t zero  v L zero X L in  f in Hz L in H

 A simple RLC series circuit RLC Series Circuits Phase differences Current in the circuit The instantaneous voltage  v R is in phase with the instantaneous current i. The instantaneous voltage  v L leads the current by 90 o. The instantaneous voltage  v C lags the current by 90 o.

 A simple RLC series circuit RLC Series Circuits Phasors It is convenient to treat a voltage across each element in a RLC circuit as a rotating vector (phasor) as shown in the phasor diagram on the right. Phasor diagram

 A simple RLC series circuit RLC Series Circuits Impedance impedance in form of Ohm’s law Note that quantities with subscript “max” is related with those with “rms”, all the results in this slide are also applicable to quantities with subscript “rms”

 Impedances and phase angles RLC Series Circuits

 Filters : Example Ex.: C = 1 μf, R = 1Ω High-pass filter Note: this is ω, ~ V out RLC Series Circuits

 Filters (cont’d) ~ V out ~ ω=0No current V out ≈ 0 ω=∞Capacitor ~ wire V out ≈ ε ~ V out ω = ∞No current V out ≈ 0 ω = 0Inductor ~ wire V out ≈ ε ω = 0No current because of capacitor ω = ∞ No current because of inductor (Conceptual sketch only) High- pass filter Low- pass filter Band-pass filter RLC Series Circuits

 Power in an AC circuit Power in an AC Circuit No power losses are associated with pure capacitors - When the current increases in one direction in an AC circuit, charge accumulates on the capacitor and the voltage drop appears across it. - At the maximum value of the voltage, the energy stored in the capacitor is: - When the current reverses direction, the charge leaves the capacitor to the voltage source and the stored energy decreases. - As long as there is no resistance, there is no energy loss. No power losses are associated with pure inductors - The source must do work against the back emf of an inductor that is carrying a current. - At the maximum value of the current, the energy stored in the inductor is: - When the current starts to decrease, the stored energy returns to the source as the inductor tries to maintain the current in the cuircuit. - As long as there is no resistance, there is no energy loss.

 Power in an AC circuit (cont’d) Power in an AC Circuit The average power delivered by the generator is converted to internal energy in the resistor. No power loss occurs in an ideal capacitor or inductor. Average power delivered to the resistor

 Resonance Resonance in a Series RLC Circuit The current in a series RLC circuit This current reaches the maximum when X L =X C (Z=R).

2 1 (primary) (secondary) ~ E N N iron V2V2 V1V1 Transformers  Transformers AC voltages can be stepped up or stepped down by the use of transformers. The AC current in the primary circuit creates a time-varying magnetic field in the iron. We assume that the entire flux produced by each turn of the primary is trapped in the iron. This induces an emf on the secondary windings due to the mutual inductance of the two sets of coils.

Transformers  Ideal transformer without a load No resistance losses All flux contained in iron Nothing connected on secondary   N 2 N 1 (primary) (secondary) iron VV 2 VV 1 The primary circuit is just an AC voltage source in series with an inductor. The change in flux produced in each turn is given by: The change in flux per turn in the secondary coil is the same as the change in flux per turn in the primary coil (ideal case). The induced voltage appearing across the secondary coil is given by: Therefore, N 2 > N 1 -> secondary V 2 is larger than primary  V 1 (step-up) N 1 > N 2 -> secondary V 2 is smaller than primary  V 1 (step-down) Note: “no load” means no current in secondary. The primary current, termed “the magnetizing current” is small!

Transformers  Ideal transformer with a load   N 2 N 1 (primary) (secondary) iron VV 2 VV 1 R What happens when we connect a resistive load to the secondary coil? Changing flux produced by primary coil induces an emf in secondary which produces current I 2 This current produces a flux in the secondary coil µ N 2 I 2, which opposes the change in the original flux -- Lenz’s law This induced changing flux appears in the primary circuit as well; the sense of it is to reduce the emf in the primary, to “fight” the voltage source. However, V 1 is assumed to be a voltage source. Therefore, there must be an increased current I 1 (supplied by the voltage source) in the primary which produces a flux µ N 1 I 1 which exactly cancels the flux produced by I 2.

Transformers  Ideal transformer with a load (cont’d) Power is dissipated only in the load resistor R. The primary circuit has to drive the resistance R of the secondary. Where did this power come from? It could come only from the voltage source in the primary:

Maxell’s Equations  Maxwell’s equations Gauss’s law Gauss’s law for magnetism Farady’s law Ampere’s law

 Oscillating electric dipole First consider static electric field produced by an electric dipole as shown in Figs. (a)Positive (negative) charge at the top (bottom) (b)Negative (positive) charge at the top (bottom) Now then imagine these two charge are moving up and down and exchange their position at every half-period. Then between the two cases there is a situation like as shown in Fig. below: What is the electric filed in the blank area? EM Waves by an Antenna

 Oscillating electric dipole (cont’d) Since we don’t assume that change propagate instantly once new position is reached the blank represents what has to happen to the fields in meantime. We learned that E field lines can’t cross and they need to be continuous except at charges. Therefore a plausible guess is as shown in the right figure. EM Waves by an Antenna

 Oscillating electric dipole (cont’d) What actually happens to the fields based on a precise calculate is shown in Fig. Magnetic fields are also formed. When there is electric current, magnetic field is produced. If the current is in a straight wire circular magnetic field is generated. Its magnitude is inversely proportional to the distance from the current. EM Waves by an Antenna

 Oscillating electric dipole (cont’d) What actually happens to the fields based on a precise calculate is shown in Fig. EM Waves by an Antenna E B B is perpendicular to E

 Oscillating electric dipole (cont’d) This is an animation of radiation of EM wave by an oscillating electric dipole as a function of time. EM Waves by an Antenna

 Oscillating electric dipole (cont’d) A qualitative summary of the observation of this example is: 1)The E and B fields are always at right angles to each other. 2)The propagation of the fields, i.e., their direction of travel away from the oscillating dipole, is perpendicular to the direction in which the fields point at any given position in space. 3)In a location far from the dipole, the electric field appears to form closed loops which are not connected to either charge. This is, of course, always true for any B field. Thus, far from the dipole, we find that the E and B fields are traveling independent of the charges. They propagate away from the dipole and spread out through space. In general it can be proved that accelerating electric charges give rise to electromagnetic waves. EM Waves by an Antenna

 Dipole antenna EM Waves by an Antenna At a location far away from the source of the EM wave, the wave becomes plane wave V(t)=V o cos(  t) time t=0 time t=  /  one half cycle later X BB

EM Waves by an Antenna  Dipole antenna (cont’d)

 Plane EM wave Properties of EM Waves y x z  Speed of light and EM wave in vacuum Speed of light Light is an EM wave!

 EM wave in matter Maxwell’s equations for inside matter change from those in vacuum by change  0 and  0 to  =  m  0 and  0 : For most of dielectrics the relative permeability  m is close to 1 except for insulating ferromagnetic materials : Index of refraction Properties of EM Waves

 Intensity of EM wave (average power per unit area) Properties of EM Waves EM waves carry energy. intensity of the EM wave

 Momentum carried by EM wave Properties of EM Waves Momentum carried by an EM wave: Momentum transferred to an area : if the wave is completely absorbed : if the wave is completely reflected :  p=mv-(-mv)=2mv Measurement of radiation pressure:

 Spectrum of EM waves Spectrum of EM Waves c : speed of light in vacuum f : frequency : wavelength

 Doppler effect Doppler Effect for EM Waves u : relative speed of the observer with respect to the source c : speed of light in vacuum f o : observed frequency, f S : emitted frequency + if the source and the observe are approaching each other - if the source and the observer are receding each other receding approaching A globular cluster