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General Physics (PHY 2140) Lecture 10 Electricity and Magnetism Induced voltages and induction Self-Inductance RL Circuits Energy in magnetic fields AC circuits and EM waves Resistors, capacitors and inductors in AC circuits The RLC circuit Power in AC circuits 5/23/2017 Chapter 20-21 http://www.physics.wayne.edu/~alan/2140Website/Main.htm 1 Reminder: Exam 2 this Wednesday 6/13 12-14 questions. Show your work for full credit. Closed book. You may bring a page of notes. Bring a calculator. Bring a pen or pencil. 5/23/2017 2 Lightning Review Last lecture: 1. Induced voltages and induction Generators and motors Self-induction mv qB BA cos I E L t r L Review Problem: Charged particles passing N I through a bubble chamber leave tracks consisting of small hydrogen gas bubbles. These bubbles make visible the particles’ trajectories. In the following figure, the magnetic field is directed into the page, and the tracks are in the plane of the page, in the directions indicated by the arrows. (a) Which of the tracks correspond to positively charged particles? (b) If all three particles have the same mass and charges of equal magnitude, which is moving the fastest? 5/23/2017 3 S S Review example v Determine the direction of current in the loop for bar magnet moving down. N Initial flux Final flux By Lenz’s law, the induced field is this change 5/23/2017 4 20.6 Self-inductance When a current flows through a loop, the magnetic field created by that current has a magnetic flux through the area of the loop. If the current changes, the magnetic field changes, and so the flux changes giving rise to an induced emf. This phenomenon is called self-induction because it is the loop's own current, and not an external one, that gives rise to the induced emf. Faraday’s law states E N t 5/23/2017 5 The magnetic flux is proportional to the magnetic field, which is proportional to the current in the circuit Thus, the self-induced EMF must be proportional to the time rate of change of the current I E L t where L is called the inductance of the device Units: SI: henry (H) If flux is initially zero, 5/23/2017 1 H 1V s A N LN I I 6 Example: solenoid A solenoid of radius 2.5cm has 400 turns and a length of 20 cm. Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75mV. N B 0 nI 0 I l NB N2A L 0 I l E L 5/23/2017 I I E t t L N B BA 0 IA l = (4p x 10-7)(160000)(2.0 x 10-3)/(0.2) = 2 mH = (75 x 10-3)/ (2.0 x 10-3) = 37.5 A/s 7 Inductor in a Circuit Inductance can be interpreted as a measure of opposition to the rate of change in the current Remember resistance R is a measure of opposition to the current As a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing current Therefore, the current doesn’t change from 0 to its maximum instantaneously Maximum current: I max 5/23/2017 E R 8 20.7 RL Circuits Recall Ohm’s Law to find the voltage drop on R V IR (voltage across a resistor) We have something similar with inductors I EL L t (voltage across an inductor) Similar to the case of the capacitor, we get an equation for the current as a function of time (series circuit). I 5/23/2017 E 1 e R Rt / L L R 9 RL Circuit (continued) V Rt / L I 1 e R 5/23/2017 10 20.8 Energy stored in a magnetic field The battery in any circuit that contains a coil has to do work to produce a current Similar to the capacitor, any coil (or inductor) would store potential energy 1 2 PEL LI 2 Summary of the properties of circuit elements. 5/23/2017 Resistor Capacitor Inductor units ohm, W = V / A farad, F = C / V henry, H = V s / A symbol R C L relation V=IR Q=CV emf = -L (I / t) power dissipated P = I V = I² R = V² / R 0 0 energy stored 0 PEC = C V² / 2 PEL = L I² / 2 11 Example: stored energy A 24V battery is connected in series with a resistor and an inductor, where R = 8.0W and L = 4.0H. Find the energy stored in the inductor when the current reaches its maximum value. 5/23/2017 12 A 24V battery is connected in series with a resistor and an inductor, where R = 8.0W and L = 4.0H. Find the energy stored in the inductor when the current reaches its maximum value. Given: V = 24 V R = 8.0 W L = 4.0 H Find: PEL =? Recall that the energy stored in the inductor is PEL 1 2 LI 2 The only thing that is unknown in the equation above is current. The maximum value for the current is I max V 24V 3.0 A R 8.0W Inserting this into the above expression for the energy gives 1 2 PEL 4.0 H 3.0 A 18 J 2 5/23/2017 13 Chapter 21 Alternating Current Circuits and Electromagnetic Waves AC Circuit An AC circuit consists of a combination of circuit elements and an AC generator or source The output of an AC generator is sinusoidal and varies with time according to the following equation Δv = ΔVmax sin 2pƒt 5/23/2017 Δv is the instantaneous voltage ΔVmax is the maximum voltage of the generator ƒ is the frequency at which the voltage changes, in Hz 15 Resistor in an AC Circuit Consider a circuit consisting of an AC source and a resistor The graph shows the current through and the voltage across the resistor The current and the voltage reach their maximum values at the same time The current and the voltage are said to be in phase 5/23/2017 16 More About Resistors in an AC Circuit The direction of the current has no effect on the behavior of the resistor The rate at which electrical energy is dissipated in the circuit is given by P = i2 R= (Imax sin 2pƒt)2 R 5/23/2017 where i is the instantaneous current the heating effect produced by an AC current with a maximum value of Imax is not the same as that of a DC current of the same value The maximum current occurs for a small amount of time 17 rms Current and Voltage The rms current is the direct current that would dissipate the same amount of energy in a resistor as is actually dissipated by the AC current Irms Imax 0.707 Imax 2 Alternating voltages can also be discussed in terms of rms values Vrms 5/23/2017 Vmax 0.707 Vmax 2 18 Ohm’s Law in an AC Circuit rms values will be used when discussing AC currents and voltages AC ammeters and voltmeters are designed to read rms values Many of the equations will be in the same form as in DC circuits Ohm’s Law for a resistor, R, in an AC circuit 5/23/2017 ΔVrms = Irms R Also applies to the maximum values of v and i 19 Example: an AC circuit An ac voltage source has an output of V = 150 sin (377 t). Find (a) the rms voltage output, (b) the frequency of the source, and (c) the voltage at t = (1/120)s. (d) Find the rms current in the circuit when the generator is connected to a 50.0W resistor. Vmax Vrms 0.707 Vmax 0.707 x 150V 106 V 2 377 rad/sec, 2p f , f / 2p 377/ 2p = 60 Hz ΔV = 150 sin (377 x 1/120) = 0 V ΔVrms = Irms R thus, Irms = ΔVrms/R = 2.12 A 5/23/2017 20 Capacitors in an AC Circuit Consider a circuit containing a capacitor and an AC source The current starts out at a large value and charges the plates of the capacitor There is initially no resistance to hinder the flow of the current while the plates are not charged As the charge on the plates increases, the voltage across the plates increases and the current flowing in the circuit decreases 5/23/2017 21 More About Capacitors in an AC Circuit The current reverses direction The voltage across the plates decreases as the plates lose the charge they had accumulated The voltage across the capacitor lags behind the current by 90° 5/23/2017 22 Capacitive Reactance and Ohm’s Law The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by 1 XC 2p ƒC When ƒ is in Hz and C is in F, XC will be in ohms Ohm’s Law for a capacitor in an AC circuit 5/23/2017 ΔVrms = Irms XC 23 Inductors in an AC Circuit Consider an AC circuit with a source and an inductor The current in the circuit is impeded by the back emf of the inductor The voltage across the inductor always leads the current by 90° 5/23/2017 24 Inductive Reactance and Ohm’s Law The effective resistance of a coil in an AC circuit is called its inductive reactance and is given by XL = 2pƒL When ƒ is in Hz and L is in H, XL will be in ohms Ohm’s Law for the inductor 5/23/2017 ΔVrms = Irms XL 25 Example: AC circuit with capacitors and inductors A 2.40mF capacitor is connected across an alternating voltage with an rms value of 9.00V. The rms current in the capacitor is 25.0mA. (a) What is the source frequency? (b) If the capacitor is replaced by an ideal coil with an inductance of 0.160H, what is the rms current in the coil? ΔVrms = Irms XC , first we find XC: ΔVrms/ Irms = 9.00V/25.0 x 10-3 A = 360 ohms Now, solve for ƒ: ƒ = 1/ 2p XC C = 0.184 Hz 1 X C 2 p ƒ C For and inductor XL = 2pƒL, try solving for Irms = ΔVrms/ XL 5/23/2017 26 The RLC Series Circuit The resistor, inductor, and capacitor can be combined in a circuit The current in the circuit is the same at any time and varies sinusoidally with time 5/23/2017 27 Current and Voltage Relationships in an RLC Circuit The instantaneous voltage across the resistor is in phase with the current The instantaneous voltage across the inductor leads the current by 90° The instantaneous voltage across the capacitor lags the current by 90° 5/23/2017 28 Phasor Diagrams To account for the different phases of the voltage drops, vector techniques are used Represent the voltage across each element as a rotating vector, called a phasor The diagram is called a phasor diagram 5/23/2017 29 Phasor Diagram for RLC Series Circuit The voltage across the resistor is on the +x axis since it is in phase with the current The voltage across the inductor is on the +y since it leads the current by 90° The voltage across the capacitor is on the –y axis since it lags behind the current by 90° 5/23/2017 30 Phasor Diagram, cont The phasors are added as vectors to account for the phase differences in the voltages ΔVL and ΔVC are on the same line and so the net y component is ΔVL - ΔVC 5/23/2017 31 ΔVmax From the Phasor Diagram The voltages are not in phase, so they cannot simply be added to get the voltage across the combination of the elements or the voltage source 2 Vmax VR ( VL VC )2 VL VC tan VR is the phase angle between the current and the maximum voltage 5/23/2017 32 Impedance of a Circuit The impedance, Z, can also be represented in a phasor diagram Z R 2 ( XL X C ) 2 XL X C tan R 5/23/2017 33 Impedance and Ohm’s Law Ohm’s Law can be applied to the impedance 5/23/2017 ΔVmax = Imax Z 34 Summary of Circuit Elements, Impedance and Phase Angles 5/23/2017 35 Problem Solving for AC Circuits Calculate as many unknown quantities as possible For example, find XL and XC Be careful of units -- use F, H, Ω Apply Ohm’s Law to the portion of the circuit that is of interest Determine all the unknowns asked for in the problem 5/23/2017 36 Power in an AC Circuit No power losses are associated with capacitors and pure inductors in an AC circuit 5/23/2017 In a capacitor, during one-half of a cycle energy is stored and during the other half the energy is returned to the circuit In an inductor, the source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuit 37 Power in an AC Circuit, cont The average power delivered by the generator is converted to internal energy in the resistor Pav = IrmsΔVR = IrmsΔVrms cos cos is called the power factor of the circuit Phase shifts can be used to maximize power outputs 5/23/2017 38 Resonance in an AC Circuit Resonance occurs at the frequency, ƒo, where the current has its maximum value To achieve maximum current, the impedance must have a minimum value This occurs when XL = XC 1 ƒo 2p LC 5/23/2017 39