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General Physics (PHY 2140) Lecture 19 Electricity and Magnetism Induced voltages and induction Energy AC circuits and EM waves Resistors in an AC circuits http://www.physics.wayne.edu/~apetrov/PHY2140/ Chapter 20-21 7/17/2015 1 Next week: Prof. Claude Pruneau Lectures on Monday and Wednesday Exam on Friday 7/17/2015 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? 7/17/2015 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 7/17/2015 4 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 7/17/2015 E R 5 20.9 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. 7/17/2015 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 6 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. 7/17/2015 7 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 th 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 7/17/2015 8 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 2ƒt Δv is the instantaneous voltage ΔVmax is the maximum voltage of the generator ƒ is the frequency at which the voltage changes, in Hz Same thing about the current (if only a resistor) I = Imax sin 2ƒt 7/17/2015 10 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 Voltage varies as ΔV = ΔVmax sin 2ƒt Same thing about the current I = Imax sin 2ƒt 7/17/2015 11 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 2ƒt)2 R 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 Averaging the above formula over one cycle we get 7/17/2015 1 2 P I max R 2 12 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 7/17/2015 Vmax 0.707 Vmax 2 13 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 ΔVrms = Irms R Also applies to the maximum values of v and i 7/17/2015 14 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 maximum current in the circuit when the generator is connected to a 50.0W resistor. 7/17/2015 15 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 7/17/2015 16 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° 7/17/2015 17 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 2 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 7/17/2015 ΔVrms = Irms XC 18 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° 7/17/2015 19 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 = 2ƒL When ƒ is in Hz and L is in H, XL will be in ohms Ohm’s Law for the inductor 7/17/2015 ΔVrms = Irms XL 20 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? 7/17/2015 21