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College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment College Physics B - PHY2054C Magnetic Flux Faraday’s Law Electrical Generator Faraday’s Law and Electrical Generators Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits 10/01/2014 LC Circuits My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building College Physics B Outline Review: Force on a Particle 1 Review: Force on a Particle Magnetic Induction 2 Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Faraday’s Experiment Magnetic Flux 3 Faraday’s Law Electrical Generator 4 Lenz’s Law Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits 5 Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits College Physics B Review Question 1 Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits A positively-charged particle undergoes circular motion as a result of the magnetic force produced by a field that is perpendicular to the plane in the figure. Does this particle move clockwise (A) or counterclockwise (B)? College Physics B Review Question 1 Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law A positively-charged particle undergoes circular motion as a result of the magnetic force produced by a field that is perpendicular to the plane in the figure. Does this particle move clockwise (A) or counterclockwise (B)? Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits B) Counterclockwise College Physics B Outline Review: Force on a Particle 1 Review: Force on a Particle Magnetic Induction 2 Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Faraday’s Experiment Magnetic Flux 3 Faraday’s Law Electrical Generator 4 Lenz’s Law Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits 5 Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits College Physics B Faraday’s Experiment Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Michael Faraday attempted to observe an induced electric field. ➜ He did not use a light bulb. A If the bar magnet is stationary, the current and the electric field are both zero. Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits B If the bar magnet was in motion, a current was observed. Inductors in AC Circuits LC Circuits Faraday’s Law: phet.colorado.edu/sims/faradays-law/faradays-law_en.html College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Another Faraday Experiment A solenoid is positioned near a loop of wire with the light bulb. Now a current is passed through the solenoid by connecting it to a battery. Magnetic Flux Faraday’s Law 1 When the current through the solenoid is constant, there is no current in the wire. 2 When the switch is opened or closed, the bulb does light up. Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits College Physics B Review: Force on a Particle Magnetic Induction Review Question 2 What is the conclusion from Faraday’s Experiment? It shows that an electric current is produced in the wire loop only when Faraday’s Experiment Magnetic Flux A the battery is directly connected to the light bulb. Faraday’s Law Electrical Generator B the magnetic field at the loop is constant. Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits C the magnetic field at the loop is changing. D No electric current is produced. A magnetic field cannot produce an electric current to power the light bulb. College Physics B Review: Force on a Particle Magnetic Induction Review Question 2 What is the conclusion from Faraday’s Experiment? It shows that an electric current is produced in the wire loop only when Faraday’s Experiment Magnetic Flux A the battery is directly connected to the light bulb. Faraday’s Law Electrical Generator B the magnetic field at the loop is constant. Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits C the magnetic field at the loop is changing. D No electric current is produced. A magnetic field cannot produce an electric current to power the light bulb. Capacitors in AC Circuits Inductors in AC Circuits LC Circuits A changing magnetic field produces an electric field. ➜ It is called induced electric field. The phenomenon is called electromagnetic induction. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Review: Magnetic Flux Faraday developed a quantitative theory of induction now called Faraday’s Law: • It uses the concept of magnetic flux that is similar to the concept of electric flux: Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits ΦB = A B cos θ (θ = ∠(normal vector, magnetic field)) College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Review: Magnetic Flux Faraday developed a quantitative theory of induction now called Faraday’s Law: • ΦB = A B cos θ • The unit of magnetic flux is the Weber (Wb) ➜ 1 Wb = 1 T · m2 College Physics B Outline Review: Force on a Particle 1 Review: Force on a Particle Magnetic Induction 2 Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Faraday’s Experiment Magnetic Flux 3 Faraday’s Law Electrical Generator 4 Lenz’s Law Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits 5 Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits College Physics B Faraday’s Law Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Faraday’s Law indicates how to calculate the potential difference that produces the induced current: Capacitors in AC Circuits Inductors in AC Circuits LC Circuits |E| = − ∆ΦB ∆t The magnitude of the induced emf equals the rate of change of the magnetic flux. College Physics B Example Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits A magnetic field B is constant and in a direction perpendicular to the plane of the rails and the bar. Assume the bar moves at constant speed: ΦB = B A = B w L ∆ΦB ∆w = |E| = B L = BLv ∆t ∆t I = BLv R College Physics B Example Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits A magnetic field B is constant and in a direction perpendicular to the plane of the rails and the bar. Assume the bar moves at constant speed: ΦB = B A = B w L ∆w ∆ΦB = |E| = B L = BLv ∆t ∆t I = BLv R College Physics B Example Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits A magnetic field B is constant and in a direction perpendicular to the plane of the rails and the bar. Assume the bar moves at constant speed: ΦB = B A = B w L ∆w ∆ΦB = |E| = B L = BLv ∆t ∆t I = BLv R College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Faraday’s Law College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Faraday’s Law The mechanical power put into the bar by the external agent is equal to the electrical power delivered to the resistor: ➜ Energy is converted from mechanical to electrical; total energy remains the same. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Electrical Generator Need to make the rate of change of the flux large enough to give a useful emf: • Use rotational motion instead of linear motion. • A permanent magnet produces a constant magnetic field in the region between its poles. • The angle between the field and the plane of the loop changes as the loop rotates. • If the shaft rotates with a constant angular velocity, the flux varies sinusoidally with time. Generator: phet.colorado.edu/en/simulation/generator College Physics B Outline Review: Force on a Particle 1 Review: Force on a Particle Magnetic Induction 2 Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Faraday’s Experiment Magnetic Flux 3 Faraday’s Law Electrical Generator 4 Lenz’s Law Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits 5 Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits College Physics B Lenz’s Law Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Lenz’s Law gives us an easy way to determine the sign of the induced emf. Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Lenz’s Law states that the magnetic field produced by an induced current always opposes any changes in the magnetic flux. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Question 3 What happens when I drop a magnet into a copper pipe (compared to a normal piece of metal)? A The magnet will fall normaly (like the metal piece). Lenz’s Law B The piece of metal will fall down, but the magnet will be repelled by the pipe and shoot up. Alternating Currents C The magnet will fall more slowly than the metal piece. Electrical Generator Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits D The magnet will fall faster than the metal piece. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Question 3 What happens when I drop a magnet into a copper pipe (compared to a normal piece of metal)? C The magnet will fall more slowly than the metal piece. College Physics B Question 4 Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law A double loop of wire (making 2 turns) is in the x-y plane centered at the origin. A uniform magnetic field is increasing at constant rate in the negative z direction. In which direction is the induced magnetic field in the loop? Electrical Generator Lenz’s Law Alternating Currents Generators A In the positive z direction. B In the negative z direction. Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits C There is no induced field because of the double loop. D There is no induced field because the rate of change of the magnetic field is constant. College Physics B Example Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux A Assume a metal loop in which the magnetic field passes upward through it. B Assume the magnetic flux increases with time. Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits C The magnetic field produced by the induced emf must oppose the change in flux. ➜ The induced magnetic field must be downward and the induced current will be clockwise (right-hand rule). College Physics B Another Example Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux A Assume a metal loop in which the magnetic field passes upward through it. B Assume the magnetic flux decreases with time. Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits C The magnetic field produced by the induced emf must oppose the change in flux. ➜ The induced magnetic field must be downward and the induced current will be counterclockwise (right-hand rule). College Physics B Question 5 Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law A double loop of wire (making 2 turns) is in the x-y plane centered at the origin. A uniform magnetic field is increasing at constant rate in the negative z direction. In which direction is the induced magnetic field in the loop? Electrical Generator Lenz’s Law Alternating Currents Generators A In the positive z direction. B In the negative z direction. Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits C There is no induced field because of the double loop. D There is no induced field because the rate of change of the magnetic field is constant. College Physics B Question 5 Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law A double loop of wire (making 2 turns) is in the x-y plane centered at the origin. A uniform magnetic field is increasing at constant rate in the negative z direction. In which direction is the induced magnetic field in the loop? Electrical Generator Lenz’s Law Alternating Currents Generators A In the positive z direction. B In the negative z direction. Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits C There is no induced field because of the double loop. D There is no induced field because the rate of change of the magnetic field is constant. College Physics B Outline Review: Force on a Particle 1 Review: Force on a Particle Magnetic Induction 2 Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Faraday’s Experiment Magnetic Flux 3 Faraday’s Law Electrical Generator 4 Lenz’s Law Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits 5 Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits DC versus AC Sources College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits DC versus AC Sources DC (direct current) Circuit Summary • Source of electrical energy is generally a battery. • If only resistors are in the circuit, the current is independent of time. • If the circuit contains capacitors and resistors, the current can vary with time but always approaches a constant value a long time after closing the switch. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators DC versus AC Sources DC (direct current) Circuit Summary • Source of electrical energy is generally a battery. • If only resistors are in the circuit, the current is independent of time. • If the circuit contains capacitors and resistors, the current can vary with time but always approaches a constant value a long time after closing the switch. Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits AC (alternating current) Circuit Introduction • The power source is a device that produces an electric potential that varies with time. • There will be a frequency and peak voltage associated with the potential. ➜ AC circuits have numerous advantages over DC circuits. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits Generators Most sources of AC voltage employ a generator based on magnetic induction. • A shaft holds a coil with many loops of wire. • The coil is positioned between the poles of a permanent magnet. • The magnetic flux through the coil varies with time as the shaft turns. • This changing flux induces a voltage in the coil. LC Circuits Generators of electrical energy convert the mechanical energy of the rotating shaft into electrical energy. ➜ Conservation of energy still applies. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Transformers Transformers are devices that can increase or decrease the amplitude of an applied AC voltage: • A simple transformer consists of two solenoid coils with the loops arranged such that all or most of the magnetic field lines and flux generated by one coil passes through the other coil. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Transformers Transformers are devices that can increase or decrease the amplitude of an applied AC voltage: 1 An AC current in one coil will induce an AC voltage across the other coil. 2 An AC voltage source is typically attached to one of the coils called the input coil. 3 The other coil is called the output coil. Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits College Physics B Review: Force on a Particle Transformers Faraday’s Law applies to both coils: Magnetic Induction Faraday’s Experiment Magnetic Flux Vin = Φin ∆t and Vout = Φout ∆t Faraday’s Law Electrical Generator Lenz’s Law If the input coil has N in turns and the output coil has N out turns, the flux in the coils is related by: Alternating Currents Generators Transformers Φout = N out Φin N in Vout = N out Vin N in Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Transformers cannot change DC voltages! College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Transformers Most practical transformers have central regions filled with a magnetic material. This produces a larger flux, resulting in a larger voltage at both the input and output coils. However: Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Vout = constant Vin Alternating Currents Generators Transformers Φout = N out Φin N in Vout = N out Vin N in Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Transformers cannot change DC voltages! College Physics B Transformers Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits At the power plant Supply voltage of about 5 000 V Cross-country lines Voltage of about 500 000 V College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Transformers and Power Transformers are used in the transmission of electric power over long distances: • Many household appliances use transformers to convert Lenz’s Law the AC voltage at a wall socket to the smaller voltages needed in many devices. Alternating Currents • The output voltage of a transformer can also be made Faraday’s Law Electrical Generator Generators much larger by arranging the number of coils. Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits According to the principle of energy conservation, the energy delivered through the input coil must either be stored in the transformer’s magnetic field or transferred to the output circuit: • The power delivered to the input coil must equal the output power. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Transformers and Power According to the principle of energy conservation, the energy delivered through the input coil must either be stored in the transformer’s magnetic field or transferred to the output circuit: • Since P = V I, if Vout is greater than Vin , then I out must be smaller than I in . • Pin = Pout only in ideal transformers ➜ In real transformers, the coils always have a small electrical resistance causing some power dissipation. ➜ For a real transformer, the output power is always less than the input power. • Power carried by the power line: Pavg = Vrms I rms College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Example An AC power line operates with a voltage Vrms = 500, 000 V and carries an AC current with I rms = 1000 A. What is the average (rms) power carried by the power line? Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Pavg = Vrms I rms = (500, 000 V) (1000 A) = 500 MW College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Example An AC power line operates with a voltage Vrms = 500, 000 V and carries an AC current with I rms = 1000 A. What is the average (rms) power carried by the power line? Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Pavg = Vrms I rms = (500, 000 V) (1000 A) = 500 MW If 10 % of the power is dissipated in the power line itself, what is the resistance of the power line? Resistors in AC Circuits 2 P line = I rms R line = (0.10) Pavg Capacitors in AC Circuits Inductors in AC Circuits LC Circuits R line = (0.10) Pavg (0.10) (5 × 108 W) = 50 Ω = 2 I rms (1000 A)2 College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Example An AC power line operates with a voltage Vrms = 500, 000 V and carries an AC current with I rms = 1000 A. The same power line is now operated with a reduced voltage of Vrms = 250, 000 V and current I rms = 2000 A. The product Vrms I rms is still the same, so the power carried by the line is the same. What percentage of this power is now dissipated in the power line? Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits 2 P line = I rms R line = (2000 A)2 (50 Ω) = 2.0 × 108 W The percentage of the total power now dissipated in the line is: Pline 2.0 × 108 W × 100 = 40 % × 100 = Ptotal 5.0 × 108 W compared to the initial 10 %. College Physics B Resistors in AC Circuits Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits Assume a circuit consisting of an AC generator and a resistor: • The voltage across the output of the AC source varies with time according to: LC Circuits V = Vmax sin (2π f t), where V is the instantaneous potential difference. College Physics B Review: Force on a Particle Magnetic Induction Resistors in AC Circuits Applying Ohm’s Law: I = Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Since the voltage varies sinusoidally, so does the current: Lenz’s Law Alternating Currents Vmax sin 2π f t R = I max sin 2π f t, I = Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits V R where I max = Vmax R . LC Circuits Since both I and V vary with time, the power also varies with time: P = Vmax I max sin2 (2π f t) College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux RMS Voltage To specify current and voltage values when they vary with time, rms values were adopted. For the voltage: Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Vrms = q Vmax (V 2 ) avg = √ 2 (RMS stands for Root Mean Square) Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits The root-mean-square value can be defined for any quantity. For current: s 1 2 I max I rms = I = √ 2 max 2 College Physics B Household Circuits Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits The standard AC voltage found in homes in the United States has a frequency of 60 Hz and is very often referred to as the “120-V power”: • The value of 120 V is the value of the rms voltage, i.e. Vrms = 120 V. • What is the corresponding value of Vmax for the voltage at an electrical outlet? College Physics B Household Circuits Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits The standard AC voltage found in homes in the United States has a frequency of 60 Hz and is very often referred to as the “120-V power”: • The value of 120 V is the value of the rms voltage, i.e. Vrms = 120 V. • What is the corresponding value of Vmax for the voltage at an electrical outlet? Vmax Vrms = √ 2 → Vmax = √ 2 Vrms = 170 V LC Circuits A DC voltage equal to the rms value of 120 V would give the same average dissipated power in a resistor. College Physics B AC Circuits with Capacitors Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Assume an AC circuit containing a single capacitor: The instantaneous charge is: Q = CV = C Vmax sin (2π f t). The capacitor’s voltage and its charge are in phase with each other. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux Faraday’s Law Electrical Generator Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits Current in Capacitors The instantaneous current is the rate at which charge flows onto capacitor plates in a short time interval: • Current is slope of the q-t plot. • Plot of current as function of time can be obtained from these slopes. • The current is a cosine function: I = I max cos (2π f t) • Equivalently, due to the relationship between sine and cosine functions: I = I max sin (2π f t + φ), where φ = π/2. College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment Magnetic Flux AC Circuits with Inductors Assume an AC circuit with a single inductor: Inductor’s voltage is proportional to the slope of the current-time relationship: Faraday’s Law Electrical Generator V = L (∆I/∆t) Lenz’s Law Alternating Currents Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits = Vmax sin (2π f t) Instantaneous current oscillates in time according to a cosine function: I = − I max cos (2π f t) = I max sin (2π f t − π/2) = I max sin (2π f t + φ), where φ = −π/2 (L = µ0 N 2 A / L). College Physics B Review: Force on a Particle Magnetic Induction Faraday’s Experiment LC Circuits Maximum energy in capacitor = maximum energy in inductor: 1 E cap = 1 Q2 2 C 2 E ind = 1 2 Magnetic Flux Faraday’s Law = LI2 = 2 1 Q max 2 C 1 2 cos2 (2π f t) 2 L I max sin2 (2π f t) Electrical Generator Lenz’s Law Alternating Currents The energy oscillates back and forth between the capacitor and its electric field and the inductor and its magnetic field: Generators Transformers Resistors in AC Circuits Capacitors in AC Circuits Inductors in AC Circuits LC Circuits f res = 2π 1 √ LC
