General Physics (PHY 2140)

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 Chapter 20-21 http://www.physics.wayne.edu/~alan/2140Website/Main.htm

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.

Lightning Review • Last lecture: • Induced voltages and induction • Generators and motors • Self-induction Review Problem: Charged particles passing 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?

change S Review example S Determine the direction of current in the loop for bar magnet moving down. v N Initial flux Final flux By Lenz’s law, the induced field is this

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

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 where L is called the inductance of the device • Units: SI: henry (H) • If flux is initially zero,

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. = (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

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:

20.7 RL Circuits • Recall Ohm’s Law to find the voltage drop on R • We have something similar with inductors • Similar to the case of the capacitor, we get an equation for the current as a function of time (series circuit). (voltage across a resistor) (voltage across an inductor)

RL Circuit (continued)

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

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.

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. Recall that the energy stored in the inductor is Given: V = 24 V R = 8.0 W L = 4.0 H Find: PEL =? The only thing that is unknown in the equation above is current. The maximum value for the current is Inserting this into the above expression for the energy gives

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 2ƒt • Δv is the instantaneous voltage • ΔVmax is the maximum voltage of the generator • ƒ is the frequency at which the voltage changes, in Hz

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

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 2ƒt)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

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 • Alternating voltages can also be discussed in terms of rms values

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

Example: an AC circuit An ac voltage source has an output of DV = 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. ΔV = 150 sin (377 x 1/120) = 0 V ΔVrms = Irms R thus, Irms = ΔVrms/R = 2.12 A

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

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°

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 • 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 • ΔVrms = Irms XC

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°

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 = 2ƒL • When ƒ is in Hz and L is in H, XL will be in ohms • Ohm’s Law for the inductor • ΔVrms = Irms XL

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/ 2 XCC = 0.184 Hz For and inductor XL = 2ƒL, try solving for Irms = ΔVrms/ XL

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

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°

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

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°

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

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 •  is the phase angle between the current and the maximum voltage

Impedance of a Circuit • The impedance, Z, can also be represented in a phasor diagram

Impedance and Ohm’s Law • Ohm’s Law can be applied to the impedance • ΔVmax = Imax Z

Summary of Circuit Elements, Impedance and Phase Angles

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

Power in an AC Circuit • No power losses are associated with capacitors and pure inductors in an AC circuit • 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

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

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

General Physics (PHY 2140)