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General Physics (PHY 2140)

Lecture 10Lecture 10Electricity and Magnetism

Induced voltages and inductionSelf-InductanceRL CircuitsEnergy in magnetic fields

AC circuits and EM wavesResistors, capacitors and inductors in AC circuitsThe RLC circuitPower in AC circuits

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

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Reminder: Exam 2 this Wednesday 6/13Reminder: Exam 2 this Wednesday 6/13

1212--14 questions.14 questions.

Show your work for full credit.Show your work for full credit.

Closed book. Closed book.

You may bring a page of notes.You may bring a page of notes.

Bring a calculator.Bring a calculator.

Bring a pen or pencil.Bring a pen or pencil.

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Lightning ReviewLightning Review

Last lecture:

1.1.

Induced voltages and inductionInduced voltages and inductionGenerators and motorsGenerators and motorsSelfSelf--inductioninduction

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?

cosBA θΦ =ILt

Δ= −

ΔE

NLIΦ

=

mvrqB

=

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N

SS

v

change

Review exampleReview example

Determine the direction of current in the loop for bar magnet moving down.

Initial flux

Final flux

By Lenz’s law, the induced field is this

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20.6 Self20.6 Self--inductanceinductance

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

Nt

ΔΦ= −

ΔE

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The magnetic flux is proportional to the magnetic field, The magnetic flux is proportional to the magnetic field, which is proportional to the current in the circuitwhich is proportional to the current in the circuitThus, the selfThus, the self--induced EMF must be proportional to the induced EMF must be proportional to the time rate of change of the currenttime rate of change of the current

where where LL

is called the is called the inductance inductance of the deviceof the device

Units: SI: Units: SI: henryhenry

(H)(H)

If flux is initially zero,If flux is initially zero,

ILt

Δ= −

ΔE

1 1V sH A⋅=

NL NI I

ΔΦ Φ= =

Δ

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Example: solenoidExample: solenoid

A solenoid of radius 2.5cm has 400 turns and a length of 20 cm. A solenoid of radius 2.5cm has 400 turns and a length of 20 cm. Find Find (a) its inductance and (b) the rate at which current must change(a) its inductance and (b) the rate at which current must change

through it to produce an through it to produce an emfemf

of 75mV. of 75mV.

0 0NB nI Il

μ μ= = 0BNBA IAl

μΦ = =

2

0BN N AL

I lμΦ

= = = (4π x 10-7)(160000)(2.0 x 10-3)/(0.2) = 2 mH

LI ILt t

Δ Δ= − → = −

Δ ΔEE = (75 x 10-3)/ (2.0 x 10-3) = 37.5 A/s

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Inductor in a CircuitInductor in a Circuit

InductanceInductance

can be interpreted as a can be interpreted as a measure of opposition to the measure of opposition to the rate rate of changeof change

in the currentin the currentRemember Remember resistance R is a measure of opposition to the currentresistance R is a measure of opposition to the current

As a circuit is completed, the current begins to increase, but tAs a circuit is completed, the current begins to increase, but the he inductor produces an inductor produces an emf that opposes the increasing currentemf that opposes the increasing current

Therefore, the current doesnTherefore, the current doesn’’t change from 0 to its maximum t change from 0 to its maximum instantaneouslyinstantaneouslyMaximum current:Maximum current:

maxIR

=E

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20.7 RL Circuits20.7 RL Circuits

Recall OhmRecall Ohm’’s Law to find the voltage drop on Rs Law to find the voltage drop on R

We have something similar with inductorsWe have something similar with inductors

Similar to the case of the capacitor, we get an equation Similar to the case of the capacitor, we get an equation for the for the currentcurrent

as a function of time (series circuit).as a function of time (series circuit).

V IRΔ =

LILt

Δ= −

ΔE

( )/1 Rt LI eR

−= −E L

Rτ =

(voltage across an inductor)

(voltage across a resistor)

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RL Circuit (continued)RL Circuit (continued)

( )/1 Rt LVI eR

−= −

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20.8 Energy stored in a magnetic field20.8 Energy stored in a magnetic fieldThe battery in any circuit that contains a coil has to do The battery in any circuit that contains a coil has to do work to produce a currentwork to produce a currentSimilar to the capacitor, any coil (or inductor) would store Similar to the capacitor, any coil (or inductor) would store potential energypotential energy

212LPE LI=

Summary of the properties of circuit elements.Resistor Capacitor Inductor

units ohm, Ω

= V / A farad, F = C / V henry, H = V s / A

symbol R C L

relation V = I R Q = C V emf = -L (ΔI / Δt)

power dissipated P = I V = I² R = V² / R 0 0

energy stored 0 PEC = C V² / 2 PEL = L I² / 2

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Example: stored energyExample: stored energy

A 24V battery is connected in series with a resistor and an induA 24V battery is connected in series with a resistor and an inductor, ctor, where R = 8.0where R = 8.0ΩΩ

and L = 4.0H. Find the energy stored in the inductor and L = 4.0H. Find the energy stored in the inductor when the current reaches its maximum value.when the current reaches its maximum value.

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A 24V battery is connected in series with a resistor and an induA 24V battery is connected in series with a resistor and an inductor, where R = ctor, where R = 8.08.0ΩΩ

and L = 4.0H. Find the energy stored in the inductor when the cand L = 4.0H. Find the energy stored in the inductor when the current urrent reaches its maximum value.reaches its maximum value.

Given:

V = 24 VR = 8.0 ΩL = 4.0 H

Find:

PEL

=?

Recall that the energy stored in the inductor is

212LPE LI=

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

max24 3.08.0

V VI AR

= = =Ω

( )( )21 4.0 3.0 182LPE H A J= =

Chapter 21Chapter 21

Alternating Current Circuits Alternating Current Circuits and Electromagnetic Wavesand Electromagnetic Waves

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AC CircuitAC Circuit

An AC circuit consists of a combination of An AC circuit consists of a combination of circuit elements and an AC generator or circuit elements and an AC generator or sourcesourceThe output of an AC generator is sinusoidal The output of an AC generator is sinusoidal and varies with time according to the and varies with time according to the following equationfollowing equationΔΔv = v = ΔΔVVmaxmax sin 2sin 2ππƒƒtt

ΔΔv is the instantaneous voltagev is the instantaneous voltageΔΔVVmaxmax is the maximum voltage of the generatoris the maximum voltage of the generatorƒƒ is the frequency at which the voltage changes, in Hzis the frequency at which the voltage changes, in Hz

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Resistor in an AC CircuitResistor in an AC CircuitConsider a circuit Consider a circuit consisting of an AC source consisting of an AC source and a resistorand a resistorThe graph shows the The graph shows the current through and the current through and the voltage across the resistorvoltage across the resistorThe current and the The current and the voltage reach their voltage reach their maximum values at the maximum values at the same timesame timeThe current and the The current and the voltage are said to be voltage are said to be in in phasephase

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More About Resistors in an AC More About Resistors in an AC CircuitCircuit

The direction of the current has no effect on The direction of the current has no effect on the behavior of the resistorthe behavior of the resistorThe rate at which electrical energy is The rate at which electrical energy is dissipated in the circuit is given bydissipated in the circuit is given by

P = iP = i22 R= R= ((IImaxmax sin 2sin 2ππƒƒt)t)22 RRwhere i is the where i is the instantaneous currentinstantaneous currentthe heating effect produced by an AC current with a the heating effect produced by an AC current with a maximum value of Imaximum value of Imaxmax is not the same as that of a DC is not the same as that of a DC current of the same valuecurrent of the same valueThe maximum current occurs for a small amount of timeThe maximum current occurs for a small amount of time

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rms Current and Voltagerms Current and Voltage

The The rms currentrms current is the direct current is the direct current that would dissipate the same amount that would dissipate the same amount of energy in a resistor as is actually of energy in a resistor as is actually dissipated by the AC currentdissipated by the AC current

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

maxmax

rms I707.02

II ==

maxmax

rms V707.02

VV Δ=Δ

=Δ

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OhmOhm’’s Law in an AC Circuits Law in an AC Circuit

rms values will be used when discussing rms values will be used when discussing AC currents and voltagesAC currents and voltages

AC ammeters and voltmeters are designed AC ammeters and voltmeters are designed to read rms valuesto read rms valuesMany of the equations will be in the same Many of the equations will be in the same form as in DC circuitsform as in DC circuits

OhmOhm’’s Law for a resistor, R, in an AC s Law for a resistor, R, in an AC circuitcircuitΔΔVVrmsrms = I= Irmsrms RR

Also applies to the maximum values of v and iAlso applies to the maximum values of v and i

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Example: an AC circuitExample: an AC circuit

maxmax0.707 0.707 x 150V 10

26 Vrms

VV VΔΔ = = Δ = =

An ac voltage source has an output of An ac voltage source has an output of ΔΔV = 150 sin (377 t).V = 150 sin (377 t). Find Find (a) the (a) the rmsrms

voltage output, voltage output, (b) the frequency of the source, and (b) the frequency of the source, and (c) the voltage at (c) the voltage at t = (1/120)st = (1/120)s. . (d) Find the (d) Find the rmsrms

current in the circuit when the generator is current in the circuit when the generator is connected to a 50.0connected to a 50.0ΩΩ

resistor. resistor.

377 rad/sec, 2 , / 2 377/ 60 H 2 z =f fω ω π ω π π= = = =

ΔV = 150 sin (377 x 1/120) = 0 V

ΔVrms

= Irms

R thus, Irms

= ΔVrms

/R = 2.12 A

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Capacitors in an AC CircuitCapacitors in an AC Circuit

Consider a circuit containing a capacitor and Consider a circuit containing a capacitor and an AC sourcean AC sourceThe current starts out at a large value and The current starts out at a large value and charges the plates of the capacitorcharges the plates of the capacitor

There is initially no resistance to hinder the flow of There is initially no resistance to hinder the flow of the current while the plates are not chargedthe current while the plates are not charged

As the charge on the plates increases, the As the charge on the plates increases, the voltage across the plates increases and the voltage across the plates increases and the current flowing in the circuit decreasescurrent flowing in the circuit decreases

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More About Capacitors in an More About Capacitors in an AC CircuitAC Circuit

The current reverses The current reverses directiondirectionThe voltage across The voltage across the plates decreases the plates decreases as the plates lose as the plates lose the charge they had the charge they had accumulatedaccumulatedThe voltage across The voltage across the capacitor lags the capacitor lags behind the current behind the current by 90by 90°°

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Capacitive Reactance and Capacitive Reactance and OhmOhm’’s Laws Law

The impeding effect of a capacitor on the The impeding effect of a capacitor on the current in an AC circuit is called the current in an AC circuit is called the capacitive capacitive reactancereactance and is given byand is given by

When When ƒƒ is in Hz and C is in F, Xis in Hz and C is in F, XCC will be in ohmswill be in ohms

OhmOhm’’s Law for a capacitor in an AC circuits Law for a capacitor in an AC circuitΔΔVVrmsrms = I= Irmsrms XXCC

Cƒ21XC π

=

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Inductors in an AC CircuitInductors in an AC Circuit

Consider an AC circuit Consider an AC circuit with a source and an with a source and an inductorinductorThe current in the The current in the circuit is impeded by circuit is impeded by the back emf of the the back emf of the inductorinductorThe voltage across the The voltage across the inductor always leads inductor always leads the current by 90the current by 90°°

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Inductive Reactance and Inductive Reactance and OhmOhm’’s Laws Law

The effective resistance of a coil in an The effective resistance of a coil in an AC circuit is called its AC circuit is called its inductive inductive reactancereactance and is given byand is given by

XXLL = 2= 2ππƒƒLLWhen When ƒƒ is in Hz and L is in H, Xis in Hz and L is in H, XLL will be in will be in ohmsohms

OhmOhm’’s Law for the inductors Law for the inductorΔΔVVrmsrms = I= Irmsrms XXLL

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Example: AC circuit with Example: AC circuit with capacitors and inductorscapacitors and inductors

A 2.40mF capacitor is connected across an alternating voltage wiA 2.40mF capacitor is connected across an alternating voltage with an th an rmsrms

value of 9.00V. The value of 9.00V. The rmsrms

current in the capacitor is 25.0mA. (a) What current in the capacitor is 25.0mA. (a) What is the source frequency? (b) If the capacitor is replaced by an is the source frequency? (b) If the capacitor is replaced by an ideal coil ideal coil with an inductance of 0.160H, what is the with an inductance of 0.160H, what is the rmsrms

current in the coil? current in the coil?

12 ƒCX

Cπ⎛ ⎞

=⎜ ⎟⎝ ⎠

ΔΔVVrmsrms

= = IIrmsrms

XXCC

, first we find , first we find XXCC

::

For and inductor XFor and inductor XLL

= 2= 2ππƒƒL, try solving for L, try solving for IIrmsrms

= = ΔΔVVrmsrms

/ / XXLL

Now, solve for Now, solve for ƒƒ::

ƒƒ = 1/ = 1/ 22ππ XXCC

CC

= = 0.184 Hz0.184 Hz

ΔΔVVrmsrms

/ / IIrmsrms

= 9.00V/25.0 x 10= 9.00V/25.0 x 10--33

A = A = 360 ohms360 ohms

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The RLC Series CircuitThe RLC Series Circuit

The resistor, The resistor, inductor, and inductor, and capacitor can be capacitor can be combined in a circuitcombined in a circuitThe current in the The current in the circuit is the same at circuit is the same at any time and varies any time and varies sinusoidally with sinusoidally with timetime

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Current and Voltage Current and Voltage Relationships in an RLC CircuitRelationships in an RLC Circuit

The instantaneous The instantaneous voltage across the voltage across the resistor is in phase with resistor is in phase with the currentthe currentThe instantaneous The instantaneous voltage across the voltage across the inductor leads the inductor leads the current by 90current by 90°°The instantaneous The instantaneous voltage across the voltage across the capacitor lags the capacitor lags the current by 90current by 90°°

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Phasor DiagramsPhasor Diagrams

To account for the To account for the different phases of the different phases of the voltage drops, vector voltage drops, vector techniques are usedtechniques are usedRepresent the voltage Represent the voltage across each element as across each element as a rotating vector, called a rotating vector, called a a phasorphasorThe diagram is called a The diagram is called a phasor diagramphasor diagram

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Phasor Diagram for RLC Phasor Diagram for RLC Series CircuitSeries Circuit

The voltage across the The voltage across the resistor is on the +x resistor is on the +x axis since it is in phase axis since it is in phase with the currentwith the currentThe voltage across the The voltage across the inductor is on the +y inductor is on the +y since it leads the since it leads the current by 90current by 90°°The voltage across the The voltage across the capacitor is on the capacitor is on the ––y y axis since it lags behind axis since it lags behind the current by 90the current by 90°°

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Phasor Diagram, contPhasor Diagram, cont

The phasors are The phasors are added as vectors to added as vectors to account for the account for the phase differences in phase differences in the voltagesthe voltagesΔΔVVLL and and ΔΔVVCC are on are on the same line and so the same line and so the net y component the net y component is is ΔΔVVL L -- ΔΔVVCC

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ΔΔVV maxmax

From the Phasor From the Phasor

DiagramDiagram

The voltages are not in phase, so they cannot The voltages are not in phase, so they cannot simply be added to get the voltage across the simply be added to get the voltage across the combination of the elements or the voltage combination of the elements or the voltage sourcesource

φφ is the is the phase anglephase angle between the current and between the current and the maximum voltagethe maximum voltage

R

CL

2CL

2Rmax

VVVtan

)VV(VV

ΔΔ−Δ

=φ

Δ−Δ+Δ=Δ

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Impedance of a CircuitImpedance of a Circuit

The impedance, Z, The impedance, Z, can also be can also be represented in a represented in a phasor diagramphasor diagram

RXXtan

)XX(RZ

CL

2CL

2

−=φ

−+=

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Impedance and OhmImpedance and Ohm’’s Laws Law

OhmOhm’’s Law can be applied to the s Law can be applied to the impedanceimpedanceΔΔVVmaxmax = I= Imaxmax ZZ

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Summary of Circuit Elements, Summary of Circuit Elements, Impedance and Phase AnglesImpedance and Phase Angles

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Problem Solving for AC Problem Solving for AC CircuitsCircuits

Calculate as many unknown quantities Calculate as many unknown quantities as possibleas possible

For example, find XFor example, find XLL and Xand XCC

Be careful of units Be careful of units ---- use F, H, use F, H, ΩΩApply OhmApply Ohm’’s Law to the portion of the s Law to the portion of the circuit that is of interestcircuit that is of interestDetermine all the unknowns asked for Determine all the unknowns asked for in the problemin the problem

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Power in an AC CircuitPower in an AC Circuit

No power losses are associated with No power losses are associated with capacitors and pure inductors in an AC circuitcapacitors and pure inductors in an AC circuit

In a capacitor, during oneIn a capacitor, during one--half of a cycle energy is half of a cycle energy is stored and during the other half the energy is stored and during the other half the energy is returned to the circuitreturned to the circuitIn an inductor, the source does work against the In an inductor, the source does work against the back emf of the inductor and energy is stored in back emf of the inductor and energy is stored in the inductor, but when the current begins to the inductor, but when the current begins to decrease in the circuit, the energy is returned to decrease in the circuit, the energy is returned to the circuitthe circuit

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Power in an AC Circuit, contPower in an AC Circuit, cont

The average power delivered by the The average power delivered by the generator is converted to internal generator is converted to internal energy in the resistorenergy in the resistor

PPavav = I= IrmsrmsΔΔVVRR = = IIrmsrmsΔΔVVrmsrms cos cos φφcos cos φφ is called the is called the power factorpower factor of the of the circuitcircuit

Phase shifts can be used to maximize Phase shifts can be used to maximize power outputspower outputs

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Resonance in an AC CircuitResonance in an AC Circuit

ResonanceResonance occurs at occurs at the frequency, the frequency, ƒƒoo, , where the current has where the current has its maximum valueits maximum value

To achieve maximum To achieve maximum current, the impedance current, the impedance must have a minimum must have a minimum valuevalueThis occurs when XThis occurs when XLL = X= XCC

LC21ƒo

π=