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Applied Physics

Lecture 19Lecture 19 Electricity and Magnetism

Induced voltages and inductionEnergy

AC circuits and EM wavesResistors in an AC circuits

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Homework Assignment

Due next classDue next class

19.7,11,3320.1,7,9,24,28,37

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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 of changerate of change in the current in the current

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

Therefore, the current doesn’t change from 0 to its maximum Therefore, the current doesn’t change from 0 to its maximum instantaneouslyinstantaneously

Maximum current:Maximum current:

maxI ER

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20.9 Energy stored in a magnetic field20.9 Energy stored in a magnetic field

The 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 current

Similar to the capacitor, any coil (or inductor) would store Similar to the capacitor, any coil (or inductor) would store potential energypotential energy

21

2LPE LI

Summary of the properties of circuit elements.

Resistor Capacitor Inductor

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

symbol R C L

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

power dissipatedP = 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 inductor, A 24V battery is connected in series with a resistor and an inductor, where R = 8.0where R = 8.0 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 inductor, where R = A 24V battery is connected in series with a resistor and an inductor, where R = 8.08.0 and L = 4.0H. Find the energy stored in the inductor when the current and L = 4.0H. Find the energy stored in the inductor when the current reaches its maximum value.reaches its maximum value.

Given:

V = 24 VR = 8.0 L = 4.0 H

Find:

PEL =?

Recall that the energy stored in th inductor is

21

2LPE 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

max

243.0

8.0

V VI A

R

214.0 3.0 18

2LPE 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 circuit elements and an An AC circuit consists of a combination of circuit elements and an AC generator or sourceAC generator or sourceThe output of an AC generator is sinusoidal and varies with time The output of an AC generator is sinusoidal and varies with time according to the following equationaccording to the following equation

ΔV = ΔVΔV = ΔVmaxmax sin 2 sin 2ƒtƒt

Δv is the instantaneous voltageΔv is the instantaneous voltage

ΔVΔVmaxmax is the maximum voltage of the generator is the maximum voltage of the generator

ƒ is the frequency at which the voltage changes, in Hzƒ is the frequency at which the voltage changes, in Hz

Same thing about the current (if only a resistor)Same thing about the current (if only a resistor)

I = II = Imaxmax sin 2 sin 2ƒtƒt

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Resistor in an AC CircuitResistor in an AC Circuit

Consider a circuit consisting of Consider a circuit consisting of an AC source and a resistoran AC source and a resistorThe graph shows the current The graph shows the current through and the voltage across through and the voltage across the resistorthe resistorThe current and the voltage The current and the voltage reach their maximum values at reach their maximum values at the same timethe same timeThe current and the voltage The current and the voltage are said to be are said to be in phasein phase

Voltage varies asVoltage varies as

ΔV = ΔVΔV = ΔVmaxmax sin 2 sin 2ƒtƒt

Same thing about the currentSame thing about the currentI = II = Imaxmax sin 2 sin 2ƒtƒt

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More About Resistors in an AC CircuitMore About Resistors in an AC Circuit

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 2 sin 2ƒt)ƒt)22 R R

where i is the where i is the instantaneous currentinstantaneous currentthe heating effect produced by an AC current the heating effect produced by an AC current with a maximum value of Iwith a maximum value of Imaxmax is not the same as is not the same as that of a DC current of the same valuethat of a DC current of the same valueThe maximum current occurs for a small The maximum current occurs for a small amount of timeamount of time

Averaging the above formula over one cycle Averaging the above formula over one cycle we getwe get

2max

1

2P I R

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rms Current and Voltagerms Current and Voltage

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

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

maxmax

rms I707.02

II

maxmax

rms V707.02

VV

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Ohm’s Law in an AC CircuitOhm’s Law in an AC Circuit

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

AC ammeters and voltmeters are designed to read rms valuesAC ammeters and voltmeters are designed to read rms values Many of the equations will be in the same form as in DC Many of the equations will be in the same form as in DC

circuitscircuits

Ohm’s Law for a resistor, R, in an AC circuitOhm’s Law for a resistor, R, in an AC circuit

ΔVΔVrmsrms = I = Irmsrms R R

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

An ac voltage source has an output of An ac voltage source has an output of V = 150 sin (377 t).V = 150 sin (377 t). Find Find (a) the rms voltage output, (a) the rms 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 maximum current in the circuit when the generator is (d) Find the maximum current in the circuit when the generator is

connected to a 50.0W resistor. connected to a 50.0W resistor.

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Capacitors in an AC CircuitCapacitors in an AC Circuit

Consider a circuit containing a capacitor and an AC sourceConsider a circuit containing a capacitor and an AC source

The current starts out at a large value and charges the plates of the The current starts out at a large value and charges the plates of the capacitorcapacitor

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

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

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More About Capacitors in an AC CircuitMore About Capacitors in an AC Circuit

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

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Capacitive Reactance and Ohm’s LawCapacitive Reactance and Ohm’s Law

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

When ƒ is in Hz and C is in F, XWhen ƒ is in Hz and C is in F, XCC will be in ohms will be in ohms

Ohm’s Law for a capacitor in an AC circuitOhm’s Law for a capacitor in an AC circuit ΔVΔVrmsrms = I = Irmsrms X XCC

Cƒ2

1XC

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Inductors in an AC CircuitInductors in an AC Circuit

Consider an AC circuit with a Consider an AC circuit with a source and an inductorsource and an inductor

The current in the circuit is The current in the circuit is impeded by the back emf of the impeded by the back emf of the inductorinductor

The voltage across the inductor The voltage across the inductor always leads the current by 90°always leads the current by 90°

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Inductive Reactance and Ohm’s LawInductive Reactance and Ohm’s Law

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

XXLL = 2 = 2ƒLƒL

When ƒ is in Hz and L is in H, XWhen ƒ is in Hz and L is in H, XLL will be in ohms will be in ohms

Ohm’s Law for the inductorOhm’s Law for the inductor ΔVΔVrmsrms = I = Irmsrms X XLL

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Example: AC circuit with capacitors and Example: AC circuit with capacitors and inductorsinductors

A 2.40mF capacitor is connected across an alternating voltage with an 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 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 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? with an inductance of 0.160H, what is the rms current in the coil?