110/16/2015 Applied Physics Lecture 19 Electricity and Magnetism Induced voltages and induction...
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Transcript of 110/16/2015 Applied Physics Lecture 19 Electricity and Magnetism Induced voltages and induction...
1104/21/2304/21/23
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?