General Physics (PHY 2140)
Lecture 19
¾ Electricity and Magnetism
9Induced voltages and induction
9Energy
9AC circuits and EM waves
9Resistors in an AC circuits
http://www.physics.wayne.edu/~apetrov/PHY2140/
Chapter 20-21
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Next week: Prof. Claude Pruneau
Lectures on Monday and Wednesday
Exam on Friday
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Lightning Review
Last lecture:
1. Induced voltages and induction
9 Generators and motors
9 Self-induction
mv
qB
Φ = BA cos θ
∆I
E = −L
∆t
r=
L=
Review Problem: Charged particles passing
NΦ
I
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?
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S
S
Review example
v
Determine the direction of current in the loop
for bar magnet moving down.
N
Initial flux
Final flux
By Lenz’s law,
the induced field
is this
change
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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:
I max
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E
=
R
5
20.9 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
1 2
PEL = LI
2
Summary of the properties of circuit elements.
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Resistor
Capacitor
Inductor
units
ohm, Ω = V / A
farad, F = C / V
henry, H = V s / A
symbol
R
C
L
relation
V=IR
Q=CV
emf = -L (∆I / ∆t)
power dissipated
P = I V = I² R
= V² / R
0
0
energy stored
0
PEC = C V² / 2
PEL = L I² / 2
6
Example: stored energy
A 24V battery is connected in series with a resistor and an inductor,
where R = 8.0Ω and L = 4.0H. Find the energy stored in the inductor
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 =
8.0Ω and L = 4.0H. Find the energy stored in the inductor when the current
reaches its maximum value.
Given:
V = 24 V
R = 8.0 Ω
L = 4.0 H
Find:
PEL =?
Recall that the energy stored in th
inductor is
PEL =
1 2
LI
2
The only thing that is unknown in
the equation above is current. The
maximum value for the current is
I max
V 24V
= =
= 3.0 A
R 8.0Ω
Inserting this into the above expression for the energy gives
1
2
PEL = ( 4.0 H )( 3.0 A) = 18 J
2
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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
„
∆V = ∆Vmax sin 2πƒt
∆v is the instantaneous voltage
∆Vmax is the maximum voltage of the generator
ƒ is the frequency at which the voltage changes, in Hz
„
Same thing about the current (if only a resistor)
„
I = Imax sin 2πƒt
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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
Voltage varies as
∆V = ∆Vmax sin 2πƒt
Same thing about the current
I = Imax sin 2πƒt
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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 2πƒt)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
Averaging the above formula over one cycle
we get
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1 2
P = I max R
2
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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
Irms
Imax
=
= 0.707 Imax
2
Alternating voltages can also be discussed in terms of
rms values
∆Vrms
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∆Vmax
=
= 0.707 ∆Vmax
2
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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
„
∆Vrms = Irms R
Also applies to the maximum values of v and i
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Example: an AC circuit
An ac voltage source has an output of ∆V = 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 maximum current in the circuit when the generator is
connected to a 50.0W resistor.
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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
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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°
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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
1
XC =
2 π ƒC
„
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
„
∆Vrms = Irms XC
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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°
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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 = 2πƒL
When ƒ is in Hz and L is in H, XL will be in ohms
Ohm’s Law for the inductor
„
∆Vrms = Irms XL
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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?
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