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8/14/2019 PC Chapter 31
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Chapter 31
Faradays Law
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Michael Faraday Great experimental
physicist
1791 1867 Contributions to early
electricity include: Invention of motor,
generator, and
transformer Electromagnetic
induction
Laws of electrolysis
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Induction An induced currentis produced by a
changing magnetic field
There is an induced emfassociatedwith the induced current
A current can be produced without a
battery present in the circuit Faradays law of induction describes the
induced emf
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EMF Produced by a Changing
Magnetic Field, 1 A loop of wire is
connected to a
sensitive ammeter When a magnet ismoved toward theloop, the ammeter
deflects The direction waschosen to be towardthe right arbitrarily
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EMF Produced by a Changing
Magnetic Field, 2 When the magnet is
held stationary,
there is nodeflection of the
ammeter
Therefore, there is
no induced current Even though the
magnet is in the loop
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EMF Produced by a Changing
Magnetic Field, 3 The magnet is moved
away from the loop
The ammeter deflects in
the opposite direction
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Active Figure 31.1
(SLIDESHOW MODE ONLY)
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EMF Produced by a Changing
Magnetic Field, Summary The ammeter deflects when the magnet is
moving toward or away from the loop
The ammeter also deflects when the loop ismoved toward or away from the magnet Therefore, the loop detects that the magnet is
moving relative to it
We relate this detection to a change in themagnetic field
This is the induced current that is produced by aninduced emf
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Faradays Experiment
Set Up A primary coil is connected to
a switch and a battery
The wire is wrapped aroundan iron ring
A secondary coil is also
wrapped around the iron ring
There is no battery present inthe secondary coil
The secondary coil is not
directly connected to the
primary coil
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Active Figure 31.2
(SLIDESHOW MODE ONLY)
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Faradays Experiment
Findings At the instant the switch is closed, the
galvanometer (ammeter) needle deflects in
one direction and then returns to zero When the switch is opened, the galvanometer
needle deflects in the opposite direction and
then returns to zero
The galvanometer reads zero when there is a
steady current or when there is no current in
the primary circuit
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Faradays Experiment
Conclusions An electric current can be induced in a circuit
by a changing magnetic field
This would be the current in the secondary circuitof this experimental set-up
The induced current exists only for a shorttime while the magnetic field is changing
This is generally expressed as: an inducedemf is produced in the secondary circuitby the changing magnetic field The actual existence of the magnetic flux is not
sufficient to produce the induced emf, the fluxmust be changing
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Faradays Law Statements Faradays law of induction states that
the emf induced in a circuit is directly
proportional to the time rate of changeof the magnetic flux through the circuit
Mathematically,
Bddt
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Faradays Law Statements,
cont RememberB is the magnetic flux
through the circuit and is found by
If the circuit consists ofNloops, all of
the same area, and ifB
is the flux
through one loop, an emf is induced in
every loop and Faradays law becomes
Bd B A
Bd
N dt
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Faradays Law Example Assume a loop
enclosing an areaAlies in a uniformmagnetic field B
The magnetic fluxthrough the loop is
B= BA cos
The induced emf is
= - d/dt(BA cos )
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Ways of Inducing an emf The magnitude ofB can change with
time
The area enclosed by the loop can
change with time
The angle between B and the normal
to the loop can change with time Any combination of the above can
occur
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Applications of Faradays Law
GFI A GFI (ground fault indicator)
protects users of electrical
appliances against electric
shock When the currents in the
wires are in opposite
directions, the flux is zero
When the return current inwire 2 changes, the flux is no
longer zero
The resulting induced emf
can be used to trigger a
circuit breaker
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Applications of Faradays Law
Pickup Coil The pickup coil of an electric
guitar uses Faradays law
The coil is placed near the
vibrating string and causes aportion of the string to become
magnetized
When the string vibrates at the
same frequency, themagnetized segment produces
a changing flux through the coil
The induced emf is fed to an
amplifier
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Motional emf A motional emfis
one induced in aconductor movingthrough a constantmagnetic field
The electrons in theconductorexperience a force,FB = qv x B that is
directed along
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Motional emf, cont. Under the influence of the force, the electrons
move to the lower end of the conductor and
accumulate there As a result of the charge separation, an
electric field E is produced inside the
conductor
The charges accumulate at both ends of the
conductor until they are in equilibrium with
regard to the electric and magnetic forces
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Motional emf, final For equilibrium, qE= qvB orE= vB A potential difference is maintained
between the ends of the conductor aslong as the conductor continues tomove through the uniform magneticfield
If the direction of the motion is reversed,the polarity of the potential difference isalso reversed
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Sliding Conducting Bar
A bar moving through a uniform field and the
equivalent circuit diagram Assume the bar has zero resistance
The work done by the applied force appears as
internal energy in the resistorR
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Active Figure 31.10
(SLIDESHOW MODE ONLY)
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Sliding Conducting Bar, cont. The induced emf is
Since the resistance in the circuit is R,
the current is
Bd dx
B B v dt dt
l l
I B v
R R
l
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Sliding Conducting Bar,
Energy Considerations The applied force does work on the conducting
bar
This moves the charges through a magnetic field The change in energy of the system during some
time interval must be equal to the transfer of
energy into the system by work
The power input is equal to the rate at which
energy is delivered to the resistor
2
appI
F v B v
R
l
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Lenzs Law Faradays law indicates that the induced
emf and the change in flux have
opposite algebraic signs This has a physical interpretation that
has come to be known as Lenzs law
Developed by German physicistHeinrich Lenz
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Lenzs Law, cont. Lenzs law: the induced current in a
loop is in the direction that creates a
magnetic field that opposes the changein magnetic flux through the area
enclosed by the loop
The induced current tends to keep theoriginal magnetic flux through the circuit
from changing
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Induced emf and Electric
Fields An electric field is created in the conductor
as a result of the changing magnetic flux
Even in the absence of a conducting loop,a changing magnetic field will generate anelectric field in empty space
This induced electric field is
nonconservative Unlike the electric field produced by stationary
charges
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Induced emf and Electric
Fields, cont. The emf for any closed path can be
expressed as the line integral ofE.ds
over the path Faradays law can be written in a
general form:
Bd
ddt
E s
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Induced emf and Electric
Fields, final The induced electric field is a
nonconservative field that is generated
by a changing magnetic field The field cannot be an electrostatic field
because if the field were electrostatic,
and hence conservative, the lineintegral ofE.ds would be zero and it
isnt
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Generators Electric generators
take in energy by work
and transfer it out byelectrical transmission
The AC generator
consists of a loop of
wire rotated by someexternal means in a
magnetic field
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Rotating Loop Assume a loop with
Nturns, all of the
same area rotatingin a magnetic field
The flux through the
loop at any time tis
B = BA cos =
BA cos t
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Induced emf in a Rotating
Loop The induced emf in
the loop is
This is sinusoidal,with max = NAB
sin
Bd Ndt
NAB t
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Active Figure 31.21
(SLIDESHOW MODE ONLY)
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Induced emf in a Rotating
Loop, cont. max occurs when t= 90
o or 270o
This occurs when the magnetic field is in
the plane of the coil and the time rate ofchange of flux is a maximum
= 0 when t= 0o or 180o
This occurs when B is perpendicular to theplane of the coil and the time rate of
change of flux is zero
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DC Generators The DC (direct
current) generatorhas essentially thesame componentsas the AC generator
The main differenceis that the contactsto the rotating loopare made using asplit ring called acommutator
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DC Generators, cont. In this configuration, the
output voltage always has
the same polarity
It also pulsates with time
To obtain a steady DC
current, commercial
generators use many coils
and commutatorsdistributed so the pulses
are out of phase
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Active Figure 31.23
(SLIDESHOW MODE ONLY)
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Motors Motors are devices into which energy is
transferred by electrical transmission
while energy is transferred out by work A motor is a generator operating in
reverse
A current is supplied to the coil by abattery and the torque acting on thecurrent-carrying coil causes it to rotate
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Motors, cont. Useful mechanical work can be done by
attaching the rotating coil to some
external device However, as the coil rotates in a
magnetic field, an emf is induced This induced emf always acts to reduce the
current in the coil The back emf increases in magnitude as
the rotational speed of the coil increases
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Motors, final The current in the rotating coil is limited
by the back emf
The term back emfis commonly used toindicate an emf that tends to reduce thesupplied current
The induced emf explains why the
power requirements for starting a motorand for running it are greater for heavyloads than for light ones
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Eddy Currents Circulating currents called
eddy currents are induced in
bulk pieces of metal moving
through a magnetic field The eddy currents are in
opposite directions as the plate
enters or leaves the field
Eddy currents are often
undesirable because they
represent a transformation of
mechanical energy into internal
energy
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Active Figure 31.26
(SLIDESHOW MODE ONLY)
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Maxwells Equations,
Introduction Maxwells equations are regarded as
the basis of all electrical and magnetic
phenomena Maxwells equations represent the laws
of electricity and magnetism that have
already been discussed, but they haveadditional important consequences
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Maxwells Equations,
Statement
Gauss's law electric
0 Gauss's law in magnetism
Faraday's law
Ampere-Maxwell lawI
oS
S
B
Eo o o
qd
d
dd
dtd
d dt
E A
B A
E s
B s
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Maxwells Equations, Details Gausss law (electrical):
The total electric flux through any
closed surface equals the net chargeinside that surface divided by o
This relates an electric field to the
charge distribution that creates it
oS
qd
E A
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Maxwells Equations, Details 2 Gausss law (magnetism): The total magnetic flux through any closed
surface is zero This says the number of field lines that entera closed volume must equal the number thatleave that volume
This implies the magnetic field lines cannotbegin or end at any point Isolated magnetic monopoles have not been
observed in nature
0S
d B A
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Maxwells Equations, Details 3 Faradays law of Induction: This describes the creation of an electric field
by a changing magnetic flux The law states that the emf, which is the lineintegral of the electric field around any closedpath, equals the rate of change of the
magnetic flux through any surface boundedby that path One consequence is the current induced in a
conducting loop placed in a time-varying B
Bdddt
E s
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Maxwells Equations, Details 4 The Ampere-Maxwell law is a generalization
of Amperes law
It describes the creation of a magnetic field
by an electric field and electric currents
The line integral of the magnetic field around
any closed path is the given sum
I Eo o o dd dt B s
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The Lorentz Force Law Once the electric and magnetic fields are
known at some point in space, the forceacting on a particle of charge q can be
calculated F = qE + qv x B This relationship is called the Lorentz force
law Maxwells equations, together with this force
law, completely describe all classicalelectromagnetic interactions
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Maxwells Equations,
Symmetry The two Gausss laws are symmetrical, apart
from the absence of the term for magnetic
monopoles in Gausss law for magnetism Faradays law and the Ampere-Maxwell law
are symmetrical in that the line integrals ofE
and B around a closed path are related to the
rate of change of the respective fluxes Maxwells equations are of fundamental
importance to all of science