Transcript Chapter 31 Faraday’s Law

Chapter 31
Faraday’s Law
Michael Faraday
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Great experimental physicist and chemist
1791 – 1867
Contributions to early electricity include:
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Invention of motor, generator, and transformer
Electromagnetic induction
Laws of electrolysis: A method of separating
bonded elements and compounds by passing
an electric current through them
Induction
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An induced current is produced by a
changing magnetic field
There is an induced emf associated with
the induced current
A current can be produced without a
battery present in the circuit
Faraday’s law of induction describes the
induced emf
EMF Produced by a Changing
Magnetic Field, 1
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A loop of wire is
connected to a
sensitive ammeter
When a magnet is
moved toward the
loop, the ammeter
deflects
EMF Produced by a Changing
Magnetic Field, 2
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When the magnet is
held stationary, there is
no deflection of the
ammeter
Therefore, there is no
induced current
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Even though the
magnet is in the loop
EMF Produced by a Changing
Magnetic Field, 3
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The magnet is moved
away from the loop
The ammeter deflects in
the opposite direction
Faraday’s Law – Statements
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Faraday’s law of induction states that
“the emf induced in a circuit is directly
proportional to the time rate of change
of the magnetic flux through the circuit”
Mathematically,
d B
ε
dt
Faraday’s Law – Statements,
cont
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Remember B is the magnetic flux
through the circuit and is found by
 B   B  dA
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If the circuit consists of N loops, all of
the same area, and if B is the flux
through one loop, an emf is induced in
every loop and Faraday’s law becomes
dB
ε  N
dt
Faraday’s Law – Example
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Assume a loop
enclosing an area A
lies in a uniform
magnetic field B
The magnetic flux
through the loop is
B = BA cos q
The induced emf is
e = - d/dt (BA cos q)
Ways of Inducing an emf
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The magnitude of B can change with time
The area enclosed by the loop can change
with time
The angle q between B and the normal to
the loop can change with time
Any combination of the above can occur
Lenz’s Law
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Faraday’s 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 Lenz’s law
Developed by German physicist Heinrich
Lenz
Lenz’s Law, cont.
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Lenz’s law: the induced current in a
loop is in the direction that creates a
magnetic field that opposes the change
in magnetic flux through the area
enclosed by the loop
The induced current tends to keep the
original magnetic flux through the circuit
from changing
Problem 1
A loop of wire in the shape of a rectangle of width w and length L and a
long, straight wire carrying a current I lie on a tabletop. Determine the
magnetic flux through the loop due to the current I. Suppose the current
is changing with time according to I = a + bt, where a and b are
constants. Determine the emf that is induced. What is the direction of
the induced current in the rectangle?
Problem 2
The square loop in the figure is made of wires with a total resistance of
20.0 W . It is placed in a uniform 0.100 T magnetic field directed
perpendicular into the plane of the paper. The loop, which is hinged at
each corner, is pulled as shown until the separation between points A
and B is 3.00 m. What is the magnitude and direction of the induced
current. [0.0301] A, clockwise
Problem 3
A 39 turn circular coil of radius 4.80 cm and resistance 1.00 W is
placed in a magnetic field directed perpendicular to the plane of
the coil. The magnitude of the magnetic field varies in time
according to the expression B = 0.0100t + 0.0400t2, where t is in
seconds and B is in tesla. Calculate the induced emf in the coil
at t = 5.60 s. [129 mV]
Applications of Faraday’s Law
– Electric Generator
An electric conductor, like a
copper wire, is moved
through a magnetic field,
which causes an electric
current to flow (be induced)
in the conductor
http://www.wvic.com/how-gen-works.htm
Generators
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Electric generators
take in energy by work
and transfer it out by
electrical transmission
The AC generator
consists of a loop of
wire rotated by some
external means in a
magnetic field
Applications of Faraday’s Law
– Pickup Coil
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The coil is placed near the
vibrating string and causes a
portion of the string to become
magnetized
When the string vibrates at the
same frequency, the
magnetized segment produces
a changing flux through the coil
The induced emf is fed to an
amplifier
Applications of Faraday’s Law
– Transformers
An alternating current in
one winding creates a
time-varying magnetic
flux in the core, which
induces a voltage in the
other windings
Is this a step-up or a stepdown transformer?
Yamanashi maglev
The magnetized coil running along the
track, called a guideway, repels the
large magnets on the train's
undercarriage, allowing the train to
levitate between 0.39 and 3.93 inches
(1 to 10 cm) above the guideway.
Sliding Conducting Bar
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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 resistor R
Sliding Conducting Bar, cont.
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The induced emf is
dB
dx
ε
 B
 B v
dt
dt
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Since the resistance in the circuit is R,
the current is
ε Bv
I 
R
R
Sliding Conducting Bar,
Energy Considerations
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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
ε
  Fappv   I B  v 
R
2
Induced emf and Electric
Fields
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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 an
electric field in empty space
This induced electric field is
nonconservative
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Unlike the electric field produced by stationary
charges
Induced emf and Electric
Fields, cont.
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The emf for any closed path can be
expressed as the line integral of E.ds
over the path
Faraday’s law can be written in a
general form:
 d B
 E .ds  dt
Maxwell’s Equations,
Statement
q
 E .dA  e
s
Gauss’s Law in Electricity
0
 B.dA  0
Gauss’s Law in Magnetism
s
 d B
 E .ds  dt
d e
 B.dl   0 I   0e 0 dt
Faraday’s Law
Ampere-Maxwell Law
Maxwell’s Equations, Details
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q
Gauss’s law (electrical):  E .dA 
e0
s
The total electric flux through any
closed surface equals the net charge
inside that surface divided by eo
This relates an electric field to the
charge distribution that creates it
Maxwell’s Equations, Details 2
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Gauss’s law (magnetism):  B.dA  0
s
The total magnetic flux through
any closed
surface is zero
This says the number of field lines that enter
a closed volume must equal the number that
leave that volume
This implies the magnetic field lines cannot
begin or end at any point
Isolated magnetic monopoles have not been
observed in nature
Maxwell’s Equations, Details 3
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 dB
Faraday’s law of Induction:  E .ds 
dt
This describes the creation of an electric field
by a changing magnetic flux
The law states that the emf, which is the line
integral of the electric field around any closed
path, equals the rate of change of the
magnetic flux through any surface bounded
by that path
One consequence is the current induced in a
conducting loop placed in a time-varying B
Maxwell’s Equations, Details 4
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The Ampere-Maxwell law is a generalization
of Ampere’s law
d e
 B.dl   0 I   0e 0 dt
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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
The Lorentz Force Law
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Once the electric and magnetic fields are
known at some point in space, the force
acting on a particle of charge q can be
calculated
F = qE + qv x B
This relationship is called the Lorentz force
law
Maxwell’s equations, together with this force
law, completely describe all classical
electromagnetic interactions
Maxwell’s Equations,
Symmetry
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The two Gauss’s laws are symmetrical, apart
from the absence of the term for magnetic
monopoles in Gauss’s law for magnetism
Faraday’s law and the Ampere-Maxwell law
are symmetrical in that the line integrals of E
and B around a closed path are related to the
rate of change of the respective fluxes
Maxwell’s equations are of fundamental
importance to all of science