Michael Faraday
Chapter 31
Faraday’s Law
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Great experimental physicist and chemist
1791 – 1867
Contributions to early electricity include:
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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
Invention of motor, generator, and transformer
Electromagnetic induction
Electrolysis: A method of separating bonded
elements and compounds by passing an
electric current through them
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
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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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The magnet is moved
away from the loop
The ammeter deflects in
the opposite direction
Even though the
magnet is in the loop
Faraday’s Law – Statements
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EMF Produced by a Changing
Magnetic Field, 3
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,
e= −
dΦB
dt
Faraday’s Law – Statements,
cont
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Remember ΦB is the magnetic flux
through the circuit and is found by
Φ B = ∫ B ⋅ dA
If the circuit consists of N loops, all of
the same area, and if ΦB is the flux
through one loop, an emf is induced in
every loop and Faraday’s law becomes
e= −N
dΦB
dt
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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
Φ B = BA cos θ
The induced emf is
ε = - d/d t (BA cos θ)
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
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 θ between B and the normal to
the loop can change with time
Any combination of the above can occur
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
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Problem 2
Problem 1
A wire loop is moving to the right as shown in the figure. What is the
direction of the induced magnetic field and current as the loop passes
through an external magnetic field pointing inward. How can we
calculate the induced emf ? B(induced) is +k. I (induced) is counterclockwise.
A 30 turn circular coil of radius 0.040m and resistance 1.00 Ω 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 + 0.0400t 2, where t is in
seconds and B is in Tesla. Calculate the induced emf in the coil
at t = 5.00 s. [61.8 mV]
Applications of Faraday’s Law
– Electric Generator
Generators
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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
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http://www.wvic.com/how-gen-works.htm
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
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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?
Induced emf and Electric
Fields
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 .
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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
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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:
∫ E .ds =
− dΦ B
dt
Maxwell’s Equations, Details
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q
Gauss’s law (electrical): ∫ E .dA =
ε
s
The total electric flux through any0
closed surface equals the net charge
inside that surface divided by εo
This relates an electric field to the
charge distribution that creates it
Maxwell’s Equations
∫ E .dA =
s
q
ε0
Gauss’s Law in Electricity
∫ B .dA = 0
s
Gauss’s Law in Magnetism
− dΦ B
∫ E .ds = dt
Faraday ’s Law
dΦ e
∫ B .dl = µ0 I + µ0ε 0 dt
Ampere-Maxwell Law
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
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Maxwell’s Equations, Details 3
Maxwell’s Equations, Details 4
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− d φB
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
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
The Ampere-Maxwell law is a generalization
of Ampere’s law
∫ B.ds = µ I + µ e
0
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0 0
dΦ e
dt
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
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
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