9/10/2016
Electromagnetic Induction
AS 102
Lecture 9-1
IPG 2016
Induced Current
Past experiments with magnetism have shown the
following
When a magnet is moved towards or away from a
circuit, there is an induced current in the circuit
This is still true even if it is the circuit that is moved
towards or away from the magnet
When both are at rest with respect to each, there is no
induced current
What is the physical reason behind this
phenomena
AS 102
Lecture 9-2
IPG 2016
Magnetic Flux
Just as was the case with electric fields, we can define a
magnetic flux
r r
dΦ B = B ⋅ dA
where dA is an incremental area
with the total flux being given by
r r
Φ B = ∫ B ⋅ dA
Note that this integral is not over a closed surface, for that
integral would yield zero for an answer, since there are no
sources or sinks for the magnetic as there is with electric
fields
AS 102
Lecture 9-3
IPG 2016
1
9/10/2016
Faraday’s Law of Induction
It was Michael Faraday who was able to link the induced
current with a changing magnetic flux
He stated that
The induced emf in a closed loop equals the negative
of the time rate of change of the magnetic flux
through the loop
ε =−
dΦ B
dt
The induced emf opposes the change that is occurring
AS 102
Lecture 9-4
IPG 2016
Faraday's law of induction is a basic law of
electromagnetism relating to the operating principles of
transformers, inductors, and many types of electrical
motors and generators.
Electromagnetic induction was discovered independently
by Michael Faraday and Joseph Henry in 1831; however,
Faraday was the first to publish the results of his
experiments.
AS 102
Lecture 9-5
IPG 2016
In Faraday's first experimental demonstration of electromagnetic
induction (August 1831),
he wrapped two wires around opposite sides of an iron torus (an
arrangement similar to a modern transformer). Based on his assessment of
recently-discovered properties of electromagnets, he expected that when
current started to flow in one wire, a sort of wave would travel through
the ring and cause some electrical effect on the opposite side. He plugged
one wire into a galvanometer, and watched it as he connected the other
wire to a battery. Indeed, he saw a transient current (which he called a
"wave of electricity") when he connected the wire to the battery, and
another when he disconnected it.
Within two months, Faraday had found several other manifestations
of electromagnetic induction. For example, he saw transient currents
when he quickly slide a bar magnet in and out of a coil of wires, and
he generated a steady (DC) current by rotating a copper disk near a bar
magnet with a sliding electrical lead ("Faraday's disk").
AS 102
Lecture 9-6
IPG 2016
2
9/10/2016
Faraday explained electromagnetic induction using a concept he called
lines of force.
However, scientists at the time widely rejected his theoretical
ideas, mainly because they were not formulated mathematically. An
exception was Maxwell, who used Faraday's ideas as the basis of his
quantitative electromagnetic theory. In Maxwell's papers, the time
varying aspect of electromagnetic induction is expressed as a
differential equation which Oliver Heaviside referred to as Faraday's
law even though it is slightly different in form from the original version
of Faraday's law, and does not describe motional EMF. Heaviside's
version is the form recognized today in the group of equations known as
Maxwell's equations.
Lenz's law, formulated by Heinrich Lenz in 1834, describes "flux
through the circuit", and gives the direction of the induced electromotive
force and current resulting from electromagnetic induction (elaborated
upon in the examples below).
AS 102
Lecture 9-7
IPG 2016
Increasing Magnetic Flux
Suppose that we are given a closed circuit through which
the magnetic field, flux, is increasing, then according to
Faraday’s Law there will be an induced emf in the loop
The sense of the emf will be so
that the induced current will set
up a magnetic field that will
oppose this increase of external
flux
Binduced
AS 102
Lecture 9-8
IPG 2016
Decreasing Magnetic Flux
Suppose now that the magnetic field is decreasing through the
closed loop, Faraday’s Law again states that there will be an
induced emf in the loop
Binduced
AS 102
The sense of the emf will be so
that the induced current will set
up a magnetic field that will try
to keep the magnetic flux at its
original value (It never can)
Lecture 9-9
IPG 2016
3
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Changing Area
Both of the previous examples were based on a changing
external magnetic flux
How do we treat the problem if the magnetic field is
constant and it is the area that is changing?
Basically the same way, it is the changing flux that will be
opposed
If the area is increasing, then the flux will be increasing and if
the area is decreasing then the flux will also be decreasing
In either case, the induced emf will be such that the change
will be opposed
AS 102
Lecture 9-10
IPG 2016
Faraday’s Experiments
N
S
v
I
Michael Faraday discovered induction in 1831.
Moving the magnet induces a current I.
Reversing the direction reverses the current.
Moving the loop induces a current.
The induced current is set up by an induced EMF.
AS 102
Lecture 9-11
IPG 2016
Faraday’s Experiments
(right)
(left)
dI/dt
EMF
S
I
Changing the current in the right-hand coil induces a current
in the left-hand coil.
The induced current does not depend on the size of the
current in the right-hand coil.
The induced current depends on dI/dt.
AS 102
Lecture 9-12
IPG 2016
4
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Magnetic Flux
A
B
In the easiest case, with a constant magnetic field B, and a
flat surface of area A, the magnetic flux is
ΦB = B . A
Units : 1 tesla x m2 = 1 weber
AS 102
Lecture 9-13
IPG 2016
Magnetic Flux
B
S
N
θ
dA
B
When B is not constant, or the surface is not flat, one must
do an integral.
Break the surface into bits dA. The flux through one bit is
dΦ
ΦB = B . dA = B dA cosθ.
θ.
Add the bits:
r r
.
Φ B = ∫ B ⋅ dA =
AS 102
Lecture 9-14
∫ Bcos θ dA
IPG 2016
Faraday’s Law
N
1)
i
S
2)
di/dt
v
EMF
S
i
Moving the magnet changes the flux Φ B (1).
Changing the current changes the flux Φ B (2).
Faraday: changing the flux induces an emf.
E = - dΦ
Φ B /dt
The emf induced
around a loop
AS 102
Faraday’s law
equals the rate of change
of the flux through that loop
Lecture 9-15
IPG 2016
5
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Lenz’s Law
All of the proceeding can be summarized as follows
The direction of any magnetic induction effect
is such as to oppose the cause of the effect
Remember that the cause of the effect could be either a
changing external magnetic field, or a changing area, or
both
r r r r
dΦ B = dB ⋅ A + B ⋅ dA
AS 102
Lecture 9-16
IPG 2016
Lenz’s Law
Faraday’s law gives the direction of the induced emf and therefore the
direction of any induced current.
Lenz’s law is a simple way to get the directions straight, with less effort.
Lenz’s Law:
The induced emf is directed so that any induced current flow will
oppose the change in magnetic flux (which causes the induced emf).
This is easier to use than to say ...
Decreasing magnetic flux ⇒ emf creates additional magnetic field
Increasing flux ⇒ emf creates opposed magnetic field
AS 102
Lecture 9-17
IPG 2016
Lenz’s Law
B
N
I
B∗
S
v
If we move the magnet towards the loop
the flux of B will increase.
Lenz’s Law ⇒ the current induced in the
loop will generate a field B∗ opposed to B.
AS 102
Lecture 9-18
IPG 2016
6
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Example of Faraday’s Law
Consider a coil of radius 5 cm with N = 250 turns.
A magnetic field B, passing through it,
changes at the rate of dB/dt = 0.6 T/s.
The total resistance of the coil is 8 Ω.
What is the induced current ?
B = 0.6 t [T] (t = time in seconds)
B
Use Lenz’s law to determine the
direction of the induced current.
Apply Faraday’s law to find the
emf and then the current.
AS 102
Lecture 9-19
IPG 2016
Example of Faraday’s Law
B
Lenz’s law: dB/dt > 0
The change in B is increasing the
upward flux through the coil.
I
So the induced current will have
a magnetic field whose flux
(and therefore field) is down.
Induced B
Hence the induced current must be clockwise
when looked at from above.
Use Faraday’s law to get the magnitude of the induced emf and current.
AS 102
Lecture 9-20
IPG 2016
B
ΦB = ∫ B . dA
I
E = - dΦ
ΦB /dt
Induced B
The induced EMF is E = - dΦ
ΦB /dt
In terms of B: ΦB = N(BA) = NB (π
πr2)
2
Therefore E = - N (π
πr ) dB /dt
E = - (250) (π
π 0.0052)(0.6T/s) = -1.18 V (1V=1Tm2 /s)
Current I = E / R = (-1.18V) / (8 Ω) = - 0.147 A
AS 102
Lecture 9-21
IPG 2016
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I a
Magnetic Flux in a Nonuniform Field
l
A long, straight wire carries a current I.
A rectangular loop (w / l) lies at a distance a, as
shown in the figure.
What is the magnetic flux through the loop?.
w
I a
l
w
AS 102
Lecture 9-22
I a
Induced emf Due to Changing Current
l
w
AS 102
IPG 2016
A long, straight wire carries a current I = I0 + i t.
A rectangular loop (w / l) lies at a distance a,
as shown in the figure.
What is the induced emf in the loop?
What is the direction of the induced current and
field?
Lecture 9-23
IPG 2016
Example
Inside the shaded region, there is a
magnetic field into the board.
If the loop is stationary, the Lorentz
force predicts:
×
B
a) A Clockwise Current b) A Counterclockwise Current
c) No Current
The Lorentz force is the combined electric and magnetic forces
that could be acting on a charge particle
r
r r r
F = q( E + v × B)
Since there is no electric field and the charges within the wire
are not moving, the Lorentz force is zero and there is no
charge motion
Therefore there is no current
AS 102
Lecture 9-24
IPG 2016
8
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Example - Continued
Now the loop is pulled to the right at a
velocity v. The Lorentz force will now give
rise to:
×
B
(a) A Clockwise Current
v
I
(b) A Counterclockwise Current
(c) No Current, due to Symmetry
The charges in the
wirerwill now experience an upward
r
r
magnetic force, F = q v × B , causing a current to flow in the
clockwise direction
AS 102
Lecture 9-25
IPG 2016
Another Example
Now move the magnet, not the loop
r r
Here there is no Lorentz force v × B
-v
×
B
I
But according to Lenz’s Law there
will still be an induced current
This “coincidence” bothered Einstein, and eventually led
him to the Special Theory of Relativity (all that matters is
relative motion).
Now keep everything fixed, but decrease the strength of B
Again by Lenz’s Law there will be an
induced current because of the
decreasing flux
×
B
I
Decreasing B
AS 102
Lecture 9-26
IPG 2016
Example
A conducting rectangular loop moves y
XXXXXXXXXXXX
with constant velocity v in the +x
XXXXXXXXXXXX
direction through a region of constant
X X X X X X X vX X X X X
magnetic field B in the -z direction as
XXXXXXXXXXXXx
shown
What is the direction of the induced current in the loop?
(a) ccw
(b) cw
(c) no induced current
There is a non-zero flux ΦB passing through the loop since B is
perpendicular to the area of the loop
Since the velocity of the loop and the magnetic field are
constant, however, the flux through the loop does not change
with time
Therefore, there is no emf induced in the loop
No current will flow!!
AS 102
Lecture 9-27
IPG 2016
9
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Example
I
y
A conducting rectangular loop moves
with constant velocity v in the -y
direction and a constant current I
flows in the +x direction as shown
Iinduced
v
x
What is the direction of the induced current in the loop?
(a) ccw
(b) cw
(c) no induced current
The flux through this loop does change in time since the loop is
moving from a region of higher magnetic field to a region of
lower magnetic field
Therefore, by Lenz’s Law, an emf will be induced which will
oppose the change in flux
Current is induced in the clockwise direction to restore the flux
AS 102
Lecture 9-28
IPG 2016
(Important) Example
Suppose we pull with velocity v a coil
of resistance R through a region of
constant magnetic field B.
a) What is the direction of the induced
current
By Lenz’s Law the induced current will be
clockwise in order to try and maintain the
original flux through the loop
xxxxxx
F
I
xxxxxx
xxxxxx
w
v
xxxxxx
x
b) What is the magnitude of the induced current
Magnetic Flux:
Φ B = B Area = B w x
ε =−
Faraday’s Law:
Then
I=
ε
R
=
dΦ B
dx
= −B w
= −B w v
dt
dt
Bwv
R
c) What is the direction of the force on the loop?
r
r r
Using F = I L × B , the force is to the left, trying to pull the loop back in!
AS 102
Lecture 9-29
IPG 2016
Motional EMF
In previous discussions we had mentioned that a charge
moving in a magnetic field experiences a force
Suppose now that we have a conducting
rod moving in a magnetic field as shown
The positive charges will experience a
magnetic force upwards, while the
negative charges will experience a
magnetic force downwards
Charges will continue to move until the magnetic force is
balanced by an opposing electric force due to the charges that
have already moved
AS 102
Lecture 9-30
IPG 2016
10
9/10/2016
Motional EMF
So we then have that
E = vB
But we also have that E = Vab / L
Vab = v B L
So
We have an induced potential difference
across the ends of the rod
If this rod were part of a
circuit, we then would
have an induced current
The sense of the induced emf can be gotten from Lenz’s Law
AS 102
Lecture 9-31
IPG 2016
Induced Electric Fields
We again look at the closed
loop through which the
magnetic flux is changing
We now know that there is an
induced current in the loop
But what is the force that is causing the charges to move in
the loop?
It can’t be the magnetic field, as the loop is not moving
AS 102
Lecture 9-32
IPG 2016
Induced Electric Fields
The only other thing that could make the charges move
would be an electric field that is induced in the conductor
This type of electric field is different from what we have dealt
with before
Previously our electric fields were due to charges and these
electric fields were conservative
Now we have an electric field that is due to a changing
magnetic flux and this electric field is non-conservative
AS 102
Lecture 9-33
IPG 2016
11
9/10/2016
Induced Electric Fields
Remember that for conservative forces, the work done in
going around a complete loop is zero
Here there is a net work done in going around the loop that is
qε
given by
But the total work done in going around the loop is also given
by
r
r
q ∫ E ⋅ dl
Equating these two we then have
AS 102
r
r
∫ E ⋅ dl = ε
Lecture 9-34
IPG 2016
Induced Electric Fields
But previously we found that the emf was related to the
negative of the time rate of change of the magnetic flux
ε =−
dΦ B
dt
So we then have finally that
r
r
∫ E ⋅ dl = −
dΦ B
dt
This is just another way of stating Faraday’s Law but for
stationary paths
AS 102
Lecture 9-35
IPG 2016
Electro-Motive Force
time
A magnetic field, increasing in time, passes through the loop
An electric field is generated “ringing” the increasing magnetic
field
The circulating E-field will drive currents, just like a voltage
difference
r
Loop integral of E-field is the “emf”:
r
∫ E ⋅ dl = ε
The loop does not have to be a wire - the emf exists even in
vacuum!
When we put a wire there, the electrons respond to the emf,
producing a current
AS 102
Lecture 9-36
IPG 2016
12
9/10/2016
Displacement Current
We have used Ampere’s Law to calculate the magnetic field
due to currents
r
r
∫ B ⋅ dl = µ0 I enclosed
where Ienclosed is the current that cuts through the area
bounded by the integration path
But in this formulation, Ampere’s Law is incomplete
AS 102
Lecture 9-37
IPG 2016
Displacement Current
Suppose we have a parallel plate capacitor being
charged by a current Ic
We apply Ampere’s Law to path
that is shown
For this path, the integral is just
µ0 I enclosed
For the plane surface which is
bounded by the path, this is just I c
But for the bulging surface which is also bounded by the
integration path, Ienclosed is zero
So how do we reconcile these two results?
AS 102
Lecture 9-38
IPG 2016
Displacement Current
But we know we have a magnetic field
Since there is charge on the
plates of the capacitor, there is an
electric field in the region
between the plates
The charge on the capacitor is related to the electric field by
q = ε E A = ε ΦE
We can define a “current” in this region, the displacement
current, by
dq
dΦ
I Displacement =
AS 102
dt
=ε
Lecture 9-39
E
dt
IPG 2016
13
9/10/2016
Displacement Current
We can now rewrite Ampere’s Law including this
displacement current
r
r

∫ B ⋅ dl = µ0  Ic + ε 0
AS 102
dΦ E 

dt 
Lecture 9-40
IPG 2016
Maxwell’s Equations
We now gather all of the governing equations together
Collectively these are known as Maxwell’s Equations
AS 102
Lecture 9-41
IPG 2016
Maxwell’s Equations
These four equations describe all of classical electric and
magnetic phenomena
Faraday’s Law links a changing magnetic field with an
induced electric field
Ampere’s Law links a changing electric field with an induced
magnetic field
Further manipulation of Faraday’s and Ampere’s Laws
eventually yield a second order differential equation which is
the wave equation with a prediction for the wave speed of
c=
AS 102
1
µ0 ε 0
= 3 × 108 m / sec = Speed of Light
Lecture 9-42
IPG 2016
14