Chapter 30
Magnetic Induction
Magnetic Induction
• Magnetic Flux
• Induced EMF & Faraday’s Law
• Lenz’s Law
• Motional EMF
• Generators & Motors
• Eddy Currents
• Inductance
• Magnetic Energy
• RL Circuits
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Magnetic Induction
A changing magnetic field that causes the magnetic flux to
change through a surface, bounded by a closed stationary
loop of wire, induces a current in that wire.
Since there is a flow of current there must be a source of
this current and it is attributed to an induced emf which
causes the induced current.
The process is called induction
If the flux stops changing then the induced emf goes to zero.
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Magnetic Induction
The induced emf is distributed over space and is the
primary effect of the changing magnetic flux. It doesn’t
need a circuit. The induced emf will be there even if there
is no wire.
The purpose of the wire coil is to allow us to measure the
effect of the induced emf and demonstrate its presence.
Most important thing to remember
The induced emf tries to maintain the existing magnetic flux
and opposes any changes in the flux.
No changes in magnetic flux ==> No induced emf
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Magnetic Flux
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Magnetic Flux
ˆ = ∫ Bn dA
φm = ∫ B ⋅ ndA
S
S
The units of magnetic flux are
Webers (W).
1 Wb = 1 Tm2
B is proportional to the number of
field lines per unit area.
B can be thought of as a flux
density in Webers/m2
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Magnetic Flux Example
Assume a flat surface area S
ˆ = BAcosθ = Bn A
φm = B ⋅ nA
φm = NBAcosθ
In the simplest case
MFMcGraw-PHY 2426
φm = NBA with nˆ B
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Induced Emf and Faraday’s Law
Stationary Wire Coil - Moving Magnet
Emf = ∫ E NC ⋅ dL
C
The induced emf is the result of the nonconservative
electric field, Enc. For a conservative electric field this
integral would be zero.
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The induced emf is such
that it produces an induced
current whose magnetic
flux opposes the change that
caused the emf.
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Induced Emf and Faraday’s Law
d ˆ
Emf = - ∫ B ⋅ ndA
dt S
Faraday’s Law
d d
ˆ = - φm
Emf = ∫ Enc idL = - ∫ B indA
dt S
dt
C
This is where we finally have to get the direction of C
correlated with the direction of unit vector normal to S.
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Faraday’s Law
Farther away
The right hand rule determines the
direction of the normal to S based
on the direction around C.
Closer
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Induced Nonconservative E-Field
Calculate the magnitude and
direction of the ENC field both
inside and outside of the ring.
Uniform magnetic field
directed into the page
everywhere inside r < R.
Conductive ring
of radius R
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Induced ENC-Field - Problem Setup
Contour C is traversed in a clockwise
direction. Whatever direction we pick
will be referred to as the positive
tangential direction.
This means that the corresponding
normal to the surface S, enclosed by C,
points into the page.
Our goal is to align this normal of S with
the direction of B.
At this point we haven’t totally determined the problem to
be solved so we do not know the final direction of ENC.
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Induced ENC-Field - Problem Setup
The direction of B will remain fixed and
its value will remain zero outside the
wire.
We will change the flux by letting the
magnitude of B change with time.
Therefore dB/dt is not zero
ε
dφm
= ∫ E NC ⋅ dl = C
dt
C
E NC ⋅ dl = E NC ∫ dl = E NC ( 2πr )
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∫
C
14
Induced ENC-Field
ε
∫
dφm
= ∫ E NC ⋅ dl = C
dt
E NC ⋅ dl = E NC ( 2πr )
C
φm = ∫
S
B⋅ˆ
ndA = ∫ Bn dA = Bn ∫ dA = Bπr 2
dφm
∫C ENC ⋅ dl = - dt
MFMcGraw-PHY 2426
S
S
d
E NC ( 2πr ) = Bπr 2
dt
1
1 dB
2 dB
E NC = πr
= r
2πr
dt
2 dt
(
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Induced ENC-Field
E NC
1 dB
=r
2 dt
The positive direction on C was
chosen to be clockwise. Therefore the
minus sign on ENC indicates its points
in a counter clockwise direction.
Trace the result intuitively.
B increases into the page.
This induces a B field out of the page.
This induced field requires a counter clockwise current
The current is driven by a counterclockwise pointing ENC.
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Induced ENC-Field for r > R
dφm
= ∫ E NC ⋅ dl = C
dt
∫ E NC ⋅ dl = ENC ( 2πr )
Same
as
before
φm = ∫ Bn dA = Bn ∫ dA = BπR 2
No B-field
past r = R
ε
C
S
dφm
∫C ENC ⋅ dl = - dt
MFMcGraw-PHY 2426
S
d
E NC ( 2πr ) = BπR 2
dt
2
1
dB
R
dB
2
E NC = πR
=2πr
dt
2r dt
(
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Induced ENC-Field for r > R
E NC
R 2 dB
=2r dt
The positive direction on C was
chosen to be clockwise. Therefore the
minus sign on ENC indicates its points
in a counter clockwise direction.
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Lenz’s Law
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Lenz’s Law
“The induced emf is in such a direction as to oppose, or
tend to oppose, the change that produces it.”
Alternatively: “When the magnetic flux through a surface
changes, the magnetic field due to any induced current
produces a flux of its own through the same surface and
opposite in sign to the initial change in flux.
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Lenz’s Law
It is as if the induced emf is producing a virtual bar magnet
to oppose the motion of the bar magnet that is causing the
inital changes.
Motion of the magnet increases the flux through the loop pointing to the right.
The induced current creates a B-field pointing to the left, i.e. a North pole, to
decrease the increased flux.
It is as if the original bar magnet sees an “image” of itself opposing it.
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The Same Motion in More Detail
The B-field increases to the right
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The Same Motion in More Detail
The ring opposes the
increasing field with a
field to the left (B2 ).
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The Same Motion in More Detail
The field B2 requires a
counter-clockwise
current (I).
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The Motion is Relative
Motion of the loop decreases the flux through the loop pointing
to the right.
The induced current creates a B-field pointing to the right, i.e. a
South pole, to increase the previously decreased flux.
It is as if the original bar magnet sees an “inverted image” of
itself replacing the decreased flux.
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Coupled Induced Current
Switch Open
The piece of iron, around which the wires are wrapped, enhances
the coupling of the magnetic flux between the two circuits.
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Coupled Induced Current
Switch Closed
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Switch Open
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Motional EMF
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Flip Coil Magnetic Field Measurement
Current Integrator
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Flip Coil Magnetic Field Measurement
dφ
ε = - dt
εdt = -dφ
m
m
dQ
IRdt = R
dt = -dφm
dt
R ∫ dQ = - ∫ dφm = -(φmf - φmi )
N ˆ f - B ⋅ nˆi A
Q=B⋅n
R
N
2NBA
Q = - ( B(-1) - B(1)) A =
R
R
QR
B=
2NA
(
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Induced Emf - Changing Flux Area
Surface area for calculating magnetic flux. Unit normal vector parallel to
field. Therefore our contour C’s positive direction is clockwise
Frictionless, conductive rod
Stationary
conductive rails
Uniform magnetic field directed
into the page
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Induced Emf - Changing Flux Area
As the bar moves to the
right the area increases.
The flux crossing the
surface S increases,
pointing into the page.
A current is induced in the blue loop such that it produces a magnetic
field out of the page to oppose the flux increase. This current is counter
clockwise.
φm = Blx ;
ε
MFMcGraw-PHY 2426
dφm
dx
= Bl
= Blv
dt
dt
dφm
== -Blv
dt
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Motional Emf
Although the bar appears after the example
of the moving rod on the rails this rod is all
by itself.
The conductive bar is originally neutral.
The conventional positive charges in the
conductor are moving through a uniform
magnetic field.
They experience an upward Lorentz force
due to the bar’s motion through the
magnetic field.
Since there is not a complete circuit the
positive charges pile up at the top and the
bottom is effectively negative.
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Motional Emf
An electric field builds up parallel to the rod
due to the separated charge called E
This field increases until it balances the
Lorentz force on the charges
qE = qvB
E = vB
∆V = El = vBl
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Magnetic Drag
ε
Blv
I= =
R
R
Blv
B 2l 2 v
dv
F = -IlB = lB = =m
R
R
dt
MFMcGraw-PHY 2426
dv
B 2l 2
=dt
v
mR
v f dv
B 2l 2 t f
∫vo v = - mR ∫0 dt
vf
B 2l 2
ln = tf
vo
mR
mR
-tτ
v(t) = vo e ; τ = 2 2
Bl
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Generators and Motors
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Generators
Motion
supplied
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Voltage output
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DC
AC
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Motors
Motion
Output
Voltage Input
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Eddy Currents
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Eddy Currents
The changing flux inside the conductor induces an emf
which gives rise to an induced current that flows in the
direction shown below. This current flow resembles the
eddies seen in the water and hence the name
Leaving the field region, the flux is decreasing therefore the
induced flux will oppose it and will be into the page.
Sometimes
these eddy
currents are
desired and
sometimes they
are not wanted.
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Dissipating Eddy Currents
If the eddy currents are not wanted they can be reduced by
breaking up the pathway of the current flow as shown
below.
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Eddy Currents - Damping Force
Sometimes the eddy
currents are wanted!
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Electromagnetic Brake
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Eddy Current Demo from Carleton Physics
http://www.youtube.com/watch?v=OJvEOXsSuaQ
http://www.hscphysics.edu.au/resource/EddyCurrents.flv
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Self Inductance
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Self Inductance
A current carrying wire, wrapped into the shape of a coil is
subject to self inductance. The magnetic field lines from one
part of the coil pass through other coils of the same wire.
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Self Inductance of a Long Solenoid
Although the geometry might be
difficult since B is proportional
to I then the flux through the coil
should also be proportional to I
Units: 1 Henry (H) = 1 Wb/A= 1Tm2/A
φm = LI
(
)
φm = NBA = N µ0 nI A =
µo N 2 IA
l
=
µo N 2 IA
l
2
l
φm = µo n 2 IAl
φm
L=
= µ on 2 Al
I
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Factors only depend on geometry
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Mutual Inductance
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Mutual Inductance
When circuits are physically close enough and/or the
currents are large enough the magnetic flux from one circuit
may pass through or couple into the second circuit. This is
referred to as mutual inductance.
Sometimes it is desirable and other times it is NOT.
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Mutual Inductance
We can write the flux of circuit 1 through circuit 2 as φm12
φm12 = M 12 I 1
We can write the flux of circuit 2 through circuit 1 as φm21
φm21 = M 21 I 2
M12 and M21 are the mutual inductances between the two circuits.
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Mutual Inductance
φm12 = M 12 I 1
φm21 = M 21 I 2
The mutual inductances M12 and M21 depend on the geometrical
arrangement of the circuits. For the mutual inductance to be
useful the circuit geometry must stay fixed.
For arbitrary circuits it is difficult to calculate the mutual
inductance. Our old friend the solenoid is one of the easier cases.
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Mutual Inductance
Calculating the mutual inductance is really an exercise in
calculating the flux
φ =M I
m12
( )
12 1
( )
( )
φm12 = N 2 B1 πr12 = n2 lB1 πr12 = n2l ( µo n1 I 1 ) πr12
φm12 = µo n1n2lπr12 I 1
φm12
M 12 =
= µ on1n 2lπr12
I1
It can be shown that
φm21
M 21 =
= µ on 2n1 lπr12
I2
MFMcGraw-PHY 2426
Therefore
M 21 = M 21 = M
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Tesla Coil: Low Voltage to High Voltage
Outer coil has
few loosely
spaced loops.
The high voltage from
the inner coil breaks
down the Argon gas in
the light bulb and
causes the streaming
arcs of current.
The inner coil
has many
closely spaced
loops.
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Commercial Handheld Tesla Coil
Tesla coils turn low voltage into relatively high voltage
Technical Details
• High-frequency transformer can step
up ordinary 110-volt electricity to
thousands of volts
• Creates impressive 3-inch sparks
• Adjustable from 10,000 to 50,000
volts
• Has momentary on/off switch in its
Bakelite handle
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Simple Tesla Coil Circuits
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DIY DC Handheld Tesla Coil
http://www.rmcybernetics.com/projects/DIY_Devices/plasma-gun.htm
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Handheld Tesla Coil
http://www.rmcybernetics.com/projects/DIY_Devices/plasma-gun.htm
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Extreme DIY Tesla Coil
http://tesladownunder.com/
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Extreme DIY Tesla Coil Circuit
http://tesladownunder.com/
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Magnetic Energy
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Stored Energy
A capacitor stores electrical energy in its electric field
An inductor stores magnetic energy in its magnetic field.
These components are referred to as reactive components
because they store energy instead of dissipating it like a
resistor.
Inductors and capacitors tend to slow down the speed
of a circuit because it takes time to build up their fields
and to let their fields decay back to zero.
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Magnetic Energy
The voltage on L is proportional to the derivative of the
current with time.
dI
VL = L
dt
A simple RL series circuit
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Magnetic Energy
dI
Vi = Vo - IR - L = 0
∑
dt
Loop
Use Kirchhoff’s Loop Rule on
this circuit.
dI
dt
Now multiply each term by I to turn
each term into a power.
Vo = IR + L
Power stored in L
dI
Vo I = I R + LI
dt
2
Power
from
battery
MFMcGraw-PHY 2426
Power
dissipated in R
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Magnetic Energy Stored in L
dI
Vo I = I R + LI
dt
2
Now power is the time rate of
change of energy
Magnetic Energy in
Inductor L
MFMcGraw-PHY 2426
dU m
dI
d 1
= LI
=  LI 2 
dt
dt dt  2

1 2
U m = LI
2
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Magnetic Energy Stored in a Solenoid
Bz = µ 0nI
L = µ 0n 2 AL
1 2
U m = LI
2
Solenoid
Magnetic Energy
Magnetic
Energy Density
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RL Circuits
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RL Circuits
When S is in position e the
current in R is building up
to steady state.
When S is in position f the
current in R is decaying to
zero.
Make before break switch
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Switches
Break-before-make
Open circuit
MFMcGraw-PHY 2426
Make-before-break
Closed circuit
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Transient Current in RL Circuit
If =
ε
o
R1 + R
The presence of the inductor slows the circuit
down. It takes a time equal to about 3-4 time
constants to achieve full steady state current after
the switch is moved to position e
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Transient Current in RL Circuit
I(t) = I o e
-tτ
The presence of the inductor slows the circuit down. It takes a
time equal to about 3-4 time constants for the current to go
back to zero after the circuit is moved to position f.
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Initial Currents & Final Currents
We want to examine the operation of this LR circuit at very
short times - transient response and at very long times steady state response.
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Transient Response
Note: This is not a
series circuit as in
the previous
example.
Rule 1: The presence of the inductor prevents the current
from changing abruptly.
Therefore since the current is zero just before the switch is
closed then it is also zero right after the switch is closed.
In the transient response it is as if the L isn’t there I3 = 0
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Transient Response
V = IR1 + IR2
V
I=
R1 + R2
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Steady State Response
When the current reaches its constant steady state level di/dt is
zero and the voltage across L is zero. This effectively reduces
L to a wire and that shorts out R2.
V = IR1
V
I=
R1
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Extra Slides
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