chapter 220
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Electromagnetic Induction
Chapter 22
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Expectations
After this chapter, students will:
Calculate the EMF resulting from the motion of
conductors in a magnetic field
Understand the concept of magnetic flux, and
calculate the value of a magnetic flux
Understand and apply Faradays Law of
electromagnetic induction
Understand and apply Lenzs Law
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Expectations
After this chapter, students will:
Apply Faradays and Lenzs Laws to some
particular devices:
Electric generators
Electrical transformers
Calculate the mutual inductance of two
conducting coils
Calculate the self-inductance of a conducting coil
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Motional EMF
A wire passes through
a uniform magnetic
field. The length of
the wire, themagnetic field, and
the velocity of the
wire are allperpendicular to
one another:
L
v
B
+
-
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Motional EMF
A positive charge in
the wire
experiences a
magnetic force,directed upward: L
v
B
+
-
qvBqvBFm 90sin
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Motional EMF
A negative charge in the
wire experiences the
same magnetic force,
but directeddownward:
These forces tend toseparate the charges.
L
v
B
+
-
qvBFm
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Motional EMF
The separation of the
charges produces an
electric field,E. It
exerts an attractiveforce on the charges: L
v
B
+
-
EqFC E
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Motional EMF
In the steady state (at
equilibrium), the
magnitudes of the
magnetic forceseparating the
chargesand the
Coulomb force
attracting themareequal.
L
v
B
+
-EqqvB
E
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Motional EMF
Rewrite the electric
field as a potential
gradient:
Substitute this result
back into our earlierequation:
L
v
B
+
-
L
EMF
L
VE
E
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Motional EMF
Substitute this result
back into our earlierequation: L
v
B
+
-
L
EMF
L
VE
E
vLBEMF
qvBqL
EMF
qvBEq
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Motional EMF
This is calledmotional
EMF. It results fromthe constant velocity
of the wire through
the magnetic field,B.
L
v
B
+
-
E
vLBEMF
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Motional EMF
Now, our moving wire slides over two other wires,
forming a circuit. A current will flow, and power
is dissipated in the resistive load:
L
v
B
+
-
R
I
R
vBLP
R
vBLvBLVIP
R
vBL
R
VI
vBLVEMF
2
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Motional EMF
Where does this power come from?
Consider the magnetic
force acting on the
current in the sliding
wire:L
v
B
+
-
R
I
R
LBvF
LB
R
vBLILBF
2
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Motional EMF
Right-hand rule #1 shows that this force opposes themotion of the wire. To move the wire at constant
velocity requires an equal and opposite force.
That force does work:
The power:
L
v
B
+
-
R
I
FvtFxW
Fvt
Fvt
t
WP
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Motional EMF
The forces magnitude was calculated as:
Substituting:
which is the same as the
power dissipated electrically.
L
v
B
+
-
R
I
R
vBLv
R
BLvFvP
22
R
BLvF
2
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Motional EMF
Suppose that, instead of being perpendicular to theplane of the sliding-wire circuit, the magnetic field
had made an angle fwith the perpendicular to that
plane.
The perpendicular
component ofB:B cos f
BB cos f
f
v
x
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Motional EMF
The motional EMF:
Rewrite the velocity:
Substitute:
BB cos f
f
v
x
fcosvLBEMF
t
xv
f
f
cos
cos
LBt
xEMF
vLBEMF
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Motional EMF
L x is simply the change in the loop area.
x
x
L
A = L x
t
ABEMF
LxA
B
t
LxEMF
f
f
cos
cos
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Motional EMF
Define a quantityF:
Then:
F is calledmagnetic
flux.
SI units: Tm2 = Wb (Weber) x
x
L
A = L xtt
ABEMF
F
fcos
fcosABF
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Magnetic Flux
Wilhelm Eduard Weber18041891
German physicist and
mathematician
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Faradays Law
In our previous result, we said that the induced EMFwas equal to the time rate of change of magnetic
flux through a conducting loop. This, rewritten
slightly, is called Faradays Law:
Why the minus sign?
tEMF
F
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Faradays Law
Michael Faraday
17911867
English physicist
and mathematician
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Faradays Law
To make Faradays Law complete, consider addingNconducting loops (a coil):
What can change the magnetic flux?
B could change, in magnitude or direction
A could change
fcould change (the coil could rotate)
t
NEMF
F
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Lenzs Law
Here is where we get the minus sign in FaradaysLaw:
Lenzs Law says that the direction of the induced
current is always such as to oppose the change in
magnetic flux that produced it.
The minus sign in Faradays Law reminds us of that.
tNEMF
F
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Lenzs Law
Heinrich Friedrich Emil Lenz
18041865
Russian physicist
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Lenzs Law
Lenzs Law says that the direction of the inducedcurrent is always such as to oppose the change in
magnetic flux that produced it.
What does that mean?
How can an induced current oppose a change in
magnetic flux?
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Lenzs Law
How can an induced current oppose a change in
magnetic flux?
A changing magnetic flux induces a current.
The induced current produces a magnetic field.
The direction of the induced current determines
the direction of the magnetic field it produces.
The current will flow in the direction (remember
right-hand rule #2) that produces a magnetic field
that works against the original change in magnetic
flux.
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Faradays Law: the Generator
A coil rotates with a constant angular speed in amagnetic field.
but fchanges
with time:
tNEMF
F
fcosABF
tf
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Faradays Law: the Generator
So the flux also changes with time:
Get the time rate of change (a calculus problem):
Substitute into Faradays Law:
tABAB f coscos F
tABt
sinF
tNABt
NEMF sinF
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Faradays Law: the Generator
The maximum voltage occurs when :
What makes the voltage larger? more turns in the coil
a larger coil area
a stronger magnetic field a larger angular speed
NABEMF max2
n
t
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Back EMF in Electric Motors
Apply Kirchhoffs loop rule:
REMFVIEMFIRV BB 0
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Mutual Inductance
A current in a coil produces a magnetic field.
If the current changes, the magnetic field changes.
Suppose another coil is nearby. Part of the magnetic
field produced by the first coil occupies the inside
of the second coil.
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Mutual Inductance
Faradays Law says that the changing magnetic fluxin the second coil produces a voltage in that coil.
The net flux in thesecondary:
PSS IN F
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Mutual Inductance
Convert to an equation, using a constant ofproportionality:
PSS
PSS
MIN
IN
F
F
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Mutual Inductance
The constant of proportionality is called themutualinductance:
P
SS
PSS
I
NM
MIN
F
F
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Mutual Inductance
Substitute this into Faradays Law:
SI units of mutual inductance: Vs / A = henry (H)
t
IM
t
MI
t
N
tNEMF PPSSSSS
F
F
PSS MIN F
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Mutual Inductance
Joseph Henry
17971878
American physicist
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Self-Inductance
Changing current in a primary coil induces a voltagein a secondary coil.
Changing current in a coil also induces a voltage inthat same coil.
This is calledself-inductance.
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Self-Inductance
The self-induced voltage is calculated fromFaradays Law, just as we did the mutual
inductance.
The result:
The self-inductance,L, of a coil is also measured inhenries. It is usually just called the inductance.
t
ILEMFself
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Mutual Inductance: Transformers
The self-induced voltage in the primary is:
Through mutual induction, and EMF appears in thesecondary:
Their ratio:
tNEMF PP
F
tNEMF SS
F
P
S
P
S
P
S
N
N
tN
tN
EMF
EMF
F
F
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Inductors and Stored Energy
When current flows in an inductor, work has beendone to create the magnetic field in the coil. As
long as the current flows, energy is stored in that
field, according to
2
2
1LIE
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Inductors and Stored Energy
In general, a volume in which a magnetic field existshas an energy density (energy per unit volume)
stored in the field:
0
2
2volume
energydensityenergy
B
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