Electromagnetic Induction
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Transcript Electromagnetic Induction
Electromagnetic Induction
Chapter 22
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 Faraday’s Law of
electromagnetic induction
Understand and apply Lenz’s Law
Expectations
After this chapter, students will:
Apply Faraday’s and Lenz’s 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
Motional EMF
A wire passes through
a uniform magnetic
field. The length of
the wire, the
magnetic field, and
the velocity of the
wire are all
perpendicular to
one another:
+
v
L
B
-
Motional EMF
A positive charge in
the wire
experiences a
magnetic force,
directed upward:
+
v
L
B
Fm qvBsin 90 qvB
-
Motional EMF
A negative charge in the
wire experiences the
same magnetic force,
but directed
downward:
+
v
L
Fm qvB
B
These forces tend to
separate the charges.
-
Motional EMF
The separation of the
charges produces an
electric field, E. It
exerts an attractive
force on the charges:
+
v
L
FC Eq
E
-
B
Motional EMF
In the steady state (at
equilibrium), the
magnitudes of the
magnetic force –
separating the
charges – and the
Coulomb force –
attracting them – are
equal.
qvB Eq
+
v
L
E
-
B
Motional EMF
Rewrite the electric
field as a potential
gradient:
V EMF
E
L
L
+
v
L
Substitute this result
back into our earlier
equation:
E
-
B
Motional EMF
V EMF
E
L
L
Substitute this result
back into our earlier
equation:
Eq qvB
EMF
q qvB
L
EMF vLB
+
v
L
E
-
B
Motional EMF
EMF vLB
This is called motional
EMF. It results from
the constant velocity
of the wire through
the magnetic field, B.
+
v
L
E
-
B
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:
EMF V vBL
V vBL
I
R
R
vBL
P VI vBL
R
I
+
v
R
L
vBL
P
B
2
R
-
Motional EMF
Where does this power come from?
Consider the magnetic
force acting on the
current in the sliding
wire:
vBL
F ILB
LB
R
2
LB
F v
R
I
+
v
R
L
B
-
Motional EMF
Right-hand rule #1 shows that this force opposes the
motion of the wire. To move the wire at constant
velocity requires an equal and opposite force.
I
+
That force does work:
W Fx Fvt
v
R
L
The power:
W Fvt
P
Fv
t
t
B
-
Motional EMF
The force’s magnitude was calculated as:
BL
F v
I
2
+
R
Substituting:
v
vBL
vBL
P Fv
v
R
R
2
2
which is the same as the
power dissipated electrically.
R
L
B
-
Motional EMF
Suppose that, instead of being perpendicular to the
plane of the sliding-wire circuit, the magnetic field
had made an angle f with the perpendicular to that
plane.
v
The perpendicular
component of B: B cos f
x
f
B
B cos f
Motional EMF
The motional EMF:
EMF vLB cos f
x
Rewrite the velocity: v
t
v
Substitute:
EMF vLB cos f
x
EMF
LB cos f
t
x
f
B
B cos f
Motional EMF
L x is simply the change in the loop area.
x L
EMF
B cos f
t
A = L x
L
A x L
B A cos f
EMF
t
x
x
Motional EMF
Define a quantity F :
F AB cos f
Then:
B A cos f F
EMF
t
t
A = L x
L
F is called magnetic
flux.
SI units: T·m2 = Wb (Weber)
x
x
Magnetic Flux
Wilhelm Eduard Weber
1804 – 1891
German physicist and
mathematician
Faraday’s Law
In our previous result, we said that the induced EMF
was equal to the time rate of change of magnetic
flux through a conducting loop. This, rewritten
slightly, is called Faraday’s Law:
F
EMF
t
Why the minus sign?
Faraday’s Law
Michael Faraday
1791 – 1867
English physicist
and mathematician
Faraday’s Law
To make Faraday’s Law complete, consider adding
N conducting loops (a coil):
F
EMF N
t
What can change the magnetic flux?
B could change, in magnitude or direction
A could change
f could change (the coil could rotate)
Lenz’s Law
Here is where we get the minus sign in Faraday’s
Law:
F
EMF N
t
Lenz’s 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 Faraday’s Law reminds us of that.
Lenz’s Law
Heinrich Friedrich Emil Lenz
1804 – 1865
Russian physicist
Lenz’s Law
Lenz’s Law says that the direction of the induced
current 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?
Lenz’s 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.
Faraday’s Law: the Generator
A coil rotates with a constant angular speed in a
magnetic field.
F
EMF N
t
F AB cos f
but f changes
with time:
f t
Faraday’s Law: the Generator
So the flux also changes with time:
F AB cosf AB cost
Get the time rate of change (a calculus problem):
F
AB sin t
t
Substitute into Faraday’s Law:
F
EMF N
NAB sin t
t
Faraday’s Law: the Generator
The maximum voltage occurs when
EMFmax NAB
What makes the voltage larger?
more turns in the coil
a larger coil area
a stronger magnetic field
a larger angular speed
n
t
2
:
Back EMF in Electric Motors
An electric motor also contains a coil rotating in a
magnetic field.
In accordance with Lenz’s Law, it generates a
voltage, called the back EMF, that acts to oppose
its motion.
Back EMF in Electric Motors
Apply Kirchhoff’s loop rule:
V EMFB
V IR EMFB 0 I
R
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.
Mutual Inductance
Faraday’s Law says that the changing magnetic flux
in the second coil produces a voltage in that coil.
The net flux in the
secondary:
NS FS I P
Mutual Inductance
Convert to an equation, using a constant of
proportionality:
NSFS IP
N S F S MI P
Mutual Inductance
The constant of proportionality is called the mutual
inductance:
N S F S MI P
NSFS
M
IP
Mutual Inductance
N S F S MI P
Substitute this into Faraday’s Law:
F S N S F S MI P
I P
EMFS N S
M
t
t
t
t
SI units of mutual inductance: V·s / A = henry (H)
Mutual Inductance
Joseph Henry
1797 – 1878
American physicist
Self-Inductance
Changing current in a primary coil induces a voltage
in a secondary coil.
Changing current in a coil also induces a voltage in
that same coil.
This is called self-inductance.
Self-Inductance
The self-induced voltage is calculated from
Faraday’s Law, just as we did the mutual
inductance.
The result:
EMFself
I
L
t
The self-inductance, L, of a coil is also measured in
henries. It is usually just called the inductance.
Mutual Inductance: Transformers
A transformer is two coils wound around a common
iron core.
Mutual Inductance: Transformers
The self-induced voltage in the primary is:
F
EMFP N P
t
Through mutual induction, and EMF appears in the
secondary:
F
EMFS N S
Their ratio:
t
F
NS
EMFS
t N S
EMFP N F N P
P
t
Mutual Inductance: Transformers
This transformer equation is normally written:
VS N S
VP N P
The principle of energy conservation requires that
the power in both coils be equal (neglecting
heating losses in the core).
PP VP I P VS I S
I S VP N P
I P VS N S
Inductors and Stored Energy
When current flows in an inductor, work has been
done to create the magnetic field in the coil. As
long as the current flows, energy is stored in that
field, according to
1 2
E LI
2
Inductors and Stored Energy
In general, a volume in which a magnetic field exists
has an energy density (energy per unit volume)
stored in the field:
energy
B2
energy density
volume 20