Electromagnetic Induction Chapter 22 Expectations After this chapter, students will: Calculate the EMF resulting from the motion of conductors in a magnetic field
Download ReportTranscript Electromagnetic Induction Chapter 22 Expectations After this chapter, students will: Calculate the EMF resulting from the motion of conductors in a magnetic field
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: + L v B -
Motional EMF
A positive charge in the wire experiences a magnetic force, directed upward:
F m
qvB
sin 90
qvB
+ L v B -
Motional EMF
A negative charge in the wire experiences the same magnetic force, but directed downward:
F m
qvB
These forces tend to separate the charges.
+ L v B -
Motional EMF
The separation of the charges produces an electric field,
E
. It exerts an attractive force on the charges:
F C
Eq
+ L E v 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
+ L E v B -
Motional EMF
Rewrite the electric field as a potential gradient:
E
V
EMF L L
Substitute this result back into our earlier equation: + L E v B -
Motional EMF
E
V L
EMF L
Substitute this result back into our earlier equation:
Eq
qvB EMF q
qvB L EMF
vLB
+ L E v B -
Motional EMF
EMF
vLB
This is called
motional EMF
. It results from the constant velocity of the wire through the magnetic field,
B
.
+ L E v 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
I +
I P P
V
vBL R R
VI
vBL
vBL
2
R vBL R
R L v B
Motional EMF
Where does this power come from?
Consider the magnetic force acting on the current in the sliding wire:
F F
ILB v
2
R
vBL R
I R + L v 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:
P
W t
Fvt
Fv t
B
Motional EMF
The force’s magnitude was calculated as: 2 I
F
v R
Substituting:
P
Fv
v R
2
v
vBL
2
R
which is the same as the power dissipated electrically.
R L + v 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 f x B B cos f
Motional EMF
The motional EMF:
EMF
Rewrite the velocity:
v
x
t
vLB
cos f Substitute:
EMF EMF
vLB
cos f
x
t LB
cos f B f B cos f v x
Motional EMF
L
x
is simply the change in the loop area.
EMF
x
t
L B
cos f
A
x
L EMF
B
A
cos f
t
A = L x x L x
Motional EMF
Define a quantity F : F
AB
cos f Then:
EMF
B
A
cos f
t
F
t
A = L x F is called
magnetic flux
.
SI units: T·m 2 = Wb (Weber) x L 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
:
EMF
F
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):
EMF
N
F
t
What can change the magnetic flux?
B
could change, in magnitude or direction
A
f could change could change (the coil could rotate)
Lenz’s Law
Here is where we get the minus sign in Faraday’s Law:
EMF
N
F
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
cos f
AB
cos Get the time rate of change (a calculus problem): F
t
AB
sin Substitute into Faraday’s Law:
EMF
N
F
t
NAB
sin
Faraday’s Law: the Generator
t
n
2
EMF
max
NAB
What makes the voltage larger?
more turns in the coil a larger coil area a stronger magnetic field a larger angular speed
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
IR
EMF B
0
I
V
EMF B 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:
N S
F
S
I P
Mutual Inductance
Convert to an equation, using a constant of proportionality:
N S
F
S
I P N S
F
S
MI P
Mutual Inductance
The constant of proportionality is called the
mutual inductance
:
N S
F
S
MI P M
N S
F
S I P
Mutual Inductance
N S
F
S
MI P
Substitute this into Faraday’s Law:
EMF S
N S
F
S
t
N
t S
F
S
MI
t P
M
I
t P
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:
EMF self
L
I
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:
EMF P
N P
F
t
Through mutual induction, and EMF appears in the secondary:
EMF S
N S
F
t
Their ratio:
EMF S EMF P
N
N P S
F
t
F
t
N S N P
Mutual Inductance: Transformers
This transformer equation is normally written:
V S V P
N N S P
The principle of energy conservation requires that the power in both coils be equal (neglecting heating losses in the core).
P P
V P I P
V S I S I S I P
V P V S
N P 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
E
1 2
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 density energy volume
B
2 2 0