Transcript i EMF

Class 31
Today we will:
• learn about EMF
• learn how Faraday’s law works
• learn Lenz’s Law and how to apply it
Last Time -- Induced Current
Accelerating charges produce electric fields in the opposite

direction to the acceleration.
E
i

E
i
Faraday’s Law
If the number of magnetic field lines through a
loop is changing, we produce a looping electric
field.
Induced Current
Current increases in a wire…
i
Induced Current
… so the magnetic field increases…
i
Induced Current
… so the number of magnetic field lines passing through the
loop (flux) increases…
i
Induced Current
… so there is an induced EMF around the loop …
EMF
i
Induced Current
… so current flows around the loop.
EMF
i
i
Faraday’s Law of Induction


B
 E  
t
d B
E    
dt
Faraday’s Law of Induction


B
 E  
t
d B
E    
dt
=EMF
What is EMF?
1) Any voltage, as from a battery.
2) An effective voltage produced by induced electric
fields.
What is EMF?
1) Any voltage, as from a battery.
2) An effective voltage produced by induced electric
fields.
I usually reserve the term EMF for induced voltages.
Two Ways to Produce Induced EMF
1) Acceleration of charges – changing current in a
circuit.
2)Motional EMF – charges in a conductor moving
in a B field.
Motional EMF
A wire of length L moves through a magnetic
field. The wire is perpendicular to B. What
happens?

B

v
Motional EMF
Charges along the wire feel a Lorentz Force.

 
F  qv  B

B



v
Motional EMF
Charges doesn’t increase indefinitely. Eventually
a voltage develops across the wire.


FB   FE  qvB  qE
E  vB
V  EL  vBL

B



v
Motional EMF
Add three other fixed wires to make a loop. Now
current will continue to flow around the loop.

B



v
Clicker Question 1
What happens if all sides
of the loop move
together?
A. Current flows.
B. Current doesn’t flow.

B

v

v
What happens both ways?

E

B
i

v

a
i

E
What happens both ways?
The magnetic flux – the number of magnetic field
lines – passing through the loop changes.

E

B
i

v

a
i

E
Faraday’s Law of Induction
…works for both motional EMF and the
EMF of accelerating charges!
What is EMF?
We can think of an induced EMF as a voltage
produced all along a wire segment.

B

v
What is EMF?
Let’s take a square loop with an increasing
magnetic field passing through it. Assume the
wire has a small resistance.
V
What is EMF?
Assume that the EMF around the loop is 40 V.
What would a voltmeter read?
V
What is EMF?
The voltmeter would read zero!
V = 0V
What is EMF?
The voltmeter would read zero because the
voltage drop due to resistance in the wire
segment is exactly the same as the voltage
increase due to induction in the wire segment.
V = 0V
What is EMF?
Another way of putting it is that the wire segment is like a
lot of little batteries and resistors in series. The voltage
goes up through each battery, but down by the same
amount through each resistor.
V = 0V
What is EMF?
Each electron that goes around the full loop
once gains 40eV of energy from the EMF and
loses 40eV of energy to heat!
V =0V
What if there’s a resistor in the loop?
The total EMF is 40V.
5
What if there’s a resistor in the loop?
The total EMF is 40V. Ohm’s Law gives i = 8A.
The voltage across the resistor is 40V.
5
Where is the EMF being produced?
5
Where is the EMF being produced?
Everywhere, including through the resistor.
5
Where is the voltage dropping?
5
Where is the voltage dropping?
Primarily in the resistor – just a little in the
wire. The resistor only lets a little current
trickle through the wire – as compared to
having no resistor.
5
Lenz’s Law
To determine the direction induced current
will flow in a circuit or to determine the
direction of the induced electric field, we use
Lenz’s Law.
Lenz’s Law
First, we ask two questions:
1) What is the direction of the external B field?
2) Is the external B field increasing or decreasing?
Lenz’s Law
1) What is the direction of the external B field?
2) Is the external B field increasing or decreasing?
3) Find the direction of the induced magnetic field.
The induced magnetic field opposes change in
the external magnetic field.
Lenz’s Law
1) What is the direction of the external B field?
2) Is the external B field increasing or decreasing?
3) Find the direction of the induced magnetic field.
The induced magnetic field opposes change in
the external magnetic field.
4) Find the direction the induced current using the
right-hand rule.
Lenz’s Law
•The external B is into the screen and increasing.
x x
x
x
x
Bexternal
x
x
x
x
x
x
x
Lenz’s Law
•The external B is into the screen and increasing.
•To oppose change, the induced B, must be out of
the screen.

Binduced

Bexternal
x
x
x
x
x
x
x
x
x
x
x
x
Lenz’s Law
•The external B is into the screen and increasing.
•To oppose change, the induced B, must be out of
the screen.
• To produce this B, the current
 is ccw.

Binduced
Bexternal
x
x
x
x
x
x
x
x
x
x
x
x
i
Lenz’s Law
• The external B is into the screen and decreasing.
x x
x
x
x
Bexternal
x
x
x
x
x
x
x
Lenz’s Law
• The external B is into the screen and decreasing.
• To oppose change, the induced B, must be into
the screen.

Binduced

Bexternal
x
x
x
x
x
x
x
x
x
x
x
x
i
Lenz’s Law
• The external B is into the screen and decreasing.
• To oppose change, the induced B, must be into
the screen.
• To produce this B, the current
 is cw.

Binduced
Bexternal
x
x
x
x
x
x
x
x
x
x
x
x
i
Class 32
Today we will:
• work several Faraday’s law problems
• learn about Eddy currents
Maxwell’s Equations in Integral Form
•Gauss’s Law of Electricity
•Gauss’s Law of Magnetism
•Ampere’s Law
•Faraday’s Law
  qenc
 E   E  dA 
0
 
 B   B  dA  0
 
d E 

 B   B  d   0  i   0

dt 

 
d B
E   E  d  
dt
Maxwell’s Equations in Integral Form
•Gauss’s Law of Electricity
  qenc
 E   E  dA 
Flux through a
Gaussian surface
 
•Gauss’s Law of Magnetism
•Ampere’s Law
•Faraday’s Law
0
 B   B  dA  0
 
d E 

 B   B  d   0  i   0

dt 

 
d B
E   E  d  
dt
Maxwell’s Equations in Integral Form
•Gauss’s Law of Electricity
  qenc
 E   E  dA 
Flux through a
Gaussian surface
 
•Gauss’s Law of Magnetism
•Ampere’s Law
•Faraday’s Law
0
 B   B  dA  0
 
d E 

 B   B  d   0  i   0

dt 

Line integral around
an Amperian
 loop
d B
E   E  d  
dt
Maxwell’s Equations in Integral Form
•Gauss’s Law of Electricity
  qenc
 E   E  dA 
Flux through a
Gaussian surface
 
•Gauss’s Law of Magnetism
•Ampere’s Law
•Faraday’s Law
0
 B   B  dA  0
 
d E 

 B   B  d   0  i   0

dt 

Line integral around
an Amperian
 loop
d B
E   E  d  
dt Flux through the
Amperian loop
Calculating Flux through a Loop
We will assume that the magnetic field is
constant over a loop. Then, the flux is:
 
 B  B  A  BA cos 

Remember that A points in the direction of
the normal to the loop.
Three Ways to Generate an EMF
 
 B  B  A  BA cos 
1) The magnetic field changes in time.
2) The area changes in time.
3) the angle changes in time.
Eddy Currents
A wire loop is moved into a region where there is a
magnetic field. In what direction does current flow?

B
Eddy Currents
A wire loop is moved into a region where there is a
magnetic field. In what direction does current flow?

B
i
Eddy Currents
Viewing the same thing from the side, the loop
becomes a magnet that is repelled by the external
field. (Remember field lines come out of the N pole.)
N
N
S
S

B
Eddy Currents
It takes force to push the loop into the
field. Where does this energy go?
N
N
S
S

B
Eddy Currents
If the wire loop is moved out of the magnetic field, in
what direction does current flow?

B
Eddy Currents
If the wire loop is moved out of the magnetic field, in
what direction does current flow?

B
i
Eddy Currents
Viewing the same thing from the side, the loop
becomes a magnet that is attracted by the external
field.
N
S
S
N

B
Eddy Currents
A disk is even more effective at producing induced
currents. Such currents are called “eddy currents.”

B
Changing Area
Put a moveable length of wire on a fixed ushaped wire in a uniform magnetic field.

B

v
R
a
x
Lenz’s law
Which way does the induced current flow?

B

v
R
a
x
Lenz’s law
Which way does the induced current flow?

B

v
R
i
x
a
Find the Flux

B

v
R
a
x
Find the Flux
 B  Bax  Bavt
as
x  vt

B

v
R
a
x
Find the EMF

B

v
R
a
x
Find the EMF
 B  Bax  Bavt
d B
 
  Bav
dt
Not worrying about the minus sign:
  Bav

B

v
R
a
x
Find the Current

B

v
R
a
x
Find the Current
  Bav
 Bav
i
R

R

B

v
R
a
x
Find the Power Dissipated in the Resistor

B

v
R
a
x
Find the Power Dissipated in the Resistor
  Bav
 Bav
i
R

R
2

Bav 
P  i 
R

B

v
R
a
x
Find the Force on the Moveable Wire

B

v
R
a
x
Find the Force on the Moveable Wire
  
F  iL  B
F  iaB
2

Ba  v
F
R

B

v
R
a
x
Find the Work Done in Moving Δx

B

v
R
a
x
Find the Work Done in Moving Δx
2

Ba  v
F
R
2

Ba  v
W  Fx 
x
R

B

v
R
a
x
Find the Mechanical Power

B

v
R
a
x
Find the Mechanical Power
2

Ba  v
W  Fx 
x
R
2
2
W Ba  v x Ba  v
P


v
t
R t
R
2

Bav 
P
R

B

v
R
a
x
Changing the Magnetic Field
B(t )  B0 e
 t /
a
b
Lenz’s law
Which way does the induced current flow?
B(t )  B0 e t /
a
b
Lenz’s law
Which way does the induced current flow?
B(t )  B0 e t /
i
b
a
Find the Flux
i
b
a
Find the Flux
B(t )  B0e
 t /
 B  BA  B0 abe
t / 
i
b
a
Find the EMF
i
b
a
Find the EMF
 B  B0 abe  t /
d B B0 abe
 

dt

t /
i
b
a
Where does the energy come from this
time?
i
b
a
Where does the energy come from this
time?
If the external field comes from a permanent
magnet, the magnetic field of the loop
attracts the permanent magnet, making it
more difficult to move away.
i
b
a
Where does the energy come from this
time?
If the external field comes from an
electromagnet, the interaction of the loop
with the electromagnet takes some energy
from the electromagnet’s circuit.
i
b
a
We can use Faraday’s Law to calculate the
electric field as well as the EMF in one
problem only!
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
B(t )  B0 e
 t /

r
R
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
2  t /
 B  B (t ) A  B0 r e
d B B0 r t /
 

e
dt

   E  2 rE
2
B0 r t /

E

e
2 r 2

r
R
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
Flux through the Amperian loop of radius r.
2  t /
 B  B (t ) A  B0 r e
d B B0 r t /
 

e
dt

   E  2 rE
2
B0 r t /

E

e
2 r 2

r
R
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
Flux through the Amperian loop of radius r.
2  t /
 B  B (t ) A  B0 r e
d B B0 r t /
 

e
dt

   E  2 rE
2
B0 r t /

E

e
2 r 2

r
R
Line integral around the Amperian loop of radius r.
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
B(t )  B0 e
 t /

r
R
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
2  t /
 B  B(t ) A  B0 R e
d B B0 R t /
 

e
dt

   E  2 rE
2
B0 R t /

E

e
2 r
2 r
2

r
R
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
Flux through the Amperian loop of radius r.
2  t /
 B  B(t ) A  B0 R e
d B B0 R t /
 

e
dt

   E  2 rE
2
B0 R t /

E

e
2 r
2 r
2

r
R
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
Flux through the Amperian loop of radius r.
2  t /  But the field stops at R!
 B  B(t ) A  B0 R e
d B B0 R t /
 

e
dt

   E  2 rE
2
B0 R t /

E

e
2 r
2 r
2

r
R
A Circular Loop in the Field of an
Electromagnet with Circular Pole Faces
Flux through the Amperian loop of radius r.
2  t /  But the field stops at R!
 B  B(t ) A  B0 R e
d B B0 R t /
 

e
dt

   E  2 rE
2
B0 R t /

E

e
2 r
2 r

r
R
2
Line integral around the Amperian loop of radius r.
Changing the Angle
Attach a handle to a circular loop of wire.
Changing the Angle
Place the loop in a magnetic field with the shaft
perpendicular to B .

B


A
Changing the Angle
We rotate the handle with angular speed  .
The flux is:
 B  BA cos t
The EMF is:
d B
 
dt
 BA  sin t

B


A
Class 33
Today we will:
•learn how motors and generators work
•learn about split commutators and their use in
DC motors and generators
Humphrey Davy
(1778-1829)
•First to isolate potassium,
sodium, barium, calcium,
strontium, magnesium, boron,
and silicon.
•1813: Discovers Michael Faraday
Michael Faraday
(1791-1867)
•1831: Discovers electromagnetic
induction independently of Henry.
•Develops the transformer, motor,
and generator.
•Discovers the Faraday Effect of
light - the rotation of the plane of
polarization in magnetic fields.
•Develops the First and Second
Laws of Electrochemistry.
•Discovers paramagnetism and
diamagnetism.
James Clerk Maxwell
b. 1831, Edinburgh, Scotland
d. 1879, Cambridgeshire,
England
•1861: Proposes
"displacement current" and
creates Maxwell's
Equations.
•Recognizes light as
electromagnetic radiation.
James Clerk Maxwell
“What is done by what is
called ‘myself’ is, I feel, done
by something that is greater
than myself within me.”
Changing the Angle
Attach a handle to a circular loop of wire.
Changing the Angle
Place the loop in a magnetic field with the shaft
perpendicular to B .

B


A
Changing the Angle
We rotate the handle with angular speed  .
The flux is:
 B  BA cos t
The EMF is:
d B
 
dt
 BA  sin t

B


A
Engineering Considerations
We usually want to get current out of the coils as
they turn, so rather than use a complete loop, we
use an open loop with each end connecting to a
wire.
Engineering Considerations
But fixed wires would twist and break.
-- So we use commutators and brushes.
Commutators and Brushes
These allow electrical connections to be made
by pressing conductors together.
commutator
shaft
shaft
wire
wire
graphite brush
spring steel clip
wire
shaft
wire
commutator
Commutators
A typical AC generator or motor uses double
commutators. Each end of the loop is connected
to a separate commutator.
Motors
A simple motor consists of a current-carrying coil
in a uniform magnetic field.
N
S
Motors
A torque on the coil tends to align the magnetic
dipole moment of the loop with the external field.
N


S
Motors
A torque on the coil tends to align the magnetic
dipole moment of the loop with the external field.
N


+
S
Motors
As the dipole moment rotates past the magnetic
field, however, the torque reverses.
N


+
S
Motors
We’ve created a vibrator instead of a motor.
N


+
S
Motors
But, we could keep the loop (armature) turning in the
same direction, if we could reverse the magnetic dipole
moment. This can be done by changing the direction of
the battery.
N


+
S
Motors
It’s a little hard to keep moving the leads on the battery
back and forth by hand. So we need a better way of doing
it.
N


+
S
AC Current
The easiest way to do this is to use AC current
instead of a battery. The direction of the current
through the loop automatically changes sign
periodically.
AC Current
Note that the speed of such a motor is closely
tied to the frequency of the AC power supply.
The Split Commutator
Another clever solution is to use a “split
commutator.” A split commutator automatically
changes the end of the loop connected to the
positive terminal of the battery every half cycle.


DC Motors
Thus, split commutators allow motors to be
operated by DC power sources.


Generators
Generators are just motors operated in reverse.
N
load
S
AC Generators
With double commutators, we get a sinusoidal
current out of a generator.
N
load
S
Quasi-DC Generators
With split commutators, we get a sinusoidal
current that changes sign each half cycle.
N
load
S
DC Generators
Adding a second loop can give something that is
closer to a DC output.
N
load
S