Transcript Slide 1

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Announcements
No announcements today!
“Wait… what, no announcements?”
“That does it, I’ve had it. I’m dropping this class.”
Today’s agenda:
Induced Electric Fields.
You must understand how a changing magnetic flux induces an electric field, and be able
to calculate induced electric fields.
Eddy Currents.
You must understand how induced electric fields give rise to circulating currents called
“eddy currents.”
Displacement Current and Maxwell’s Equations.
Displacement currents explain how current can flow “through” a capacitor, and how a timevarying electric field can induce a magnetic field.
Back emf.
A current in a coil of wire produces an emf that opposes the original current.
Time-Varying Magnetic Fields
and Induced Electric Fields
A Changing Magnetic Flux Produces an Electric Field?
dB
ε = -N
dt
V = Vb - Va = - Ed
ε = V = - Ed
-N
d B
= - Ed
dt
This suggests that a changing magnetic flux produces an
electric field. This is true not just in conductors, but anywhere in space where there is a changing magnetic flux.
The previous slide uses an equation
(Mr. Ed’s) valid only for a uniform
electric field. Let’s see what a more
general analysis gives us.
Consider
conducting
loopdown
of and
Put yourthe
pens
and pencils
radiusjust
r around
(buta not
a region
listen for
fewin)
minutes!
where the magnetic field is into the
page and increasing (e.g., a solenoid).

 
 


 
r
 

This could be a wire loop around
the outside of a solenoid.
The charged particles in the conductor are not in a magnetic
field, so they experience no magnetic force.
But the changing magnetic flux induces an emf around the
loop.
dB
ε = dt
The induced emf causes a counterclockwise current (charges move).
But the magnetic field did not
accelerate the charged particles
(they aren’t in it). Therefore, there
must be a tangential electric field
around the loop.
E

 
 

I
E
E

 
r
 

E
B is increasing.
The work done moving a charged particle once around the
loop is.
W = qV = qε
The sign is positive because the particle’s kinetic energy
increases.
We can look at work from a different
point of view.
ds
E

 
 

I
The electric field exerts a force qE
on the charged particle. The
instantaneous displacement is
always parallel to this force.
E
E

 
r
 

E
Thus, the work done by the electric field in moving a charged
particle once around the loop is.
W   F  ds  q  E  ds = qE  ds  qE  2r 
The sign is positive because the particle’s displacement and
the force are always parallel.
Summarizing…
dB
ε = dt
W = qV = qε
ds
E

 
 

I
E
W = q  E  ds
q  E  ds = qε
dB
 E  ds = ε = - dt
dB
 E  ds = - dt
E

 
r
 

E
Generalizing still further…
The loop of wire was just a
convenient way for us to visualize
the effect of the changing magnetic
field.
The electric field exists whether or
not the loop is present.
ds
E

 
 

I
E
E

 
r
 

E
dB
 E  ds = - dt
A changing magnetic flux gives rise to an electric field.
Was there anything in this discussion that bothered you?
q
E = k 2 , away from +
r
ds
E

 
 

I
-
+
E
This should bother you: where are
the + and – charges in this picture?
E

 
r
 

E
Answer: there are no + and – charges. Instead, there are
electric field lines that form continuous, closed loops.
Huh?
But wait…there’s more!
A potential energy can be defined
only for a conservative force.*
A potential energy is a single-valued
function.

 
 


 
 

E
If this electric field E is due to a
conservative force, then the
potential energy of a charged
particle must be unchanged when it
goes once around the loop.
*The work done by the force is independent of path.
But the work done is
I and F
W = q  E  ds = qE  2r 

 
 

Work depends on the path!

 
r
 

If we tried to define a potential
energy, it would not be singlevalued:
F
UF - UI = -q E  ds
I
UF - UI = -q  E  ds = -
dB
0
dt
even if I = F
U is not single-valued! We can’t define a U for this E!
(*%&^#!)
E
One or two of you might not have followed the discussion
on the previous 9 slides. Did I confuse anybody?
You can start taking notes again, if you want.
Induced Electric Fields: a summary of the key ideas
A changing magnetic flux induces an electric field, as given by
Faraday’s Law:
dB
 E  ds = - dt
This is a different kind of electric field than the one you are
familiar with; it is not the electrostatic field caused by the
presence of stationary charged particles.
Unlike the electrostatic electric field, this “new” electric field is
nonconservative.
E =EC +ENC
“conservative,” or “Coulomb”
“nonconservative”
Stated slightly differently: we have “discovered” two different
ways to generate an electric field.
Coulomb Electric Field
q
E = k 2 , away from +
r
“Faraday” Electric Field
dB
 E  ds = - dt
Both “kinds” of electric fields are part of Maxwell’s
Equations.
Both “kinds” of electric fields exert forces on charged particles.
The Coulomb force is conservative, the “Faraday” force is not.
Direction of Induced Electric Fields
The direction of E is in the direction a positively charged
particle would be accelerated by the changing flux.
dB
 E  ds = - dt
Use Lenz’s Law to determine the direction the changing
magnetic flux would cause a current to flow. That is the
direction of E.
Example—to be worked at the blackboard in lecture
A long thin solenoid has 500 turns per meter and a radius of
3.0 cm. The current is decreasing at a steady rate of 50 A/s.
What is the magnitude of the induced electric field near the
center of the solenoid 1.0 cm from the axis of the solenoid?
Example—to be worked at the blackboard in lecture
A long thin solenoid has 500 turns per meter and a radius of
3.0 cm. The current is decreasing at a steady rate of 50 A/s.
What is the magnitude of the induced electric field near the
center of the solenoid 1.0 cm from the axis of the solenoid?
“near the center”
radius of
3.0 cm
1.0 cm from
the axis
this would not really
qualify as “long”
Image from: http://commons.wikimedia.org/wiki/User:Geek3/Gallery
Example—to be worked at the blackboard in lecture
A long thin solenoid has 500 turns per meter and a radius of
3.0 cm. The current is decreasing at a steady rate of 50 A/s.
What is the magnitude of the induced electric field near the
center of the solenoid 1.0 cm from the axis of the solenoid?
dB
 E  ds = - dt
ds
E
A
r
B
B is decreasing
d BA 
dB
dB
E  2r  = =
=A
dt
dt
dt
d I
2 d  0nI 
2
E  2r  = r
= r 0n
dt
dt
r
dI
E =  0n
2
dt
V
E = 1.57x10
m
-4
Some Revolutionary Applications of Faraday’s Law
 Magnetic Tape Readers 
 Phonograph Cartridges 
 Electric Guitar Pickup Coils
 Ground Fault Interruptors
 Alternators
 Generators
 Transformers
 Electric Motors
Application of Faraday’s Law (MAE Plasma Lab)
From Meeks and Rovey, Phys. Plasmas 19, 052505 (2012); doi: 10.1063/1.4717731. Online at
http://dx.doi.org/10.1063/1.4717731.T
“The theta-pinch concept is one of the most widely used inductive plasma source designs ever
developed. It has established a workhorse reputation within many research circles, including
thin films and material surface processing, fusion, high-power space propulsion, and
academia, filling the role of not only a simply constructed plasma source but also that of a key
component…
“Theta-pinch devices utilize relatively simple coil
geometry to induce electromagnetic fields and
create plasma…
“This process is illustrated in Figure 1(a), which
shows a cut-away of typical theta-pinch operation
during an initial current rise.
“FIG. 1. (a) Ideal theta-pinch field topology for an
increasing current, I.”
The Big Picture
You have now learned Gauss’s Law for both electricity and
magnetism… and Faraday’s Law of Induction:
q enclosed
 E  dA  o
d B
 E  ds   dt
 B  dA  0
These equations can also be written in differential form:

E 
0
B
×E=t
B  0
You are ¾S&T
of the
way to of
being
qualified
to wear…
The Missouri
Society
Physics
Student
T-Shirt!
This will not be tested on the exam.
Today’s agenda:
Induced Electric Fields.
You must understand how a changing magnetic flux induces an electric field, and be able
to calculate induced electric fields.
Eddy Currents.
You must understand how induced electric fields give rise to circulating currents called
“eddy currents.”
Displacement Current and Maxwell’s Equations.
Displacement currents explain how current can flow “through” a capacitor, and how a timevarying electric field can induce a magnetic field.
Back emf.
A current in a coil of wire produces an emf that opposes the original current.
Eddy Currents
You have seen how a changing magnetic field can induce a
“swirling” current in a conductor (the beginning of this lecture).
If a conductor and a magnetic field are in relative motion, the
magnetic force on charged particles in the conductor causes
circulating currents.
These currents are called “eddy currents.”
Eddy currents give rise to magnetic fields that oppose any
external change in the magnetic field.
Eddy Currents
Eddy currents are useful in generators, microphones, metal
detectors, coin recognition systems, electricity meters, and
roller coaster brakes (among other things).
However, the I2R heating from eddy currents causes energy
loss, so if you don’t want energy loss, you probably think eddy
currents are “bad.”
Eddy Current Demos
cylinders falling through a tube
magnetic “guillotine”
hopping coil
coil launcher
magnetic flasher
Conceptual Example: Induction Stove
An ac current in a coil in the stove
top produces a changing
magnetic field at the bottom of a
metal pan.
The changing magnetic field gives
rise to a current in the bottom of
the pan.
Because the pan has resistance, the current heats the pan. If
the coil in the stove has low resistance it doesn’t get hot but
the pan does.
An insulator won’t heat up on an induction stove.
Remember the controversy about cancer from power lines a few years back? Careful studies
showed no harmful effect. Nevertheless, some believe induction stoves are hazardous.
Today’s agenda:
Induced Electric Fields.
You must understand how a changing magnetic flux induces an electric field, and be able
to calculate induced electric fields.
Eddy Currents.
You must understand how induced electric fields give rise to circulating currents called
“eddy currents.”
Displacement Current and Maxwell’s Equations.
Displacement currents explain how current can flow “through” a capacitor, and how a timevarying electric field can induce a magnetic field.
Back emf.
A current in a coil of wire produces an emf that opposes the original current.
Displacement Current
Apply Ampere’s Law to a
charging capacitor.
 B  ds = μ I
0 C
ds
IC
+
+q
-q
IC
The shape of the surface used for Ampere’s Law shouldn’t
matter, as long as the “path” is the same.
Imagine a soup bowl
surface, with the + plate
resting near the bottom of
the bowl.
Apply Ampere’s Law to the
charging capacitor.
ds
IC
+
+q
-q
IC
 B  ds = 0
The integral is zero because no current passes through the
“bowl.”
 B  ds = μ I
0 C
 B  ds = 0
Hold it right there pal! You can’t have it both ways. Which is it
that you want? (The equation on the right is actually incorrect, and the equation on the left is incomplete.)
As the capacitor charges, the electric field between the plates
changes.
A

q = CV =  ε 0  Ed
d

= ε0EA = ε0E
ds
IC
As the charge and electric field
change, the electric flux changes.
E
+
+q
-q
IC
dq d
d
=  ε 0 E  = ε 0  E 
dt dt
dt
This term has units of current.
 Is the dielectric constant of the medium in between the capacitor plates. In the diagram, with an air-filled capacitor,  = 1.
We define the displacement current to be
d
ID = ε 0  E  .
dt
The changing electric flux
through the “bowl” surface
is equivalent to the current
IC through the flat surface.
ε = ε 0
ds
IC
E
+
+q
-q
IC
The generalized (“always correct”) form of Ampere’s Law is
then
dE
 B  ds = μ0  IC  ID encl = μ0Iencl  μ0 ε dt .
Magnetic fields are produced by both conduction currents and
time varying electric fields.
The “stuff” inside the gray boxes serves as your official starting
equation for the displacement current ID.
dE
 B  ds = μ0  IC  ID encl = μ0Iencl  μ0 ε dt .
 is the relative dielectric constant; not emf.
In a vacuum, replace  with 0.
ID
Why “displacement?” If you put an insulator in between the plates of the capacitor, the atoms of the insulator are “stretched”
because the electric field makes the protons “want” to go one way and the electrons the other. The process of “stretching” the
atom involves displacement of charge, and therefore a current.
Homework Hints
dE
Displacement current: ID = ε
dt
where ε = ε0
 is the relative dielectric constant; not emf
Homework Hints
For problem 9.45a, recall that
I
E = J =
A
For problem 9.45c, displacement current density is calculated
the same as conventional current density
ID
JD =
A
Not applicable Spring 2015.
The Big Picture
You have now learned Gauss’s Law for both electricity and
magnetism, Faraday’s Law of Induction,
Induction…and the generalized
form of Ampere’s Law:
q enclosed
 E  dA  o
d B
 E  ds   dt
 B  dA  0
 B  ds=μ 0 Iencl +μ 0ε 0
dΦ E
.
dt
These equations can also be written in differential form:

E 
0
B  0
dB
×E=dt
1 dE
  B= 2
+μ 0 J
c dt
Congratulations! It is my great honor
The Missouri
S&T Society
of qualified
PhysicstoStudent
to pronounce
you fully
wear… T-Shirt!
…with all the rights and privileges thereof.*
This will not be tested on the exam.
*Some day I might figure out what these rights and privileges are. Contact me if you want to buy an SPS t-shirt for the member price of $10.
Today’s agenda:
Induced Electric Fields.
You must understand how a changing magnetic flux induces an electric field, and be able
to calculate induced electric fields.
Eddy Currents.
You must understand how induced electric fields give rise to circulating currents called
“eddy currents.”
Displacement Current and Maxwell’s Equations.
Displacement currents explain how current can flow “through” a capacitor, and how a timevarying electric field can induce a magnetic field.
Back emf.
A current in a coil of wire produces an emf that opposes the original current.
back emf (also known as “counter emf”) (if time permits)
A changing magnetic field in wire produces a current. A
constant magnetic field does not.
An electrical current produces a magnetic field, which by
Lenz’s law, opposes the change in flux which produced the
current in the first place.
http://campus.murraystate.edu/tsm/tsm118/Ch7/Ch7_4/Ch7_4.htm
The effect is “like” that of friction.
The counter emf is “like” friction that opposes the original
change of current.
Motors have many coils of wire, and thus generate a large
counter emf when they are running.
Good—keeps the motor from “running away.” Bad—”robs”
you of energy.
If your house lights dim when an appliance starts up, that’s
because the appliance is drawing lots of current and not
producing a counter emf.
When the appliance reaches operating speed, the counter emf
reduces the current flow and the lights “undim.”
Motors have design speeds their engineers expect them to run
at. If the motor runs at a lower speed, there is less-thanexpected counter emf, and the motor can draw more-thanexpected current.
If a motor is jammed or overloaded and slows or stops, it can
draw enough current to melt the windings and burn out. Or
even burn up.