Today

1
Todays 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
Maxwells Equations. Displacement currents
explain how current can flow through a
capacitor, and how a time-varying electric field
can induce a magnetic field. Back emf. A current
in a coil of wire produces an emf that opposes
the original current.
2
Time-Varying Magnetic Fields and Induced Electric
Fields
A Changing Magnetic Flux Produces an Electric
Field?
This suggests that a changing magnetic flux
produces an electric field. This is true not just
in conductors, but any-where in space where there
is a changing magnetic flux.
3
The previous slide uses an equation (Mr. Eds)
valid only for a uniform electric field. Lets
see what a more general analysis gives us.
Consider a conducting loop of radius r around
(but not in) a region where the magnetic field is
into the page and increasing (e.g., a solenoid).
Put your pens and pencils down and just listen
for a few minutes!
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.
4
The induced emf causes a counter-clockwise
current (charges move).
E
E
I
But the magnetic field did not accelerate the
charged particles (they arent in it). Therefore,
there must be a tangential electric field around
the loop.
r
E
E
B is increasing.
The work done moving a charged particle once
around the loop is.
The sign is positive because the particles
kinetic energy increases.
Remember, the magnetic force does no work when it
accelerates a charged particle. If the loop has
no resistance, the work done by the electric
field goes into increasing the charged particles
speed (and therefore kinetic energy). If the loop
has resistance, the work done by the electric
field is dissipated in the resistance (energy
leaves the system).
5
We can look at work from a different point of
view.
ds
E
E
I
The electric field exerts a force qE on the
charged particle. The instantaneous displacement
is always parallel to this force.
r
E
E
Thus, the work done by the electric field in
moving a charged particle once around the loop is.
The sign is positive because the particles
displacement and the force are always parallel.
6
Summarizing
ds
E
E
I
r
E
E
7
Generalizing still further
ds
E
E
The loop of wire was just a convenient way for us
to visualize the effect of the changing magnetic
field.
I
r
E
E
The electric field exists whether or not the loop
is present.
A changing magnetic flux gives rise to an
electric field.
Was there anything in this discussion that
bothered you?
8
ds
E
E
I
r
E
E
This should bother you where are the and
charges in this picture?
Answer there are no and charges. Instead,
there are electric field lines that form
continuous, closed loops.
Huh?
9
But waittheres more!
E
A potential energy can be defined only for a
conservative force.
A potential energy is a single-valued function.
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.
10
But the work done is
I and F
E
r
Work depends on the path!
If we tried to define a potential energy, it
would not be single-valued
U is not single-valued! We cant define a U for
this E!
(!)
11
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.
12
Induced Electric Fields a summary of the key
ideas
A changing magnetic flux induces an electric
field, as given by Faradays Law
This is a different manifestation of the 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.
conservative, or Coulomb
nonconservative
It is better to say that there is an electric
field, as described by Maxwells equations. We
saw in lecture 17 that what an observer measures
for the magnetic field depends on the motion of
the observer relative to the source of the field.
We see here that the same is true for the
electric field. There arent really two different
kinds of electric fields. There is just an
electric field, which seems to behave
differently depending on the relative motion of
the observer and source of the field.
13
Stated slightly differently we have discovered
two different ways to generate an electric field.
Coulomb Electric Field
Faraday Electric Field
Both kinds of electric fields are part of
Maxwells Equations.
Both kinds of electric fields exert forces on
charged particles. The Coulomb force is
conservative, the Faraday force is not.
It is better to say that there is an electric
field, as described by Maxwells equations. We
saw in lecture 17 that what an observer measures
for the magnetic field depends on the motion of
the observer relative to the source of the field.
We see here that the same is true for the
electric field. There arent really two different
kinds of electric fields. There is just an
electric field, which seems to behave
differently depending on the relative motion of
the observer and source of the field.
14
Direction of Induced Electric Fields
The direction of E is in the direction a
positively charged particle would be accelerated
by the changing flux.
Use Lenzs Law to determine the direction the
changing magnetic flux would cause a current to
flow. That is the direction of E.
15
Exampleto 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?
16
Exampleto 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
17
Exampleto 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?
A
ds
r
E
B
B is decreasing
18
Some Revolutionary Applications of Faradays Law
? Magnetic Tape Readers ?
? Phonograph Cartridges ?
? Electric Guitar Pickup Coils
? Ground Fault Interruptors
? Alternators
? Generators
? Transformers
? Electric Motors
19
Application of Faradays 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.