Transcript Induced Electric Fields.

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
dB
-N
= - 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 a conducting loop of radius
Put your pens and pencils down and
r around (but not in) a region
just listen for a few 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.
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
particle’s 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).
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?
E= k
q
r
2
, away from +
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 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.
E  EC +ENC
“conservative,” or “Coulomb”
“nonconservative”
It is better to say that there is an electric field, as described by Maxwell’s 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 aren’t 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.
Stated slightly differently: we have “discovered” two different
ways to generate an electric field.
Coulomb Electric Field
E= k
“Faraday” Electric Field
q
r
2
, away from +
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.
It is better to say that there is an electric field, as described by Maxwell’s 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 aren’t 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.
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.”