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Announcements
No announcements today!
“Wait… what, no announcements?”
“That does it, I’ve had it. I’m dropping my 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 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.
Time-Varying Magnetic Fieldsand Induced Electric Fields
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.
A Changing Magnetic Flux Produces an Electric Field?
Bd = - N
dtε
b aV=V - V - Ed=
V= - Edε=
Bd - N = - Ed
dt
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 r around (but not in) a region where the magnetic field is into the page and increasing (e.g., a solenoid).
But the changing magnetic flux induces an emf around the loop.
r
Bd = -
dtε
The charged particles in the conductor are not in a magnetic field, so they experience no magnetic force.
This could be a wire loop around the outside of a
solenoid.
Put your pens and pencils down and just listen for a few minutes!
r
The induced emf causes a counter-clockwise 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.
The work done moving a charged particle once around the loop is.
r
W=q =qV ε
The sign is positive because the particle’s kinetic energy increases.
I
E
EE
E
B is increasing.
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).
r
We can look at work from a different point of view.
The electric field exerts a force qE on the charged particle. The instantaneous displacement is always parallel to this force.
Thus, the work done by the electric field in moving a charged particle once around the loop is.
r
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.
I
E
EE
Eds
r
Summarizing…
r
I
E
EE
Eds
W=q =qV ε
Bd = -
dtε
q E ds=qε
Bd
E ds= - dt
Bd
E ds= = - dt
ε
W=q E ds
Generalizing still further…
r
I
E
EE
Eds
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.
Was there anything in this discussion that bothered you?
Bd
E ds= - dt
A changing magnetic flux gives rise to an electric field.
r
I
E
EE
Eds
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.
2
qE= k , away from +
r
+-
Huh?
But wait…there’s more!
A potential energy can be defined only for a conservative force.*
*The work done by the force is independent of path.
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.
E
But the work done is
If we tried to define a potential energy, it would not be single-valued:
W=q E ds= qE 2 r
Work depends on the path!
F
F I IU -U =-q E ds
B
F I
dU -U =-q E ds = - 0 even if I =F
dt
U is not single-valued! We can’t define a U for this E!
(*%&^#!)
EI and F
r
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:
BdE 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.
C NCE E +E
“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.
2
qE= k , away from +
r
Bd
E ds= - dt
Coulomb Electric Field
“Faraday” Electric Field
Both “kinds” of electric fields exert forces on charged particles. The Coulomb force is conservative, the “Faraday” force is not.
Both “kinds” of electric fields are part of Maxwell’s Equations.
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.
Bd
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
Image from: http://commons.wikimedia.org/wiki/User:Geek3/Gallery
1.0 cm from the
axis
this would not really qualify as “long”
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?
0
r dIE= n
2 dt
-4 VE= 1.57x10
m
Bd
E ds= - dt
B
B is decreasing
E
dsr
Bd BAd dB
E 2 r = - = = Adt dt dt
02 2
0
d nI d IE 2 r = r = r n
dt dt
A
Some Revolutionary Applications of Faraday’s Law
Electric Guitar Pickup Coils
Magnetic Tape Readers
Phonograph Cartridges
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
enclosed
o
qE dA
These equations can also be written in differential form:
0
E
B dA 0
B 0
BdE ds
dt
B×E=-
t
You have now learned Gauss’s Law for both electricity and magnetism… and Faraday’s Law of Induction:You have now learned Gauss’s Law for both electricity and magnetism…
This will not be tested on the exam.
The Missouri S&T Society of Physics Student T-Shirt!You are ¾ of the way to being qualified to wear…
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 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.
Eddy Currents
You have seen how a changing magnetic flux can induce a “swirling” current in a conductor (the beginning of this lecture).
These currents are called “eddy currents.”
If a conductor and a magnetic field are in relative motion, the magnetic force on charged particles in the conductor causes circulating 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
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.
Conceptual Example: Induction Stove
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 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.
Displacement Current
+ --q+q
ICIC
dsApply Ampere’s Law to a charging capacitor.
μ
0 CB ds= I
Apply Ampere’s Law to the charging capacitor.
B ds=0
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.
The integral is zero because no current passes through the “bowl.”
+ --q+q
ICIC
ds
μ
0 CB ds= I
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.)
+ --q+q
ICIC
ds
As the capacitor charges, the electric field between the plates changes.
E
0ε
Aq=C V= Ed
d
0 0ε ε E= EA=
As the charge and electric field change, the electric flux changes.
0 0ε ε E E
dq d d= =
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.
+ --q+q
ICIC
ds
We define the displacement current to be
E
0ε . D E
dI =
dt
The changing electric flux through the “bowl” surface is equivalent to the current IC through the flat surface.
The generalized (“always correct”) form of Ampere’s Law is then
μ μ μ ε .
E0 C D 0 encl 0encl
dB ds= I I = I
dt
Magnetic fields are produced by both conduction currents and time varying electric fields.
0ε ε=
is NOT emf!
The “stuff” inside the gray boxes serves as your official starting equation for the displacement current ID.
μ μ μ ε .
E0 C D 0 encl 0encl
dB ds= I I = I
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
Displacement current:
where
εE
D
dI =
dt
ε ε 0=
is the relative dielectric constant; not emf
Homework Hints
Displacement current density is calculated the same as conventional current density
DD
IJ =
A
This hint is not needed every semester.
The Big Picture
E0 encl 0 0
dΦB ds=μ I +μ ε .
dt
enclosed
o
qE dA
These equations can also be written in differential form:
0
E
02
1 dEB= +μ Jc dt
B dA 0
B 0
BdE ds
dt
dB×E=-
dt
You have now learned Gauss’s Law for both electricity and magnetism, Faraday’s Law of Induction…You have now learned Gauss’s Law for both electricity and magnetism, Faraday’s Law of Induction, and the generalized form of Ampere’s Law:
This will not be tested on the exam.
The Missouri S&T Society of Physics Student T-Shirt!
Congratulations! It is my great honorto pronounce you fully qualified to wear…
…with all the rights and privileges thereof.*
*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 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.
A changing magnetic field in wire produces a current. A constant magnetic field does not.
A changing 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
back emf (also known as “counter emf”) (if time permits)
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.
Motors have design speeds their engineers expect them to run at. If the motor runs at a lower speed, there is less-than-expected counter emf, and the motor can draw more-than-expected current.
When the appliance reaches operating speed, the counter emf reduces the current flow and the lights “undim.”
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.