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PES 1120 Spring 2014, Spendier
Lecture 33/Page 1
Today: continue chapter 30
- Induction and energy transfer
- Eddy Currents
 
Magnetic flux:  B   B  dA   B dA cos( )
units: [Wb] = [T m2]
weber (Wb)
d B
(induced emf in single loop!)
dt
For a coil that consists of N loops, the total induced emf would be N times as large:
dB
 ind   N
dt
Lenz’s Law
The direction of the induced current is determined by
Lenz’s law: The induced magnetic field opposes the change in flux that created it
Faraday’s Law :  ind  
We now have an equation that relates a change in magnetic flux through a surface to an
induced electromotive force around the surface.
Thus, we see that an emf may be induced in the following ways:

(i) by varying the magnitude of B with time

(ii) by varying the magnitude of A ,


(iii) varying the angle between B and the area vector A with time  Generators!
Generator: Alternating current generator
One of the most important applications of Faraday’s law of induction is to generators and
motors. A generator converts mechanical energy into electric energy, while a motor
converts electrical energy into mechanical energy.
A simple generator consists of an N-turn loop rotating in a magnetic field which is
assumed to be uniform. The magnetic flux varies with time, thereby inducing an emf.
From Figure at the right, we see that the magnetic flux through the loop may be written as
 B  BA cos q  BA cos wt
ω angular velocity
The rate of change of magnetic flux is
d B
eind  
 BAw sin(wt )
dt
PES 1120 Spring 2014, Spendier
Lecture 33/Page 2
For N turns in the loop, the total induced emf across the two ends of the loop is
eind  NBAw sin(wt )
For a circular coil with radius R=2.5cm and 150 turns of wire, B = 0.060 T, and coil
rotating at 440 rev/min, the maximum induced emf is:
If we connect the generator to a circuit which has a resistance R, then the current
generated in the circuit is given by
I
e
NBAw

sin(wt )
R
R
Motional EMF: Energy transfer

Imagine pulling a closed conducting loop out of a magnetic field at a constant velocity v .
While the loop is moving, a clockwise current Iind is induced in the loop. This means that
in turn the loop segments still within the magnetic field will now experience forces
 

F1 , F2 , and F3 .

 
Force on a wire segment: F1  I L  B
Pushing the loop into the field therefore requires work to be done by the arem, even to
maintain constant velocity
PES 1120 Spring 2014, Spendier
Lecture 33/Page 3
v = constant
a = 0;
 F  Fpush  F1  0
Where does this work (mechanical energy) go?
It is converted to electrical energy in the loop!
When the loop’s motion is stopped, what happens?
The induced emf goes to zero and the induced current dies away. In this case electrical
energy is converted to thermal energy due to the resistance of the wire.
So we have: Mechanical energy  electrical energy  thermal energy (or heat)
The only time we don’t have transfer to thermal energy is for materials with resistance
R=0, i.e. superconductors!
From the analysis above, in order for the bar to move at a constant speed, an external
agent must constantly supply a force. What happens if at t = 0, the speed of the rod is v0,
and the external agent stops pushing? In this case, the bar will slow down because of the
magnetic force directed to the left. Using Newton’s laws, one can shoe that the speed
decreases exponentially in the absence of an external agent doing work. In principle, the
bar never stops moving. However, one may verify that the total distance traveled is finite.
DEMO: Lenz's Law with Copper Pipe
A magnet is dropped down a conducting copper pipe and feels a resistive force. The
falling magnet induces a current in the copper pipe and, by Lenz's Law, the current
creates a magnetic field that opposes the changing field of the falling magnet. Thus, the
magnet is "repelled" and falls more slowly.
Example: A metal rod is forced to move with constant velocity along two parallel metal
rails, connected with a strip of metal at one end. A magnetic field of magnitude
B = 0.350 T points out of the page.
a) In which direction is the induced current?
PES 1120 Spring 2014, Spendier
Lecture 33/Page 4
b) If the rails are separated by L = 25.0 cm and the speed of the rod is 55.0 cm/s, what is
the magnitude of the induced emf?
c) If the rod has a resistance of 18.0  and the rails and connector have negligible
resistance, what is the current in the rod?
d) At what rate is energy being transferred to thermal energy?
PES 1120 Spring 2014, Spendier
Lecture 33/Page 5
Eddy current
We have seen that when a conducting loop moves through a magnetic field, current is
induced as the result of changing magnetic flux. What if instead of a loop, we move a
conducting plate through the uniform magnetic field? The induced current appears to be
circulating and is called an eddy current. A typical loop of eddy current is shown.
eddy current loop
The region of the conducting plate that is not in the field provides return conducting paths
for charges displaced in the region experiencing a magnetic field. The result is a
circulating eddy current in the conducting plate.
By Lenz’s Law
- eddy currents flow oppose the motion of metal
- eddy currents act as an effective brake to its motion
- mechanical work done is converted into heat inside the metal
DEMO:
The eddy current damping of a swinging plate (one solid metal, one with slits, one nonmetal).
Braking effect
- swinging a thick copper plate between poles of magnets
- changes in magnetic flux linked with the plate induce large eddy current to produce
braking effect on plate
- plate oscillation is heavily damped
There are important applications of eddy currents. For example, the currents can be used
to suppress unwanted mechanical oscillations. Another application is the magnetic
braking systems in high-speed transit cars.
PES 1120 Spring 2014, Spendier
Lecture 33/Page 6
When the magnetic flux created at the coil passes through the bottom of the pot placed on
the top plate, eddy currents are generated. This eddy current then is transformed into heat
by the resistant element of the pan and heats up the pan for cooking. This eddy current
then is transformed into heat by the resistant element of the pan and heats up the pan for
cooking. Induction cooktops use the high resistance of irom to their advantage,
converting a little current into a lot of heat -- highly conductive materials like copper or
aluminum would require much higher frequencies to make their small resistances add up
to much heat. With this in mind, some manufacturers have begun adding an iron plate to
the bottom of their nonferrous cookware to allow them to work with induction cooktops.
Induced Electric fields: We will skip this section