L101

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L10-1
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Chapter 29 Electromagnetic Induction
Goals
- Electromagnetic brings electric and magnetic
together - Learn how a changing magnetic flux
creates an induced voltage. Faradays Law - find
the direction of the induced voltage. Lenzs
Law - applications inductors and circuits
This material is particularly new and interesting
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Meta-comments
Heading towards closure in EM - static
charges and static E fields - constant currents
and static B fields - now time-varying
fields Very interesting/unexpected results -
B fields generating E fields and E generating B
- leads to waves of E and B fields -
light Leads to complete set of Maxwells
Equations
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Three Experiments
Experiment one (top) Move a conducting circuit
near a stationary magnet.
Current flows due to force on electrons F qv?B
Not new , but interesting.
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Experiment two (center) Move a magnet near a
stationary conducting circuit.
The electrons are not moving initially, yet they
feel a force.
- new physical law!
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Experiment three (bottom) Dont move anything.
Instead, change the current in a coil near a
conducting circuit.
Our new physical law says a changing magnetic
field creates an electric field.
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Faradays Law of Induction
The results of all three experiments may be
expressed as
Faradays Law of Induction The induced voltage
(EMF) in a circuit is equal to the time rate of
change of the magnetic flux through the circuit
Vind e -dFB/dt.
? The minus sign is traditionally put in to
express Lenzs law, which we will discuss soon.
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Faradays Law of Induction
The results of all three experiments may be
expressed as
Faradays Law of Induction The induced voltage
(EMF) in a circuit is equal to the time rate of
change of the magnetic flux through the circuit
Vind ? -dFB/dt.
? The text uses ? and EMF, or we can say Vind
and induced voltage. ? They are measured
in volts . ? They can be used as the
voltage in Ohms law, I ?/R Vind/R
? The minus sign is traditionally put in to
express Lenzs law, which we will discuss soon.
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Magnetic Flux
The magnetic flux FB measures how many lines of
B pass through the circuit.
A is the area of the circuit ? is the angle
between the field and the normal to the circuit.
The units of magnetic flux are T-m2 Wb (weber).
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Faradays Law Vind e -d F B/dt
Faradays law says that we are interested in
the CHANGE of the magnetic flux
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The three experiments are all explained by
Faradays law Vind e -d F B/dt
Note If N loops Vind e -Nd F B /dt
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A loop of wire of area 2m2 is placed with its
plane perpendicular to a magnetic field of 3T.
If in 2 seconds the loop is flipped over one half
rotation so that the plane is again
perpendicular to the magnetic field, what is the
induced EMF? 1) 2V 2) 3V 3) 6V 4) 12V 5) 24 V
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FB BA cos?
Vind e -NdFB/dt

• A flat loop of wire consisting of a single turn
  of area 8.00 cm2 is perpendicular to a magnetic
  field that increases uniformly in magnitude from
  0.500 T to 2.500 T in 2.00 sec.
• What is the resulting induced current if the loop
  has a resistance of 2.00 O?

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?B BA cos?
Vind ? -dFB/dt

• A flat loop of wire consisting of a single turn
  of area 8.00 cm2 is perpendicular to a magnetic
  field that increases uniformly in magnitude from
  0.500 T to 2.500 T in 2.00 sec.
• What is the resulting induced current if the loop
  has a resistance of 2.00 O?

Solution. Perpendicular ? FB BA. Starting flux
FB1 (0.5 T)(8e-4 m2) 4e-4 Wb. Ending flux
FB2 (2.5 T)(8e-4 m2) 2e-3 Wb. Induced voltage
Vind (FB2-FB1)/(2.00 sec) 8.0e-4 V. (we
dont care about sign for this problem) Induced
current I Vind /R (8.0e-4 V)/(2.00 O)
4.00e-4 A.
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Electric Guitar
Permanent magnet magnetizes wire Vibrating wire
creates varying flux in coil Induced EMF is
signal - amplified
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Lenzs Law
The direction of induced current is to oppose the
change in flux
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Other examples
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• A square loop of wire lies in the
• plane of the page. A decreasing magnetic
• field is directed into the page. The
• direction of the induced current is
• 1. counterclockwise
• 2. clockwise
• 3. depends upon whether B is decreasing at a
  constant rate
• 4. clockwise in two sides of the loop and
  counterclockwise in the other two
• 5. impossible to determine from the information
  given

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• As an externally generated magnetic field
  through a certain conducting loop increases in
  magnitude, the field produced at points inside
  the loop by the current induced in the loop must
  be
• 1. increasing in magnitude
• 2. decreasing in magnitude
• 3. in the same direction as the magnetic field
• 4. directed opposite to the applied field
• 5. perpendicular to the applied field

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• A bar magnet is dropped from above and falls
  through a
• loop of wire as shown. The north pole of the
  magnet
• points downward as the magnet falls. Which of
  the statements
• is correct.
• The current in the loop always flows in a
• clockwise direction.
• 2. The current in the loop always flows in a
• counter-clockwise
• direction, then in a counterclockwise
  direction.
• 3.The current in the loop flows first in a
• counterclockwise
• direction, then in a clockwise direction.

4. No current flows in the loop because both ends
of the magnet move through the loop.
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A current-carrying wire is pulled away from a
conducting loop in the direction shown. As the
wire is moving, what is the current in the loop?
1. There is a clockwise current around the
loop. 2. There is a counterclockwise current
around the loop. 3. There is no current around
the loop.
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Thinking about forces
A conducting loop is halfway into a magnetic
field. Suppose the magnetic field begins to
increase rapidly in strength. What happens to the
loop?
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• A wire loop of mass m, width w, and length
  has resistance R. The rectangle falls through a
  magnetic field as shown, with the top not yet in
  the field. The terminal velocity is vt. In
  terms of these quantities, find the magnetic
  field strength B.

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• A wire loop of mass m, width w, and length ?
  has resistance R. The rectangle falls through a
  magnetic field as shown, with the top not yet in
  the field. The terminal velocity is vt. In
  terms of these quantities, find the magnetic
  field strength B.

Solution. Induced voltage is Vind dFB/dt
Bwvt. Induced current is Iind Vind/R, and force
on the loop is F IindwB (Bw)2vt/R. Set F
equal to the weight of the loop mg, and solve for
B.
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1. F2 F4 gt F1 F3 2. F3 gt F2 F4 gt F1 3.
F3 gt F4 gt F2 gt F1 4. F4 gt F2 gt F1 F3 5. F4 gt
F3 gt F2 gt F1
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Eddy Currents
Move a conducting circuit in a static magnetic
field, or move a magnet near a conducting circuit.
In either case currents (eddy currents) will be
induced in the circuit, and energy will be
dissipated in the circuit (turned into heat), at
the rate P Iind2R.
The energy must come from somewhere! There must
be a drag force opposing the motion, due to the
induced currents.
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What is really going on with Faradays law?
currents driven by E field JsE gt changing B
field induces an E field!
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The wire itself is not the important part - E
fields generated everywhere
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A more general expression of Faradays Law
Faradays Law is Vind -dFB/dt
Faradays Law can be expressed more fundamentally
by Eds -dFB/dt This relates the line integral
of E around any arbitrary loop to the rate of
change of flux of B through that loop.
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Faradays Law E ds -dFB/dt
Unlike the electric fields produced by static
charges, the induced electric field lines can
form CLOSED LOOPS.
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AC flows in a loop around an iron core, creating
a large alternating B field in the iron. A light
bulb is connected first to a loop of wire with a
few turns around the iron core, then to a loop
with many more turns. The light bulb
1
2
B
B

• shines more brightly with loop 1
• shines more brightly with loop 2
• shines equally bright with both
• jumps up and hovers in midair
• does nothing (lame demonstration)

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E
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E

d?E/dt
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A capacitor with circular plates is being
charged. The magnetic field at b is ________ that
at a the magnetic field at d is ________ that at
c.

• larger than larger than
• smaller than smaller than
• the same as the same as
• larger than the same as
• larger than smaller than
• smaller than the same as
• smaller than larger than

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As the capacitor is charged with a constant
current I, at point P (which lies in the z
direction relative to the wire axis) there is a

• constant electric field
• changing electric field
• constant magnetic field
• changing magnetic field
• both (1) and (3) above
• both (1) and (4) above
• both (2) and (3) above
• both (2) and (4) above
• none of the above

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Maxwells Equations of Electromagnetism
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As the capacitor is charged with a constant
current I, at point P (which lies in the z
direction relative to the wire axis) there is a

• constant electric field
• changing electric field
• constant magnetic field
• changing magnetic field
• both (1) and (3) above
• both (1) and (4) above
• both (2) and (3) above
• both (2) and (4) above
• none of the above