Transcript AP C UNIT 10 - student handout

AP C UNIT 11
ELECTROMAGNETIC
INDUCTION
Recall Electric Flux
E  EA cos
Welcome our newest concept…
Magnetic Flux
Faraday’s Observations,1830
• When a magnet moves
toward a loop of wire, the
ammeter
• When the magnet is held
stationary,
• When the magnet moves
away from the loop, the
ammeter
• If the loop is moved
instead of the magnet,
Experimental Conclusions
• A current is set up in the circuit as long as
there is ______________between the magnet
and the loop
– The same experimental results are found whether the loop
moves or the magnet moves
• The current is called ____________since
there is no power source.
• An EMF is actually
Faraday’s Law &
Electromagnetic Induction
• The instantaneous emf induced in a circuit
equals the time rate of change of magnetic flux
through the circuit.
EM induction refers to electricity deriving from
magnetism whereas electromagnetism is the
opposite.
Traffic light sensors
There is an
inductive loop
at intersection
of Ft Wash &
Susquehanna.
Electric Guitar
• A vibrating string induces
an emf in a coil
• A permanent magnet
inside the coil magnetizes
a portion of the string
nearest the coil
• As the string vibrates at
some frequency, its
magnetized segment
produces a changing flux
through the pickup coil
• The changing flux
produces an induced emf
that is fed to an amplifier
Apnea Monitor
• The coil of wire
attached to the chest
carries an alternating
current
• An induced emf
produced by the varying
field passes through a
pick up coil
• When breathing stops,
the pattern of induced
voltages stabilizes and
external monitors sound
an alert
Applications of Faraday’s Law –
Ground Fault Interrupters
• The ground fault interrupter (GFI) is a
safety device that protects against
electrical shock
– Wire 1 leads from the wall outlet to
the appliance
– Wire 2 leads from the appliance
back to the wall outlet
– The iron ring confines the magnetic
field, which is generally 0
– If a leakage occurs, the field
is no longer 0 and the
induced voltage triggers a
circuit breaker shutting off the
current
• Faraday's Law is the basic principle behind the
microphone. In a microphone there is a
diaphragm, around which a coil is wrapped,
which can move back and forth in response to
sound waves. A stationary bar magnet, placed
near the coil, induces current in the coil which
can then be transmitted (with amplification) to the
speaker.
Example
i
d
w
l
A long wire carries current i a distance ‘d’ from a
rectangular wire loop as shown above. Determine
an expression for the flux through loop.
Suppose that i(t) = 3t +1. Find
induced voltage in loop
Negative sign explained in
Faraday’s Law
The negative sign in Faraday’s Law is included
to indicate the polarity of the induced emf,
which is found by Lenz’ Law:
If i(t) = 3t +1, what was direction of
induced current in loop as t increases?
i
d
w
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Lenz’s Law examples
Determine
direction of
induced
current in loop
Determine
direction of
induced
current in loop
as magnet
approaches
loop area.
Determine direction of
induced current in loop
as loop gets smaller.
When switch is closed,
describe current flow in R
IRON
CORE
Describe current through R
when I goes to zero.
Which situation(s) cause(s)
induced current?
Example
A conducting rectangular loop
moves with constant velocity v in
the -y direction and a constant
current I flows in the +x direction
as shown
I
y
Iinduced
v
x
What is the direction of the induced current in the loop?
(a) ccw
(b) cw
(c) no induced current
Generator
A coil of wire
turns in a
magnetic field.
The flux in the
coil is constantly
changing,
generating an
emf in the coil.
Converts
mechanical energy
to electrical energy
If loop is made to rotate at constant rate ω
in uniform B, we have from Faraday’s Law:
(e)Motional EMF
• A straight conductor of
length ℓ moves
perpendicularly with
constant velocity
through a uniform field
ℓ
A conducting bar is placed across conducting path and
pulled to right with speed v as shown.
As bar moves, a change in flux occurs which induces CCW
current.
Also, a magnetic force on bar arises which acts as a
resistance to the motion of the bar as it is pulled to
the right
Lenz’ Law Revisited,
Conservation of Energy
Consequence
• Assume the induced current is
clockwise instead…
– The magnetic force on the bar
would be to the right
– The force would cause an
acceleration and the velocity
would increase
– This would cause the flux to
increase and the current to
increase and the velocity to
increase…
example
A metal rod of mass 0.22kg lies across two parallel conducting
rails which sits on a tabletop as shown. The rod and rails have
negligible resistance but significant friction where uk=0.20. A field
of 0.80T points into page. A string pulls the rod to right at a
constant speed of 1.8m/s.
a) Calculate the force needed to pull
rod at constant speed.
b) Calculate the energy dissipated in the resistor in 2.0s.
c) Calculate the work done by string in 2.0s.
Example
A conducting rod with mass m and length L moves on two
frictionless parallel rails in the presence of a uniform magnetic
field. The bar is given an initial velocity vi at time t=0. Calculate
the velocity of the bar as a function of time. Bar will slow down
due to resistive force.
The magnitude of the
magnetic force is given by
Now, using Newton's second
law, we can write the net force
on the conducting rod as
Eddy currents
Eddy currents are small circular or swirling currents that
arise in conductors like a sheet of metal.
Eddy currents lead to heat being generated in the
conductor
This is the basic principle behind induction stoves. Safe to
touch unless you are metallic. Eddy currents are
established in cookware causing metal to heat up.
Magnetic Braking
Rollercoaster brakes
Analog speedometers
Metal Detectors
Transformers
A transformer is a device used to change the voltage in a
circuit. AC currents must be used.
75,000 V
in the
power
lines
120 V in
your
house
Electrical Power Transmission
• When transmitting electric power over long distances,
it is most economical to use high voltage and low
current, which minimizes I2R power losses.
Induced E-fields
Ampere's Law has shown us how currents, moving electric
charges, can create magnetic fields.
Faraday's Law has shown us how changing magnetic fields
can induce an emf in a closed loop.
Consider a loop of wire outside of a solenoid. Current
is flowing through solenoid from back to front where
B-field from coils is into page.
Side View
Front View
If the current is dereased gradually, the magnetic field
in the solenoid's core decreases and an emf & Iinduced
will be induced in the wire loop (LENZ LAW).
The force that pushes
the charges around the
wire is F = qE, where E
is the induced electric
field.
E-fields produced by either static charge or induction both
exert forces on charged particles, however, there is an
important difference.
Static E
Induced E
Recall we previously learned that
the potential difference between 2
points in a static field is given by:
 
Vab   E  d 
b
a
However, in a changing B-field case…
As charge makes journey around
closed loop, it must be experiencing
an emf, however, to interpret that as a
changing potential, it doesn’t make
sense. Why?
B
time
As magnetic field increases in time through loop, an
electric field is generated
Consider B-field between pole of electromagnet. Assume
B to be uniform at any instant over a circular radius R. The
current in the windings of the electromagnet increase with
time. Beyond the circular region (r > R), assume B=0. Find
E at any distance r from the center.
S
N
side view
top view, looking down on N pole
E
r
r
E
EXAMPLE: If B = 0.75T and B is decreasing at a rate
of -0.035T/s.
A) What is the shape of the induced
E-field on the conducting ring?
B) What are the magnitude and
direction of this field at any point
on the conducting ring with radius
0.1m?
c) Determine the current in the ring if
the resistance is 4.0Ω.
d) What is the EMF between a and b?
e) If the ring is cut at some point and the ends are
separated slightly, what will be the EMF between the
ends?
Inductance
Similar to the idea of capacitance (holding onto
charge), inductance deals with how well an inductor
holds onto a magnetic field.
Energy Density of solenoid
RL CIRCUIT
Close switch
As soon as current
appears at the first coil of
the inductor, a change in
magnetic flux is created,
and therefore an EMF.
This EMF pushes opposite
to the EMF causing the
flux in the first place,
according to Lenz's Law
We expect that the current in the inductor, and hence in the
entire circuit, must increase over time until it reaches its
maximum value of imax = ε / R where ε is the voltage provided
by the EMF source. Resistance of inductor goes to zero
whereas in capacitor we say it becomes infinite R.
di/dt is positive since current is increasing, however, the EMF
induced across the inductor is negative since it is pushing
current in opposite direction to oppose change, therefore a
minus sign is added
Going CW around circuit:
Current versus time in an R-L circuit
63%Imax
Flipping switch back
causes decrease in
current. Similar analysis
yields:
37%Imax
Example: Close S1.
After a long time:
a) Find value of current in inductor
at moment S1 is closed.
b) Find value of current in inductor
after long time.
Simultaneously open
S1 and close S2.
20V
100Ω
0.10H
At that moment, what is
current in inductor?
Find current at t=2.0x10-3s
in inductor.
Example 2
40V
Determine the work done by the
battery from t = 0s to 0.0165s.
160Ω
2.2H
LC Circuits
Analogous to an oscillating mass on a spring
The Oscillation Cycle
Prior to discharge, all energy
resides in E-field of capacitor.
When capacitor discharges,
current flows CW and gets
larger, B-field emerges in
inductor resisting change.
At t=T/4, capacitor has zero
charge, current has max
value, B-field is max with all
energy now residing in
inductor’s field.
Eventually charge starts to
accumulate on capacitor,
current dies, B-field
decreases to zero. At t=T/2,
all energy is back in E-field
with polarity of capacitor
reversed.
Process repeats itself
returning to state it was at
t = T/4. At t = 3T/4 energy is
back in B-field. This is called
electrical oscillation.
LC Circuit Analysis:
What differentiates an LC circuit from the
RC or RL circuit is the fact that current in
both the RC and RL circuits changes
exponentially towards a steady state.
However, in the LC circuit, the current
oscillates, never reaching steady state.
L C Circuit Analogy to SHM
Current and
Charge variations
with time for LC
circuit
System oscillates
according to:
Just like resonance in SHM, LC circuit has electrical
resonance. Your radio is tuned in to a resonant
frequency by changing the capacitance
Maxwell’s Equations
We now gather all of the governing equations together
The net magnetic flux is zero
through a closed surface. B field
lines cannot begin or end at any
point. If they did, monopoles
would exist
Maxwell retooled
Ampere’s law which
is only good for static
case, not oscillating
situations
Maxwell’s Equations
These four equations describe all of classical electric
and magnetic phenomena
Maxwell’s own contribution is just the last term of the
last equation but realizing the necessity of that term
had dramatic consequences. It made evident for the
first time that varying electric and magnetic fields could
feed off each other & these fields could propagate
indefinitely through space, far from the varying charges
and currents where they originated. Previously, the
fields had been envisioned as tethered to the charges
and currents giving rise to them. Maxwell’s new term
(he called it the displacement current) freed them to
move through space in a self-sustaining fashion, and
even predicted their velocity it was the velocity of light!
As the E-field starts to change, that in turn induces a B-field
which in turns induces E and so on.