chapter 23
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Chapter 23. Faraday’s Law and Induction. Michael Faraday. 1791 – 1867 Great experimental physicist Contributions to early electricity include Invention of motor, generator, and transformer Electromagnetic induction Laws of electrolysis. Induction. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 23
Faraday’s Law
and
Induction
Michael Faraday 1791 – 1867 Great experimental
physicist Contributions to early
electricity include Invention of motor,
generator, and transformer
Electromagnetic induction
Laws of electrolysis
Induction An induced current is produced by a
changing magnetic field There is an induced emf associated with
the induced current A current can be produced without a
battery present in the circuit Faraday’s Law of Induction describes
the induced emf
EMF Produced by a Changing Magnetic Field, 1
A loop of wire is connected to a sensitive ammeter When a magnet is moved toward the loop, the
ammeter deflects The deflection indicates a current induced in the wire
EMF Produced by a Changing Magnetic Field, 2
When the magnet is held stationary, there is no deflection of the ammeter
Therefore, there is no induced current Even though the magnet is inside the loop
EMF Produced by a Changing Magnetic Field, 3
The magnet is moved away from the loop The ammeter deflects in the opposite
direction
EMF Produced by a Changing Magnetic Field, Summary The ammeter deflects when the magnet is
moving toward or away from the loop The ammeter also deflects when the loop is
moved toward or away from the magnet An electric current is set up in the coil as long
as relative motion occurs between the magnet and the coil This is the induced current that is produced by an
induced emf
Faraday’s Experiment – Set Up
A primary coil is connected to a switch and a battery
The wire is wrapped around an iron ring
A secondary coil is also wrapped around the iron ring
There is no battery present in the secondary coil
The secondary coil is not electrically connected to the primary coil
Faraday’s Experiment – Findings At the instant the switch is closed, the
galvanometer (ammeter) needle deflects in one direction and then returns to zero
When the switch is opened, the galvanometer needle deflects in the opposite direction and then returns to zero
The galvanometer reads zero when there is a steady current or when there is no current in the primary circuit
Faraday’s Experiment – Conclusions
An electric current can be produced by a time-varying magnetic field This would be the current in the secondary circuit of this
experimental set-up The induced current exists only for a short time
while the magnetic field is changing This is generally expressed as: an induced emf is
produced in the secondary circuit by the changing magnetic field The actual existence of the magnetic field is not sufficient
to produce the induced emf, the field must be changing
Magnetic Flux To express
Faraday’s finding mathematically, the magnetic flux is used
The flux depends on the magnetic field and the area:
B d B A
Flux and Induced emf An emf is induced in a circuit when the
magnetic flux through the surface bounded by the circuit changes with time This summarizes Faraday’s experimental
results
Faraday’s Law – Statements Faraday’s Law of Induction states that
the emf induced in a circuit is equal to the time rate of change of the magnetic flux through the circuit
Mathematically,
Bd
dt
Faraday’s Law – Statements, cont If the circuit consists of N identical and
concentric loops, and if the field lines pass through all loops, the induced emf becomes
The loops are in series, so the emfs in the individual loops add to give the total emf
BdN
dt
Faraday’s Law – Example Assume a loop
enclosing an area A lies in a uniform magnetic field
The magnetic flux through the loop is B = B A cos
The induced emf is cos
dBA
dt
Ways of Inducing an emf The magnitude of the field can change
with time The area enclosed by the loop can
change with time The angle between the field and the
normal to the loop can change with time Any combination of the above can occur
Applications of Faraday’s Law – Pickup Coil
The pickup coil of an electric guitar uses Faraday’s Law
The coil is placed near the vibrating string and causes a portion of the string to become magnetized
When the string vibrates at the some frequency, the magnetized segment produces a changing flux through the coil
The induced emf is fed to an amplifier
Motional emf A motional emf is one
induced in a conductor moving through a magnetic field
The electrons in the conductor experience a force that is directed along l
B q F v B
Motional emf, cont Under the influence of the force, the electrons
move to the lower end of the conductor and accumulate there
As a result of the charge separation, an electric field is produced inside the conductor
The charges accumulate at both ends of the conductor until they are in equilibrium with regard to the electric and magnetic forces
Motional emf, final For equilibrium, q E = q v B or E = v B A potential difference is maintained
between the ends of the conductor as long as the conductor continues to move through the uniform magnetic field
If the direction of the motion is reversed, the polarity of the potential difference is also reversed
Sliding Conducting Bar
A bar moving through a uniform field and the equivalent circuit diagram
Assume the bar has zero resistance The work done by the applied force appears as
internal energy in the resistor R
Sliding Conducting Bar, cont The induced emf is
Since the resistance in the circuit is R, the current is
Bd dxB B v
dt dt
B vI
R R
Sliding Conducting Bar, Energy Considerations The applied force does work on the
conducting bar This moves the charges through a magnetic
field The change in energy of the system during
some time interval must be equal to the transfer of energy into the system by work
The power input is equal to the rate at which energy is delivered to the resistor
2
appF v I B vR
AC Generators Electric generators take
in energy by work and transfer it out by electrical transmission
The AC generator consists of a loop of wire rotated by some external means in a magnetic field
Induced emf In an AC Generator The induced emf in
the loops is
This is sinusoidal, with max = N A B
sin
BdN
dtNAB t
Lenz’ Law Faraday’s Law indicates the induced
emf and the change in flux have opposite algebraic signs
This has a physical interpretation that has come to be known as Lenz’ Law
It was developed by a German physicist, Heinrich Lenz
Lenz’ Law, cont Lenz’ Law states the polarity of the
induced emf in a loop is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop
The induced current tends to keep the original magnetic flux through the circuit from changing
Lenz’ Law – Example 1
When the magnet is moved toward the stationary loop, a current is induced as shown in a
This induced current produces its own magnetic field that is directed as shown in b to counteract the increasing external flux
Lenz’ Law – Example 2
When the magnet is moved away the stationary loop, a current is induced as shown in c
This induced current produces its own magnetic field that is directed as shown in d to counteract the decreasing external flux
Induced emf and Electric Fields An electric field is created in the conductor
as a result of the changing magnetic flux Even in the absence of a conducting loop,
a changing magnetic field will generate an electric field in empty space
This induced electric field has different properties than a field produced by stationary charges
Induced emf and Electric Fields, cont The emf for any closed path can be
expressed as the line integral of over the path
Faraday’s Law can be written in a general form
Bdd
dt
E s
dE s
Induced emf and Electric Fields, final The induced electric field is a
nonconservative field that is generated by a changing magnetic field
The field cannot be an electrostatic field because if the field were electrostatic, and hence conservative, the line integral would be zero and it isn’t
Self-Induction When the switch is
closed, the current does not immediately reach its maximum value
Faraday’s Law can be used to describe the effect
Self-Induction, 2 As the current increases with time, the
magnetic flux through the circuit loop due to this current also increases with time
The corresponding flux due to this current also increases
This increasing flux creates an induced emf in the circuit
Self-Inductance, 3 The direction of the induced emf is such that
it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field
The direction of the induced emf is opposite the direction of the emf of the battery Sometimes called a back emf
This results in a gradual increase in the current to its final equilibrium value
Self-Induction, 4 This effect is called self-inductance
Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself
The emf L is called a self-induced emf
Self-Inductance, Equations An induced emf is always proportional
to the time rate of change of the current
L is a constant of proportionality called the inductance of the coil It depends on the geometry of the coil and
other physical characteristics
L
dIL
dt
Inductance of a Coil A closely spaced coil of N turns carrying
current I has an inductance of
The inductance is a measure of the opposition to a change in current Compared to resistance which was
opposition to the current
BNL
I dI dt
Inductance Units The SI unit of inductance is a Henry (H)
Named for Joseph Henry
AsV
1H1
Joseph Henry 1797 – 1878 Improved the design of
the electromagnet Constructed one of the
first motors Discovered the
phenomena of self-inductance
Inductance of a Solenoid Assume a uniformly wound solenoid
having N turns and length l Assume l is much greater than the radius
of the solenoid The interior magnetic field is
o o
NB nI I
Inductance of a Solenoid, cont The magnetic flux through each turn is
Therefore, the inductance is
This shows that L depends on the geometry of the object
B o
NABA I
2oB N AN
LI
RL Circuit, Introduction A circuit element that has a large self-
inductance is called an inductor The circuit symbol is We assume the self-inductance of the
rest of the circuit is negligible compared to the inductor However, even without a coil, a circuit will
have some self-inductance
RL Circuit, Analysis An RL circuit contains
an inductor and a resistor
When the switch is closed (at time t=0), the current begins to increase
At the same time, a back emf is induced in the inductor that opposes the original increasing current
RL Circuit, Analysis, cont Applying Kirchhoff’s Loop Rule to the
previous circuit gives
Looking at the current, we find
0dI
IR Ldt
1 Rt LI eR
RL Circuit, Analysis, Final The inductor affects the current
exponentially The current does not instantly increase
to its final equilibrium value If there is no inductor, the exponential
term goes to zero and the current would instantaneously reach its maximum value as expected
RL Circuit, Time Constant The expression for the current can also be
expressed in terms of the time constant, , of the circuit
where = L / R Physically, is the time required for the
current to reach 63.2% of its maximum value
1 tI t eR
RL Circuit, Current-Time Graph, 1
The equilibrium value of the current is /R and is reached as t approaches infinity
The current initially increases very rapidly
The current then gradually approaches the equilibrium value
RL Circuit, Current-Time Graph, 2 The time rate of
change of the current is a maximum at t = 0
It falls off exponentially as t approaches infinity
In general, tdI
edt L
Energy in a Magnetic Field In a circuit with an inductor, the battery
must supply more energy than in a circuit without an inductor
Part of the energy supplied by the battery appears as internal energy in the resistor
The remaining energy is stored in the magnetic field of the inductor
Energy in a Magnetic Field, cont Looking at this energy (in terms of rate)
I is the rate at which energy is being supplied by the battery
I2R is the rate at which the energy is being delivered to the resistor
Therefore, LI dI/dt must be the rate at which the energy is being delivered to the inductor
2 dII I R LI
dt
Energy in a Magnetic Field, final Let U denote the energy stored in the
inductor at any time The rate at which the energy is stored is
To find the total energy, integrate and UB = ½ L I2
BdU dILI
dt dt
Energy Density of a Magnetic Field Given U = ½ L I2,
Since Al is the volume of the solenoid, the magnetic energy density, uB is
This applies to any region in which a magnetic field exists not just the solenoid
2 221
2 2oo o
B BU n A A
n
2
2Bo
U Bu
V
Inductance Example – Coaxial Cable Calculate L and
energy for the cable The total flux is
Therefore, L is
The total energy is
ln2 2
bo o
B a
I I bBdA dr
r a
ln2
oB bL
I a
221
ln2 4
o I bU LI
a
Magnetic Levitation – Repulsive Model A second major model for magnetic
levitation is the EDS (electrodynamic system) model
The system uses superconducting magnets
This results in improved energy effieciency
Magnetic Levitation – Repulsive Model, 2 The vehicle carries a magnet As the magnet passes over a metal plate that
runs along the center of the track, currents are induced in the plate
The result is a repulsive force This force tends to lift the vehicle
There is a large amount of metal required Makes it very expensive
Japan’s Maglev Vehicle The current is
induced by magnets passing by coils located on the side of the railway chamber
EDS Advantages Includes a natural stabilizing feature
If the vehicle drops, the repulsion becomes stronger, pushing the vehicle back up
If the vehicle rises, the force decreases and it drops back down
Larger separation than EMS About 10 cm compared to 10 mm
EDS Disadvantages Levitation only exists while the train is in
motion Depends on a change in the magnetic flux Must include landing wheels for stopping and
starting The induced currents produce a drag force as
well as a lift force High speeds minimize the drag Significant drag at low speeds must be overcome
every time the vehicle starts up