23.5 self-induction
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23.5 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,. As the current increases with time, the magnetic flux through the circuit loop also increases with time - PowerPoint PPT PresentationTRANSCRIPT
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23.5 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
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Self-Induction, As the current increases with time, the magnetic
flux through the circuit loop also increases with time
This increasing flux creates an induced emf in the circuit
The direction of the induced emf is opposite to that of the emf of the battery
The induced emf causes a current which would establish a magnetic field opposing the change in the original magnetic field
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Equation for Self-Induction This effect is called self-inductance and the
self-induced emf Lis 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
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Inductance Units The SI unit of inductance is a Henry (H)
AsV
1H1
Named for Joseph Henry 1797 – 1878 Improved the design of the
electromagnet Constructed one of the first motors Discovered the phenomena of self-
inductance
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Inductance of a Solenoid having N turns and Length l
B o
NABA I
o o
NB nI I
The interior magnetic field is
The magnetic flux through each turn is
The inductance is
This shows that L depends on the geometry of the object
2oB N AN
LI
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23.6 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
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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
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The current in RL Circuit Applying Kirchhoff’s Loop Rule to the
previous circuit gives
The current
where = L / R is the time required for the current to reach 63.2% of its maximum value
0dI
IR Ldt
1 tI t eR
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RL Circuit, Current-Time Graph
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
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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
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Open the RL Circuit, Current-Time Graph
The time rate of change of the current is a maximum at t = 0
It falls off exponentially as t approaches infinity
In general,
tdIe
dt L
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23.7 Energy stored 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
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Energy in a Magnetic Field 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
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Energy in a Magnetic Field 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
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Energy Density in 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 in the solenoid
2 221
2 2oo o
B BU n A A
n
2
2Bo
U Bu
V
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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
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23.8 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
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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
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Japan’s Maglev Vehicle The current is
induced by magnets passing by coils located on the side of the railway chamber
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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
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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
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Exercises of Chapter 23 5, 9, 12, 21, 25, 32, 35, 39, 42, 47, 52,
59, 65, 67