Download self-inductance

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 eLis always proportional to
the time rate of change of the current
dI
e L  L
dt

L is a constant of proportionality called the
inductance of the coil

It depends on the geometry of the coil and other
physical characteristics
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Inductance Units

The SI unit of inductance is a Henry (H)
V s
1H  1
A





Named for Joseph Henry
1797 – 1878
Improved the design of the
electromagnet
Constructed one of the first motors
Discovered the phenomena of selfinductance
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Inductance of a Solenoid having
N turns and Length l
The interior magnetic field is
N
B  o nI  o
I
The magnetic flux through each turn is
 B  BA  o
The inductance is
NA
I
N  B o N 2 A
L

I
This shows that L depends on the geometry of
the object
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23.6 RL Circuit, Introduction



A circuit element that has a large selfinductance 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
6
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
dI
e  IR  L  0
dt
The current
I t  

e

R
1  e t t

where t = L / R is the time required for the
current to reach 63.2% of its maximum
value
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RL Circuit,
Current-Time Graph



The equilibrium value
of the current is e/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,
dI e t t
 e
dt L
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12
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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)
dI
2
Ie  I R  LI
dt



Ie 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
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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
dUB
dI
 LI
dt
dt

To find the total energy, integrate and
UB = ½ L I2
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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
U
B2
uB  
V 2 o
This applies to any region in which a
magnetic field exists

2
2


1
B
B
U  o n 2 A 
A
 
2
2o
 o n 

not just in the solenoid
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Inductance Example –
Coaxial Cable


Calculate L and
energy for the cable
The total flux is
 B   BdA  
b
a
o I
 I b
dr  o ln  
2 r
2  a 

Therefore, L is

The total energy is
 B o
b
L

ln  
I
2  a 
1 2 o I 2  b 
U  LI 
ln  
2
4
a
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23.8 Magnetic Levitation –
Repulsive Model


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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

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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

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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
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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


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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

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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
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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
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5, 9, 12, 21, 25, 32, 35, 39, 42, 47, 52,
59, 65, 67
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