INDUCTANCE

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• INDUCTANCE
• AND
• INDUCTORS

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Inductor is a device for storing energy in a
magnetic field. eg coil or solenoid serves the
purpose of inductor
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Whats the point of Faraday?
Take a circuit Pass a current through it Magnetic
field is created (Ampere) Put another circuit
nearby If the induced magnetic field changes
with time, Faradays Lawcauses an emf and
current to appear This is Magnetic Inductance
orMutual Inductance between twocircuits
expresses the strength withwhich they couple
inductively.
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Take a circuit Pass a current through it Magnetic
field is created (Ampere) This field passes
through the circuit that created it If the
magnetic field is time-varying,it induces an emf
and thusa current in the circuit. This emf
opposes the change inmagnetic field that caused
it and thus induces a current in the
oppositedirection from the current that
causedthe magnetic field in the first
place. This is called (self-) inductance
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Self-Induction

• When a current is induced by a changing magnetic
  field, that current itself produces its own
  magnetic field. This effect is called
  self-induction.

Primary Magnetic Field
Current
Self-Induced Magnetic Field
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Self-Inductance
Magnetic flux due to magnetic field B produced
due to a current in conductor is given by
We know induced emf due to changing flux is given
by
When current change, flux will also change. Let
in a single coil current i is changing, The
change in flux will be proportional to change in
current.
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or
or we know
so
Here L is known as coefficient of self inductance
or self inductance of the coil.
L is a constant, depends on the geometry of the
circuit and the permeability of the medium in
which coil is placed.
L can also be defined as
Inductance L of an inductor is equal to the ratio
of its total magnetic flux linking to the current
i through the inductor.
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Self-Inductance of two coaxial cylinders
The field B due to current i in the inner
cylinder at a distance r
The flux is given by
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And we know,
This is the self inductance of two coaxial
cylinders per unit length.
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Self-Inductance of Solenoid
Let the length of solenoid with an air core is
l Total number of turns N
Let current flowing through it is i
The magnetic field inside it will be
If area of the each turn is A
So magnetic flux through each turn will be
Total flux through the solenoid (due to N turns)
When current i varies, the flux changes and
induces emf
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Therefore
If number of turns per unit length is n i.e.
nN/l
Then,
If the solenoid is wound over a core of constant
permeability ?
Then,
Where
Where is relative permeability
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Self-Inductance of toroidal coil
Let the think a toroid with an air core with mean
radius R Total number of turns N
Let current flowing through it is i
The magnetic field is given by
So total magnetic flux through toroid
Where A is the cross sectional area of each turn
of the coil.
If current is changing, then
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Therefore
Foe a toroid wound on a core of constant
permeability ?
Where
Where is relative permeability
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Mutual-Inductance
Let there are two fixed coils.
Let i1 and i2 currents are flowing in 1st
(primary) coil and in 2nd (secondary) coil
respectively.
Current i1 in 1st coil will produce magnetic
field B1 when current is constant.
Magnetic flux due to magnetic field B1 through
the secondary coil
This flux depends on B1 and B1 depends on i1 so
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Suppose is changes with time, the flux will also
change and it will produce an induced emf in
secondary coil According to Faradays law
The ve sign indicated the direction of induced
emf.
The value of M depends on the geometry of coils.
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Let us now pass the current through the secondary
coil and calculate the induced emf in the primary
coil in the same way.
Unit of mutual inductance is henrys or
(newtons.meters/amperes2).
For any two coils or circuits
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Mutual-Inductance between two circuits
All conditions are same as before
We know and so
Where A1 is the vector potential at the circuit 2
due to current in circuit 1.
Using the Stokes theorem
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We know
So
or
or
This eq is known as Neumann equation.
where
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Proceeding in the same manner we can get
where
and
So
This relation is called theorem of reciprocity.
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Mutual-Inductance of concentric solenoids
Let first solenoid has np number of turns per
unit length and second solenoid has ns number of
turns per unit length
Magnetic field due to current i in the first
solenoid
Magnetic flux B ? area of cross section
This flux is linked with each turn of the
secondary coil, therefore total flux linked with
ns turns
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Induced emf
so