### Induction and Inductance

```Induction and Inductance
• When a bar magnet moves
towards the loop, there is a
deflection on the ammeter and
when the magnet is moved away,
there is also a deflection in the
reverse direction although no
battery is connected. This noticed
current is called induced current (
it is induced by the relative motion
between the loop and the magnet)
The work done per unit charge to produce(induce)
this current is called induced current. An emf is
induced in the loop when the number of magnetic
field lines that pass through the loop is changing.
We therefore define the magnetic flux density through
the loop as
If the magnetic field is perpendicular to the plane
of the loop then we can write
And if the magnetic field is uniform then
The unit of magnetic flux is tesla-square meter called
Weber(Wb)
The magnitude of the emf ε induced in a conducting loop is
equal to the rate at which the magnetic flux ,
through that loop changes with time
The –ve sign only indicates that the emf usually opposes
the magnetic flux change. If the loop comprises of a coil of
N turns then
Note that the magnetic flux through a coil
depends on
 The magnitude B of the magnetic field within the coil
 The total area of the coil or the portion that lies within
the magnetic field
 The angle between the direction of the magnetic field
and the plane of the coil
 The flux through each turn of coil depends on the area
A and orientation of that turn in the solenoid’s
magnetic field . Because is uniform and directed
perpendicular to area A.
 The magnitude B of the magnetic field in the interior of
a solenoid’s current i and its number n turns per unit
length
LENZ’S LAW
• Heinrich Friedrich lenz devised a rule for
determining the direction of an induced
current in a loop.
• An induced current has a direction such that
the magnetic field due to the current opposes
the change in the magnetic flux that induces
the current
INDUCED ELECTRIC FIELD
An emf is induced by a changing magnetic flux
even if the loop through which the flux is
changing is not a physical conductor but an
imaginary line. The changing magnetic field
induces an electric field E at every point of
such a loop. The induced emf is related to by
Where the integraton is taken around the loop. In most
general form, Faraday’s law can be written as
a changing magnetic field induces
an electric field
INDUCTORS AND INDUCTANCE
• Just as capacitors can be used to produce
electric field, an inductor
can be
used to produce a desired magnetic field.
• The inductance of an inductor is given as
(Tesla sq meter per Ampere)
where N=no of turns, i= current in the
windings.
• The product
= magnetic flux leakage
Hence Inductance per unit length is given by
inductance thus depends only on the geometry of
the devise.
SELF INDUCTANCE
• When two coils (inductors) are beside each other, a
current I in one coil produces a magnetic flux
through the second coil. If we change this flux by
changing the current, an induced emf appears in
the second coil according to Faraday’s law.
• “an induced emf εL appears in any coil in which the
current is changing”
• From these eqns (
) we have
(self induced emf)
in any inductor (coil,solenoid or a toroid) a self
induced emf appears whenever the current
changes with time.
RL circuit
i
x
y
z
*
(RL circuit eqn)
• For a circuit containing resistor,
inductor and emf source, the loop
rule is applied.
• From x to y in the directon of
current, there is a voltage drop
across R is
• From y to z, there is a self induced
emf across the inductor given by
( the direction opposes
the loop current)
There is a potential difference of
due to the emf source
From the loop rule
The solution to eqn * when i(0) =0 is
where
• When the emf source is removed, current I does
not suddenly goes to zero.
• Setting the RL circuit eqn to zero we have
• The solution which satisfies the initial condition
is
decay current
ENERGY STORED IN A MAGNETIC FIELD
• From the RL circuit eqn
• Multiplying by i we have
1st term
represents the rate at which the emf
devise delivers energy to the rest of the circuit
3rd term i2R is the rate at which energy appears as
thermal energy in the resistor
2nd term rep the energy that does not appear as thermal
energy which implies it is stored in the magnetic field
of the inductor.
rate at which energy is stored up in
the magnetic field of the inductor and
Sample problem
In the figure below, R1=8 Ω, R2= Solution
10Ω, L1=0.3 H, L2=0.2 H and Just when the circuit is connected,
the ideal battery has ε=6V (a) t-0, no current flows through the
just after switch S is closed, inductor and also through the
at what rate is the current in resistor so that equation of LR
inductor changing? (b) When circuit becomes
the circuit is in steady state,
what is the current in
inductor 1?
When the circuit is in steady state,
inductors L1 and L2 just act connecting
wire. If we neglect the resistivity of the
wires, then no current passes through
R2 such that
Sample problem
• A coil is connected in series with a 10Kilo-ohm
resistor, an ideal 50 V battery is applied across
the two devices, and the current reaches a value
of 2mA after 5 ms. (a) find the inductance of the
coil. (b) How much energy is stored in the coil at
this moment?
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