13. Self Induction ---The production of induced emf, in the circuit (coil) itself, on
account of a change in the current in it, is termed as the phenomenon of self-
inductance.
Let at any instant, the value of magnetic flux linked with the circuit itself be φ.
The property of a coil which enables to produce an opposing induced emf in it when the
current in the coil changes is called self induction. A coil is connected in series with a battery
and a key (K)
On pressing the key, the current through the coil increases to a maximum value and
correspondingly the magnetic flux linked with the coil also increases. An induced current flows
through the coil which according to Lenz’s law opposes the further growth of current in the
coil.
On releasing the key, the current through the coil decreases to a zero value and the magnetic
flux linked with the coil also decreases. According to Lenz’s law, the induced current will
oppose the decay of current in the coil.
Coefficient of self induction When a current I flows through a coil, the magnetic flux (φ) linked
with the coil is proportional to the current.
14. where L is a constant of proportionality and is called coefficient of self induction or self
inductance.
If I = 1A,
φ = L × 1, then L = φ coefficient of self induction of a coil is numerically equal to the magnetic
flux linked with a coil when unit current flows through it.
According to laws of electromagnetic induction.
E = -
dφ
dt
= -
d
dt
( LI )
= - L
dI
dt
If
dI
dt
= 1
E = - L
The coefficient of self induction of a coil is numerically equal to the opposing emf induced in
the coil when the rate of change of current through the coil is unity.
The unit of self inductance is henry (H).
Self induction is also called inertia of electricity as it opposes the growth or decay of current.
15. One henry --One henry is defined as the self-inductance of a coil in which a change in current
of one ampere per second produces an opposing emf of one volt.
Self inductance is said to be 1 henry when 1 A current in a coil links magnetic
flux of 1 weber.
Or
Self inductance is said to be 1 henry when unit rate of change of current (1 A / s) induces emf of 1
volt in the coil.
Self inductance of a long solenoid
Let us consider a solenoid of N turns with length l and area of cross section A.
It carries a current I.
If B is the magnetic field at any point inside the solenoid, then B =
𝝁𝟎 𝑵 𝑰
𝒍
Magnetic flux per turn = B × area of each turn
Thus Magnetic flux per turn = ∅ =
𝝁𝟎 𝑵 𝑰
𝒍
A
Hence, the total magnetic flux (φ) linked with the solenoid is given by the product of flux
through each turn and the total number of turns. ∅ =
𝝁𝟎 𝑵 𝑰
𝒍
A N
Magnetic Flux linked across N turns of the coil is
16. Φ=
µ0N2IA
𝒍
But, Φ = LI
L =
µ0N2IA
𝒍
= µ0n2A 𝒍 = = µ0n2 V
If L is the coefficient of self induction of the solenoid, then
We know that φ = LI
From above equations If the core is filled with a magnetic material of permeability μ, then
L =
µN2IA
𝒍
= µn2A 𝒍
= µn2l(
𝝅d2
𝟒
)
This implies that inductance of a solenoid L ∝ n2 , L ∝ d2 , L ∝ V .
As n is a number per unit length, inductance can be written as a product of permeability
constant µ0 / µ and a quantity with dimension of length.
This implies that µ0 / µ can be expressed in henry/ meter (
H
m
).
volume
V = A 𝒍
17.
18.
19. An inductor --- An inductor is a passive circuit element.
The inductor is basically the coil of wires which concentrates the magnetic field
into the circuit.
Inductances----The Inductance is the property of a material by virtue of which it
opposes any change of magnitude and direction of electric current passing
through the conductor.
In series----inductors are connected in series, the equivalent inductance of the
combination will be the sum of all individual inductors’ inductance.
If several inductances are connected in series L = L𝟏 + L𝟐 + L𝟑 + L𝟒 ……….
When inductors are connected in series, the total inductance is more than any one
of the series inductors' inductances.
In Parallel: --inductors are connected in parallel, the reciprocal of the equivalent
inductance of the combination will be the sum of individual inductances’ reciprocal.
When inductors are connected in parallel, the total inductance is less than any one
of the parallel inductors' inductances.
in parallel, then the total inductance is
1
L
=
1
L𝟏
+
1
L𝟐
+
1
L𝟑
+………
20.
21.
22.
23.
24. Energy associated with an inductor --------
Whenever current flows through a coil, the self−inductance opposes the growth of the current.
Hence, some work has to be done by external agencies in establishing the current.
If e is the induced emf then,
e = - L
dI
dt
The small amount of work dw done in a time interval dt is
dw = e I dt
dw = - L
dI
dt
I dt
The total work done when the current increases from 0 to maximum value (𝑰𝟎) is
dw = 𝟎
𝑰𝟎
𝑳𝑰𝒅𝑰
w = -
i
2
L 𝑰𝟎
𝟐
This work done is stored as magnetic potential energy in the coil Energy stored in the coil
U = -
i
2
L 𝑰𝟎
𝟐
P =
w
t
P =
dw
dt
P = VI = eI
25. w = -
i
2
L 𝑰𝟎
𝟐
w if 𝑰𝟎 = 1,
Then w =
i
2
L or L = + 2W (numerically)
Hence self-inductance of a circuit is numerically equal to twice the work done in
establishing the magnetic flux associated with unit current in the circuit.
This work done W, will represent the energy of the circuit.
∴ Energy of the circuit U = -
i
2
L 𝑰𝟎
𝟐
26. Energy density ---The energy density is the energy stored per unit volume of
the space .
It is denoted by letter 𝝁𝑩.
Magnetic and electric fields can also store energy.
The energy density formula in case of magnetic field or inductor is given by , or the energy
stored, per unit volume, in the magnetic field is
𝝁𝑩 =
𝑼𝑩
𝑨𝒍
𝑼𝑩 =
𝑳𝒊𝟐
𝟐
𝝁𝑩 =
𝑳𝒊𝟐
𝟐𝑨𝒍
Volume of the solenoid = Al
𝑳 = 𝝁𝟎 𝒏𝟐Al
𝝁𝑩 =
𝝁𝟎 𝒏𝟐Al 𝒊𝟐
𝟐𝑨𝒍
𝝁𝑩 =
𝝁𝟎 𝒏𝟐 𝒊𝟐
𝟐
B = 𝝁𝟎 n i
𝝁𝑩 =
𝑩𝟐
𝟐 𝝁𝟎
