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ELECTROMAGNETIC INDUCTION
Introduction :
Electricity and magnetism were considered separate and unrelated phenomena for a long
time. In 1820 Oersted discovered that electric current produces magnetic field. Ampere and
others found that moving charges produces magnetic fields. Thus it was established that
electricity and magnetism are interrelated. Then scientists were think of the reverse effect. Is
it possible to get electric current by magnetic fields. Michael Faraday and Joseph Henry in
1930 demonstrated that electric current was induced in a closed circuit when subjected to a
changing magnetic fields. The phenomenon in which electric current is generated in a closed
circuit by varying magnetic fields is called electromagnetic induction.
EXPERIMENTS OF FARADAY AND HENRY :
The phenomenon of electromagnetic induction was discovered and understood on the
basis of the following experiments performed by Faraday and Henry.
Experiment-1 :
Consider a coil C of Several turns of insulated copper wire connected to a sensitive
galvanometer G
i) When the N-pole of the bar magnet is moved
towards the coil, the galvanometer shows a
sudden deflection in one direction indicating
that a current is induced in the coil,
ii) When the N-pole of the magnet is moved away
from the coil, the galvanometer again shows
deflection but now it is in opposite direction
which indicates that the induced current in opposite direction
iii) When the S-Pole of the magnet is moved towards or away from the coil the
deflection in the galvanometer are opposite to those observed with N-pole for
similar movements
iv) The deflection of the galvanometer is found to be large when the magnet is
moved faster towards or away from the coil.
v) When the magnet is kept stationary and the coil is moved towards or away from
the magnet same effects are observed
vi) When the magnet is kept stationary any where near or inside the coil the
galvanometer does not show any deflection
This shows that the relative motion between the magnet and the coil is responsible for
the generation of current in the coil.
Experiment-2 :
Consider a coil C1of few turns of insulated copper
wire which is connected to a sensitive galvanometer G.
Another coil C2 is connected to a battery is placed such
that they have common axis. A steady current in C2
produces a magnetic field around it.
i) When the coil C2 is moved towards the coil C1 the
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galvanometer shows deflection in one direction.
ii) When the coil C2 is moved away from the coil, the galvanometer shows deflection
in the opposite direction.
iii) If the coil C2 is kept stationary and the coil C1 is moved the same effects are
observed.
iv) The deflection is observed as long as there is a relative motion between C 1 and C2.
No deflection is observed if there is no relative motion between them.
v) The deflection in the galvanometer is found to be larger when the coils are moved
faster towards or away from each other.
Experiment-3 :
Let C1 and C2 be two coils placed coaxially. C1 is connected to a sensitive
galvanometer G and C2 is connected to a battery through a tapping key K.
i) When the tapping key K is pressed the
galvanometer shows momentary
deflection and returns to zero
immediately.
ii) When the key K is kept pressed
continuously there is no deflection in
the galvanometer.
iii) When the key is released the galvanometer shows momentary deflection in
opposite direction and returns to zero immediately
iv) Deflection of the galvanometer increases a lot when the iron rod is inserted into
the coils along the axis and the key is pressed or released.
MAGNETIC FLUX () :
The term magnetic flux is used to explain the
results of Faraday’s experiments.
“Magnetic flux through any surface is defined as
the total number of magnetic lines passing through that
surface”
If a uniform magnetic field B passes normally
through a plane surface area A then the magnetic flux
through this area is B=BA
If the field B makes an angle  with normal drawn to the surface area A then the
magnetic flux through the surface area is given by B = B A = BA Cos
= (B Cos)A
Where BCos is the component of B normal to A. Flux B is maximum when the field is
normal to the surface (=0) and is zero when the field is parallel to the surface (=90).
When the magnetic field is normal to the surface B = BA
or
Electromagnetic Indication
B
B
A
-2-
Thus “magnetic flux density (B) is defined as the magnetic flux per unit area held
normal to the magnetic field”
When the surface is in a magnetic field the outward flux is taken to be positive and the
inward flux is taken as negative
The SI unit of magnetic flux is Weber (Wb)
It is a scalar quantity.
Weber :
One Weber is the amount of magnetic flux produced when a uniform magnetic field of
one tesla acts normally over an area of 1 metre2.
ie. 1 Weber = 1tesla  1m2.
FARADAY’S LAWS OF ELECTROMAGNETIC INDUCTION :
On the basis of his experiments Faraday stated the laws of electromagnetic induction.
I Law : Whenever the magnetic flux linked with the coil changes an emf and hence current is
induced in it. The induced emf lasts as long as there is a change in magnetic flux
II Law : The magnitude of the induced emf is directly proportional to the rate of change of
magnetic flux linked with the coil
ie. e 
d
dt
If the magnetic flux linked with the coil changes from 1 to 2 in a time interval ‘t’ the
  1
rate of change of magnetic flux = 2
.
t
If e is the induced emf then according to Faraday’s laws
e
 2  1
t
or e = k
 2  1
t
Where k is a constant of proportionality and in SI K = 1
e 
 2  1
t
If dB is the change in magnetic flux in a small interval of time dt the induced emf is
d
given by e   B
dt
The negative sign indicates that the direction of the induced emf is such that it opposes
the change in magnetic flux.
If the coil consisting of N turns then the total emf induced is given by e   N
d B
dt
By increasing the number of turns in the coil, we can increase the induced emf.
Lenz’s Law : Lenz’s law gives the direction of the induced emf.
Statement : “The direction of the induced emf or current in a circuit is such that it opposes
the cause which produces it ie. it opposes the change in magnetic flux”.
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Explanation : When the N-pole of a magnet is moved towards the coil the induced current in
the coil flows in anticlockwise direction so that the face of the coil towards the magnet
develops north polarity, which opposes the motion of the magnet towards the coil, which is
actually the cause of the induced current in the coil.
When the N-pole of the magnet is moved away from the coil the induced current in the
coil flows in clockwise direction the face of the coil towards the magnet develops S-polarity
and attracts N-pole of the magnet ie the motion of the magnet away from the coil is opposed
which is really the cause of the induced current.
LENZ’S LAW AND CONSERVATION OF ENERGY :
Whether the magnet is moved towards or away from the coil, the induced current
always opposes the motion of the magnet as predicted by Lenz’s law. For example if the
N-pole of the magnet moves towards the coil, its face towards the magnet develops Npolarity and repels the N-pole of magnet. Work has to be done in moving the magnet towards
the coil against the force of repulsion. This work done against the force of repulsion appears
as electrical energy in the form of induced current.
If Lenz’s law is not true, then on moving N-pole of a magnet towards the coil and pulls
the magnet into the coil.
This means once the magnet is set into motion it moves on its own with increasing speed
and gains kinetic energy without expending an equivalent amount of energy. This is not
observed in the experiment, because it is against the law of conservation of energy. Hence
Lenz’s law is valid and is a consequence of the law of conservation of energy.
Motional emf :
Consider a conductor XY of length l placed in a uniform magnetic field B acting
perpendicular to the plane of paper and into it. The conductor moves with a velocity V in a
direction r to the magnetic field B.Let the conductor moves from XY to XY ı through a
small distance dx in a time dt.
The area swept by the conductor is dA = l. dx.
The change in magnetic flux across the conductor is
dф = B dA
= B l dx
The magnitude of the induced emf in the conductor is
e
d Bldx

 Blv
dt
dt
dx 

 v  
dt 

e  Blv
This is the expression for motional emf.
“The emf induced in the conductor due to its motion perpendicular to the uniform
magnetic field is called motional emf”.
Motional emf e=Blv
The direction of induced emf or current is given by Flemings right hand rule or motor
rule.
FLEMING’S RIGHT HAND RULE :
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Statement :
“If the first three fingers of the right hand are held mutually perpendicular to each other
such that the fore finger points the direction of magnetic field the thumb points the direction
of motion of the conductor and the central finger gives the direction of induced current in the
conductor”
ENERGY CONSIDERATION IN MOTIONAL EMF :
When a straight conductor of length l is moved with a velocity v in a direction
perpendicular to the magnetic field B, the motional emf produced in the conductor is e = Blv.
Let R be the resistance of the conductor and the two ends of the conductor be connected
externally by a wire of negligible resistance. Then the current through the conductor is
e Blv
I 
R
R
The magnetic force on the conductor moving r to the magnetic field B is
B 2l 2 v
 Blv 
F  BIl  
 Bl 
R
 R 
The direction of this force is opposite to the velocity of the conductor
The
power required to
 B 2l 2 v 
B 2l 2 v 2


P  F v  
v R
 R 
push
the
conductor
against
this
force
is
As the conductor is pushed mechanically, the mechanical energy dissipated per second
2
B 2l 2 v 2
 Blv 
R

is given by (Joules heating loss) PJ  I 2 R  

R
 R 
Clearly PJ = P. Thus the mechanical energy expended to maintain the motion of the
conductor is first converted into electrical energy (ie. induced emf) and then to thermal
energy.
EDDY CURRENTS :
Eddy currents are the induced currents in a metal block, due to the change in magnetic
flux linked with it.
The induced currents circulates throughout the volume of the metal which looks like
eddies in water and so they are called eddy currents. It was discovered by Focault in 1895 and
hence they are also called Focault currents.
EXPERIMENT TO DEMONSTRATE EDDY CURRENTS :
Consider a metal block in the form of a plate. It is free to oscillate between the pole
pieces of an electromagnet. In the absence of the magnetic field it is found to oscillate for a
long time. As the electromagnet is switched on, the plate oscillates in a strong magnetic field.
Due to the continuous change in magnetic flux linked with the plate, eddy currents are setup
in the metal plate. Since the eddy currents are produced due to the oscillations of plate.
According to Lenz’s law, the eddy currents oppose the motion of the plate and finally bring it
to rest. This kind of opposition offered by eddy currents to the motion of a system is called
electromagnetic damping.
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Since the resistance of the metal block is very small, generally the eddy currents are very
large in magnitude and produce considerable amount of heat. Thus energy is wasted in the
form of heat.
The heating effect of eddy currents is undesirable in devices like dynamos, motors and
transformers. Eddy currents cause unnecessary heating and wastage of power. To minimize
the energy loss, laminated cores are used. Laminated cores are thin sheets of metal separated
by an insulating material. This increases the resistance and hence reduces the strength of eddy
currents. It must be noted that eddy currents can only minimized but never completely
eliminated.
APPLICATIONS OF EDDY CURRENTS :
1. Induction furnace :
Induction furnace is based on the heating effect of eddy currents. The metal
block to be melted is placed in rapidly changing magnetic field. Strong eddy currents
are induced in the block and a large amount of heat is produced in it. This heat
produced is sufficient to melt the metal. This process is used to separate metals from
their ores and to make alloys by melting the constituent metals
2. Electromagnetic brakes :
Now a day’s trains have electromagnetic brakes in addition to the usual vacuum
brakes. A metallic drum is coupled to the wheels of the train so that when the train
runs, the drum also rotates. Strong electromagnets are situated near the metal drums.
When electromagnets are activated, eddy currents are setup in the drums which
oppose the motion of the train. As the speed is reduced eddy currents also reduced and
the braking action is smooth.
Eddy currents are also used in speedometer, electromagnetic damping, induction
motor, energy meters etc.
Self induction :
“The phenomenon in which an emf is induced in a coil due to the
change of current in the same coil is called self induction”.
Consider a coil connected to a battery through a tap key K. when
the key is pressed the current through the coil raises from zero to
maximum as a result the flux linked with the coil increases.
According to Faraday’s law an emf is induced in it which opposes the
growth of current in the coil. This delays the current to attain the
maximum value.
When the key is released the current falls from maximum to zero
which causes a decreases in the magnetic flux. Hence an emf is induced in the coil which
opposes the decay of current. So the current does not become zero instantaneously but takes
some time.
Coefficient of self induction :
At any instant, the magnetic flux linked with a coil is proportional to the current through it.
ie   I or  = LI where L is a constant of proportionality known as coefficient of self
induction or self inductance of the coil.
The emf induced in the coil is given by
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e
If
d
dI
 L
dt
dt
dI
 1 then L = e
dt
Thus “self inductance of a coil is numerically equal to the emf induced in the coil due to
a unit rate of change of current through it”
Unit of self inductance :
The SI unit of self inductance is Henry (H)
L
e
dI
dt
L = 1H, if e = 1v and
dI
 1A / s
dt
The self inductance of a coil is said to be one henry if one volt of emf is induced in the
coil when the current in it changes at rate of one ampere per second.
Self inductance of a solenoid :
Consider a long solenoid of length ‘l’ area of cross section A and having n turns per unit
length. Let I be the current flowing through it.
The magnetic field inside the solenoid is
uniform and is given by B= n I
The total magnetic flux linked with the
solenoid is
 = NB A where N = total number of turns =
nl
 = n l (on I)
But
A = on2AlI
 = LI
LI = on2Al
or L = o n2Al
 N2 
 A
 L   o 
 l 
n 
N
l
If the solenoid is wound over a material of relative permeability r (soft iron) then
 N2 
 A  r L
L  r o 
 l 
Thus self inductance of a solenoid depends on :
1. Total number of turns of the solenoid (N)
2. Length of the solenoid (l)
3. Area of cross section (A) and
Electromagnetic Indication
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4. Nature of the material of the core of solenoid.
ENERGY STORED IN AN INDUCTOR :
Consider a source of emf connected to an inductor L. As the current starts growing, an
emf is induced in it. The self induced emf opposes the growth of current through it. Some
work to be done or energy spent by the source in sending the current through it against the
induced emf. This work done or energy spent by the source is stored as magnetic potential
energy.
Let dw be the small amount of work done in sending a current I through the coil in a
time ‘dt’ then
dw = e I dt where e is the induced emf e  L
 dw  L
dI
dt
dI
I dt  LI dt
dt
The total work done by the source to increase the current from zero to I is
w   dw
I
  L I dI
o
I
I 2 
 L 
 2 o

1
LI2
2
Thus magnetic potential energy stored in an inductor of self inductance L carrying
current I is :
U 
1
LI2
2
This equation gives the energy stored in an inductor.
MUTUAL INDUCTION :
The phenomenon in which an emf is induced in one coil due to the change of current in
the neighbouring coil is called mutual induction.
Consider two coils P and S placed close to
each other. A battery and a tapping key K is
connected to primary coil and a galvanometer is
connected across secondary coil.
When the key K is pressed, the current in the
primary increases. As a result of this, magnetic
field produced around P increases. So the
magnetic flux that linked with secondary coil
increases. Therefore an emf is induced in the
secondary coil which opposes the growth of current in the primary.
Similarly when the key is released, the current in the primary coil decreases so the
magnetic flux linked with the primary coil as well as secondary coil decreases. This change in
magnetic flux induces an emf in the secondary coil which opposes the decay of current in the
primary coil. This phenomenon of inducing emf is called mutual induction
COEFFICIENT OF MUTUAL INDUCTION :
At any instant, magnetic flux linked with the secondary coil is directly proportional to
the current in the primary coil.
Electromagnetic Indication
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ie.   I or  = MI
Where M is a constant of proportionality called coefficient of mutual induction or
mutual inductance.
The emf induced in the secondary coil is
e
If
d
d
  MI 
dt
dt
 dI 
 e  M  
 dt 
dI
 1 then M  e
dt
The mutual inductance of two coils is equal to the emf induced in one coil (S) when the
current in the other coil (P) changes at the rate of unity.
The SI unit of mutual inductance is henry (H) :
M
e
dI
dt
If e  1v,
dI
 1 A/s then M = 1 H
dt
The mutual inductance of two coils is said to be one henry if one volt of emf is induced
in one coil (S) when the current in the other coil (P) changes at the rate of one ampere per
second.
MUTUAL INDUCTANCE OF THE LONG COAXIAL SOLENOIDS :
Consider two long coaxial solenoids S1 and S2
each of length l. Let n1 be the number of turns per unit
length and r1 be the radius of the inner solenoid S1.
Let n2 be the number of turns per unit length and r2 be
the radius of the outer solenoid S2
Imagine a time varying current I2 pass through
S2
The magnetic field setup inside S2 due to I2 is
B2 = o B2
A =
 o N1 N 2 I 2 , A
l
Mutual inductance of coil L W.r. to coil 2 is
M 12 
1
I2

o N1 N 2 A
l
Now consider the current I, pass through S1 produces a magnetic field B1 inside S1
ie, B1 = o n, I
Where n1 
N1
number of turns per unit length of S1
l
Total flux linked with the outer solenoid S2 is
 2  N 2 B1 A N 2  o n1 I , A 
 o N1 N 2 I 1 , A
l
 Mutual inductance of coil 2 w.r.to coil 1 is
Electromagnetic Indication
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M 21 
2
I1

o N1 N 2 A
l
Clearly, M12 = M21 = M
M 
 o N1 N 2 A
l
  o n1n2 l A   o n1n2 l  r12
Thus mutual inductance of two solenoids depends on :
1. The no. of turns in the two solenoids
2. Cross sectional area
3. Relative separation and permeability of the core of material
If a magnetic material of relative permeability r is present inside the solenoids then
M   r  o n1n2 l  r12
AC GENERATOR (AC Dynamo) :
An A.C. generator is a device used to convert mechanical energy in to electrical energy.
Principle :
It works on the principle of electromagnetic induction ie. When a coil is rotated in
uniform magnetic field an induced emf is produced in it.
Construction :
The A.C. generator consists of the following parts.
1. Armature :
A rectangular coil ABCD consisting of large number
of turns of copper wire wound over a soft iron core
is called armature.
2. Field magnet :
It is usually a strong permanent magnet having
concave poles. The coil is rotated between the poles
of the magnet so that its axis is perpendicular to the
magnetic field lines.
3. Slip rings :
The two ends of the armature coil connected to two brass slip rings R 1 and R2. These
rings rotate along with the armature coil.
4. Brushes :
Two carbon brushes B1 and B2 are pressed against the slip rings. The brushes remain
fixed but they are always in contact with the rotating rings. These brushes are
connected to the load through which the output is obtained.
Working :
When the armature rotates with its axis r to the magnetic field, the magnetic flux
linked with the coil changes and an emf is induced in the coil. Therefore an induced current
flows in the coil. The direction of the induced current is given by Fleming’s right hand rule.
Electromagnetic Indication
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The current flows out through the brush B1 in one direction of half of the revolution and
through the brush B2 in the next half revolution in the reverse direction. Thus in one complete
revolution the current changes its direction once. This process repeats and alternating current
flows in the external circuit.
Theory :
Consider a coil of n turns and area of cross section A placed
with its plane perpendicular to the magnetic field so that  = o at
t=o
Let the coil be rotated in anticlockwise direction with a
constant, angular velocity . Then the angle between the normal to the plane of the coil and
B at any instant‘t’ is given by  = t
 The component of magnetic field normal to the plane of the coil = B cos
= B cost
 The total magnetic flux linked with the coil is given by  = nAB Cost
From Faraday’s law the induced emf in the coil is given by
e
d
d
  nAB Cost    nAB   Sint  or e  nAB  Sin t
dt
dt
The maximum value of the induced emf is eo = nAB if Sint
 Instantaneous value of emf is e  eo Sint
Instantaneous value of current in the circuit is given by I 
or
Where
e eo Sin t

R
R
I = Io Sint
eo
 I o is the maximum value of the induced current
R
[When =0 , e=0, when =90 e=eo, When =180, e=0 When =270 e=– eo and
When =360 e=0]
Thus when the coil is rotated from its position r
to the magnetic field through 180, the induced emf
and current increase from zero to maximum and then
decreases from maximum to zero in the same direction.
When the coil rotates through next 180 emf and
current rises from zero to maximum and then decreases
from maximum to zero in opposite direction. Thus
current supplied by an A.C. generator is sinusoidal.
The variation of induced emf is as shown in figure.
ALTERNATING CURRENTS :
An alternating current is that current whose magnitude changes continuously with time
and direction reverses periodically
At any instant the alternating induced emf in the coil is given by e = eo Sint
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Suppose this emf is applied to a circuit of resistance R then by Ohm’s law the current in
e e Sin  t
or I  I o Sin t
the circuit is I   o
R
R
Thus the current in the circuit varies sinusoidally with time and is called alternating
current. Here I = Instantaneous value of A.C
Io =
eo
= Peak value or maximum value of an A.C
R
Time period (T) :
The time taken by an A.C to complete one cycle of its variation is called its time period
ie. T =
2

Frequency (f) :
The number of cycles completed per second by an A.C. is called its frequency
Average value or mean value of an A.C :
It is defined as that value of steady current which sends the same charge in a circuit in
the same time as is sent by the given A.C in half time period.
It is denoted by Iav or Im
It can be shown that I av 
similarly
2

I o  0.637 I o
Vav 
2

Vo  0.637Vo
Root mean square value of A.C (rms) :
It is defined as that value of a direct current which produces the same heating effect in a
given resistor in a given time as produced by the given A.C when passed through the same
resistor for the same time.
It is denoted by Irms
It can be shown that I rms 
Io
 0.707 I o
2
Also rms value of alternating voltage is given by Vrms 
Vo
 0.707 Vo
2
Phasors and Phasor diagram :
A rotating vector which represents a sinusoidally varying quantity is called a phasor
This vector is imagined to rotate with angular velocity equal to the angular frequency of
that quantity. Its length represents the amplitude of that quantity and its projection upon a
fixed axis gives the instantaneous value of the quantity. The phase angle between two
quantities is shown as the phase angle between their phasors.
A diagram that represents alternating current and voltage of
the same frequency as rotating vectors along with proper phase
angle between them is called a phasor diagram
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Suppose alternating voltage and current in a circuit is given by V = Vo Sint and I = Io
Sin (t + ) Where  is the angle between V and I.
To represent these quantities as phasors we draw circles of radii Vo and Io as shown in
figure. Let AOˆ X  t and BOˆ X  t  
Then vector OA represents phasor V of magnitude Vo and vector OB represents
phasor I of magnitude Io both rotating with the same angular velocity  in the anticlockwise
direction. The projection OM(v) of OA on the vertical axis represents the instantaneous
value of alternating voltage. The projection ON(I) of OB on the vertical axis represents the
instantaneous value of alternating current. The angle   AOˆ B represents the phase angle
between the phasors V and I . In this case a current leads the voltage by phase angle .
A.C. Voltage applied to a resistor :
Consider a resistor of resistance R is connected to a source which produces alternating
voltage across its terminals let the alternating voltage be given by
V = Vo Sin t
Where Vo is the maximum value of alternating voltage and  is its angular frequency.
If I is the current through R at any instant of time t then the P.d. across R is VR = IR
 The P.d across R is equal to the p.d. across the source
IR = Vo Sint
or I =
Vo
Sint
R
or I = Io Sint
Where I o 
------ -----
(2)
Vo
is the maximum or peak value of A.C
R
From equations (1) and (2) it is found that both V and I are the functions of Sint hence
V and I are in the same phase this means both V and I reach zero, minimum and maximum
values at the same time
The phase relationship is shown graphically in figure (i)
Figure (ii) shows the phasor diagram for a resistance A.C circuit both V and I are in
the same direction making same angle t with X-axis. The phase angle between them is zero.
Alternating voltage applied to an inductor :
Electromagnetic Indication
- 13 -
Consider an inductor of inductance L and negligible resistance connected to an A.C
source.
The alternating voltage across the inductor is given by V = Vo Sint
As the applied voltage varies with time, the current through it also varies. As a result an
emf is induced in the coil which opposes the change of current in it
The induced emf e   L
dI
dt
Since there is no resistance in the circuit, using Kirchhoff’s voltage law V+e = o
ie, Vo Sint  L
L
dI
o
dt
dI
 Vo Sin t
dt
or dI =
Vo
Sin t dt
L
Integrating on both sides.
Vo
 dI  L  Sin t dt
I=
Vo  Cos t 


L
 
–Cost = Sin (t –/2)
=
Vo
 Cos t 
L
 Cos  = Sin (/2 – )
=
Vo
Sin t  
2
L


I  Io Sin t  
Where I o 
2

 
– Sin  = Sin (–)
(2)
Vo
is the peak value of current
L
Phase relationship between V and I :
From equation (1) and (2) it is observed that the current in the circuit lags behind the
applied voltage by an angle 
2
In the phasor diagram the phasor I lagging behind the phosor V
Inductive reactance :
Electromagnetic Indication
- 14 -
Vo
V
with Ohmic relation Io  o . We find that L plays the
L
R
same role as resistance R in resistive case. L is called inductive reactance and denoted by x.
Compare the equation I o 
 X L  L 
Vo Vrms

I o I rms
“Inductive reactance is the effective opposition offered by the inductor to the flow of
A.C. It is defined as the ratio of rms value of voltage across the inductor to the rms value of
current flowing through it”.
ie, XL = L  2fL . Where f is the frequency of A.C. supply.
Note :
i) For A.C. X L  f and X L  L
ii) For D.C f  o X L  o Thus pure inductor offers no opposition to direct current
AC Voltage applied to a Capacitor :
Consider a capacitor of capacitance C is connected to an A.C source
At any instant
V  Vo Sint .....(1)
the
alternating
voltage
applied
across
the
capacitor
is
Let q be the charge on the capacitor at any instant then the P.d across the capacitor is
Vc 
q
But
e
Vc = V (Voltage across the source)
q
  Vo Sin t
c
or q  Vo c Sin t
The current in the circuit at any instant of time t is
I
dq d
 Vo Sin t 
dt dt
 Vo c Cos t

I  Vo c  Sin t  

I  I o Sin t  
2

 ........................(2)
2
Where I o 
Vo
1
c
Phase relation between V and I : Electromagnetic Indication
- 15 -
From equation (1) and (2) it is found that the current leads the applied voltage by and
angle 
2
The phase relationship between V and I is shown in figure. In the phasor diagram the
phasor I leads the phasor V
Capacitive reactance:
In the equation I o 
 The term
Vo
1
,
is similar to the resistance
1
c
c
1
is known as capacitive reactance (Xc)
c
ie, Xc =
1
1

c 2fc
Thus Capacitive reactance is the effective opposition offered by the capacitor to the flow
of A.C in the circuit
The SI unit of capacitive reactance is Ohm ().
The capacitive reactance depends on the capacity of the capacitor and the frequency of
AC.
ie. X c 
1
f
For d.c. f = o
and
Xc 
 Xc 
1

0
1
c
Thus capacitor offers infinite opposition for the flow of D.C. So direct current cannot
pass through a capacitor. In other words a capacitor blocks the flow of DC.
For AC f = finite  X c 
1
= Smaller value
finite
Thus capacitor offers small opposition for the flow of AC.
Series
LCR
circuit :
Electromagnetic Indication
- 16 -
Consider an A.C circuit containing an inductor of inductance L, capacitor of capacitance
C and resistance of R connected in series across an A.C source
At any instant the voltage across the combination is V=Vo Sin t. As all the components
are in series, the current (I) through all of them is same. Let VL, VC and VR be the voltages
across L, C and R then VL = IXL, VC = IXC and VR = IR
In order to find the relation between V, VL, VC and VR let us draw the phasor diagram
for current and voltages.
Phasor diagram solution for series LCR circuit :
In an A.C circuit VR is in phase with I. Therefore both the phasors V and I are
represented along OX. VL leads the current I by an angle 
then the phasor VL is
2
represented along OY. VC lags the current I by an angle  then VC is represented along
2
OY.
Suppose OA, OB and OC represents the magnitudes of VR, VL and Ve respectively.
If VL > VC then the resultant of VL and VC represented by OD in the diagram
The diagonal OE of the parallelogram OAED gives the resultant voltage V across the
combination. Let  is the phase angle between V and I.
In the right angled triangle OAE we have
OE 2  OA 2  AE2
V 2  VR2  VL  VC 
2
V 2  IR   IX L  IX C 
2
2

V 2  I 2 R2  X L  X C 
Or V  I
2

R2  X L  X C 
2
2
V
 R2  X L  X C 
I
2
V
 Z where Z  R 2   X L  X C  is called impedance of the series LCR circuit.
I
Impedance :
The effective opposition offered by the series LCR circuit for the flow of current
through it is called its impedance.
Phase difference between V and I :
From the phasor diagram it follows that in series LCR circuit resultant voltage leads the
current I by an angle 
Then tan  
VL  VC
VR

Electromagnetic Indication
IX L  IX C
IR

XL  XC
R
or
 XL  XC 

R


  tan 1 
- 17 -
 X  XL 
If VC > VL then   tan 1  C
 ie, voltage lags the current by an angle 
R


Analytical solution for series LCR circuit :
Let the instantaneous value of alternating current is given by I = Io Sin t ……..(1)
dI
is the rate of
dt
change of current in the circuit. Then the instantaneous emf induced in the inductor is
dI
e  L .
dt
Suppose V is the emf of source, q is the charge on the capacitor and
At any instant the P.d across the capacitor is VC =
q
c
At any instant the P.d across the resistor is VR=IR
 Net emf of the circuit = P.d. across R + P. d across C
dI 
q

V    L   IR 
dt 
c

dI q
V  IR  L 
dt c
I  I o Sin t
dI
 I o Cost
dt
Also I 
dq
 or dq  I dt  I o Sin t dt
dt
Integrating on both sides we get
 dq  I  Sin t dt
o
I
  Cos t 
q  Io 
   o Cos t

 

Substituting the values of I ,
dI
and q in equation (2)
dt
V=(Io Sin t) R + L (Io Cos t) +
1   Io

Cos t 

c 

1


 I o  R sin t  L Cost  Cost 
c




1 

V  I o  R Sin t   L 
 Cos t 
c 



But L = XL and
1
 Xc
c
V = Io [R Sin t + (XL –XC) Cos t]
Electromagnetic Indication
- 18 -
R 2   X L  X C  we get
2
Multiply and divide the above equation by
V  I o R2   X L  X C 



XL  XC 
R

Sin t 
Cost 
2
2
2


2
R  X L  X C 
 R  X L  X C

R
If we put
And
2
R2  X L  X C 
2
X L  XC
 Cos  ......(3)
 Sin  .............(4)
R2  X L  X C 
2
V  I o R 2   X L  X C  Cos  Sin t  Sin Cos t 
2
V  I o R 2   X L  X C  Sin (t   
2
Now R 2   X L  X C  has got dimensions of resistance and is called impedance of
series LCR circuit and denoted by Z
2
 Z  R2  X L  X C 
2
Further Io R 2   X L  X C 2  Vo the maximum value of alternating emf
V Vo Sin t    ...............(5)
From equation (1) and (5) it follows that in a series LCR circuit the voltage leads the
current by a angle  if XL > XC
To find  :
Dividing equation (4) by equation (3) we get
Sin 

Cos 
X L  XC
R2  X L  X C 

R
2
R2  X L  X C 
2
Sin  X L  X C

Cos 
R
tan  
X L  XC
R
 X  XC 
or   tan 1  L

R


Electrical resonance :
Current
in
a
series
LCR
circuit
Z  R 2   X L  X C  Where X L  2fL . X C 
2
Electromagnetic Indication
is
given
by
I
V
Z
and
1
2fc
- 19 -
In an AC circuit, R is independent of frequency but XL and XC depends on the
frequency. For a given values of R L and C impendence depends only on the frequency of the
applied A.C
As the frequency increases XC decreases and XL increases. At a particular frequency
f=fo, XL=XC , then Z is minimum and the current is maximum. The circuit is purely resistive.
Now the current and voltage are in the same phase and the phase difference   o . This
frequency is called resonant frequency.
The phenomenon of impendence in a series LCR circuit becomes minimum and the
current in the circuit is maximum at a particular frequency of the applied alternating voltage
is called electrical resonance
Expression for resonant frequency :
Resonant frequency is the frequency of the applied AC at which the current through the
circuit is maximum.
The current in an LCR circuit is given by I 
I
V
Z
V
R2  X L  X C 
2
I is maximum, when Z is minimum, Z is minimum if XL =XC then the circuit is in
resonance
At resonance XL = XC
Or
L =
1
c
2 =
1
LC
1
LC
=
At resonance  = 2fo
 2f o 
Or f o 
1
LC
1
2 LC
Figure shows the variation of current with frequency in a series LCR circuit. It follows
that current in the circuit is maximum for f = fo. For values of f less than or greater then fo
comparatively small current flows in the circuit
Sharpness of resonance and Q – factor :
At resonant frequency o, the current in the LCR
circuit is maximum. Figure shows the variation of current
with angular frequency  for two different values of
resistance R1 and R2 (R1 < R2)
Electromagnetic Indication
- 20 -
For a given values of L and C the resonant frequency does not depend on R. hence
current is maximum in the two cases at the same value of resonant frequency o. But the Imax
decreases with the increase in the value R. For smaller value of R, the resonance curve is
more sharp. For large value of R the resonance curve is less sharp. The width and height of
the curve varies with R. The sharpness of resonance curve is determined by quality factor or
Q-factor.
Q-factor in terms of band width of the circuit :
Let 1 and 2 be the angular frequencies below and above resonant angular frequency.
1
At which the current in the circuit is
times the maximum value of current at the resonant
2
frequency.
Q- factor is defined as the ratio of the resonant frequency to the band width.
Let 1 = o –  and 2 = o + 
Then 2–1 = 2 is called the band width of the circuit.
Q –factor =

resonant frequency
ie, Q  o
band width
2
Expression for Q factor :
We know that at resonant angular frequency the current in the circuit is maximum and is
given by
I
V
1 

R2   L 

c 

ie, I max 
2
Now at 2 = o+, the current is

V

1 

R 2   2 L 
2 c 


2
V
R
I max
V

2
2R
V
2R
2
Or

1 
  2R
R   2 L 
2c 

2
2

1 
  2 R 2
Or R    2 L 
2c 

2
 2 L 
1
2c
R
ie, o    L 
1
R
o   c
Electromagnetic Indication
- 21 -

o L 1 

 
1
 
R
o 
  

o c1 
o 

1
  
  
  o L1 
  R
o L 1 
 o 
 o 
Since

1 
 : o L 

o c 

  
  
  o L1 
R
1, o L1 
o
o 
o 



ie , o L 
or 2 
2
o
R
R
L
Thus Q – factor is given by, Q 
o
L
 o
2
R
Thus smaller the value of resistance R in the circuit smaller will be the band width 2
and larger will be the Q-factor of the circuit. Larger value of Q means sharpness of resonance
is higher.
Since oL =
1
o c
we can also write
Q=
o L
R

1
o cR
Power in AC circuit :
The rate at which the electrical energy is consumed in an electrical circuit is called its
power.
In a DC circuit, power is given by the product of voltage and current. But in an AC
circuit both voltage and current varies sinusoidally with time and generally they are not in
phase. So for an AC circuit the average power is calculated by defining the instantaneous
power of the circuit.
Instantaneous power is defined as the product of the instantaneous voltage and
instantaneous current
Suppose in an AC circuit the voltage and current at any instant are given by
V=Vo Sin t and I = Io Sin (t+)
Where  is the phase angle between voltage and current.
The instantaneous power is given by
P = VI = Vo Sint  Io Sin (t+)
= Vo Io Sint (Sint. Cos + Cost.Sin)
= Vo Io(Sin2t Cos  + Sin t Cos t Sin )
= Vo Io (Sin2. Cos  +
Electromagnetic Indication
Sin 2t
Sin 
2
- 22 -
But Sin2t =
1  Cos 2t
2

 1  Cos2t 
Sin 2t 
  Sin 
 P  Vo I o Cos 

2
2 



P
Vo I o
Cos  Cos2t Cos  Sin Sin 2t 
2
If we assume the instantaneous power to remain constant for a small time dt, the work
done during this time is dw = P.dt = VI dt.
T
Total work done over a complete cycle is W   VIdt
o
Hence average power consumed in the circuit over a complete cycle can be obtained by
integrating the above equation between t = 0 to t = T
T
 Pav 
VI
1
VIdt  0 0

To
2T
T
 Cos  Cos.CosT  Sin Sin 2t dt
0
T
T
T

Vo I o 
Pav 
Cos  dt  Cos  Cos 2t dt  Sin  Sin 2t dt 
2T 
o
o
o

T
T
T
o
o
o
Now  dt  T and it can be shown that  Cos2t dt   Sin 2t dt
 Pav 
or
Pav 
Pav 
Vo I o
Cos (T )  Cos  (o)  Sin  (o)
2T
Vo I o
VI
CosT  o o Cos
2T
2
Vo I o
Cos
2 2
 Pav  Vrms I rms Cos
Here Pav is called true power and Vrms Irms is called apparent power or virtual power and
the quantity Cos is called the power factor
Special Cases :
1. AC circuit having R only : here V and I are in same phase
ie,  = o and Cos  = 1
 Pav  Vrms I rms 
2
Vrms
2
 I rms
R
R
True power = apparent power
2.
AC circuit having L only : Here voltage leads the current by an angle

2
ie,
Electromagnetic Indication
  2
- 23 -
 Pav  Vrms I rms Cos 
2
0
Thus the average power consumed in an inductive circuit over a complete cycle is zero
3. AC circuit having C only : In this case  = 
 Pav  Vrms I rms Cos 
2
2
0
Thus the average power consumed in a capacitive circuit over a complete cycle is zero
Power factor :
The power factor is defined as the ratio of true power to the apparent power of an A.C
circuit
Its value varies from 0 to 1
The power factor of a series LCR circuit is given by
Cos 
R

Z
R
R  X L  X C 
2
2
For a purely inductive or capacitive circuit  = 90
 power factor = Cos 0 = 1
Wattless current :
The current flowing through a circuit containing only an inductor or a capacitor in which
no power is consumed or dissipated is called wattless current or idle current
Transformer :
A transformer is a device used to change alternating voltage to any desired value.
It works on the principle of mutual induction.
It consists of two separate coils wound on a rectangular laminated iron core. One of the
coil is called primary (p) and the other secondary (s). The primary coil is connected to the
source of alternating voltage to be varied. The output voltage is obtained across the terminals
of secondary.
When an alternating voltage is applied to the primary, current flows through it varies
and magnetic flux linked to the primary and hence that linked to the secondary also varies. As
a result an alternating emf of the same frequency as that of the applied voltage is induced
across the secondary coil. The magnitude of the induced output voltage depends on the
number of turns in the primary and secondary coil.
Electromagnetic Indication
- 24 -
Consider an ideal transformer in which primary has negligible resistance and all the
magnetic flux in the core links both primary and secondary. Let  be the magnetic flux in
each turn in the core at time t due to current in the primary when voltage Vp is applied to it
The induced emf in the secondary with Ns turns is es  
d
d
The magnetic flux  also induces a back emf in the primary and is given by
d
ep   N p
where Np= no. of turns in primary
dt
If Vp is the applied voltage across primary and Vs is the induced output voltage across
the secondary. Then e p  V p and es  Vs
ie, Vs   N s

d
d
and V p   N p
dt
dt
Vs N s

Vp N p
For an ideal transformer, the input power = output power
V p I p  Vs I s
or
Vs I p N s


Vp I s N p
If Ns > Np, Vs >Vp such a transformer is called step up transformer. In this case Is < Ip ie,
step up transformer increase the voltage but decrease current.
If Ns>Np, Vs<Vp such a transformer is called step down transformer. In this case Is>Ip
ie, step down transformer decreases the voltage but increase current.
But in actual transformers, the efficiency varies from 90 to 99%. This indicates that
there are some energy losses in the transformer.
Energy losses in transformers :
1. Hysterisis loss : When an alternating current carries the iron core through cycles of
magnetization and demagnetization some energy is lost and appears in the form of
heat. This is called hysterisis loss and can be minimized by using a suitable material
having narrow hysterisis loop
2. Loss due to flux leakage : The magnetic flux produced by the primary may not fully
pass through the secondary. Some of the flux may leak into air. This loss can be
minimized by winding the primary and secondary coils over one another.
3. Loss due to eddy currents : The varying magnetic flux due to alternating current
induces eddy currents in the iron core and leads to some energy loss in the form of
heat. This loss can be minimized by using laminated iron core.
4. Humming loss : As the transformer works, its core lengthens and shortens during
each cycle of the alternating voltage due to the phenomenon called magnetostriction.
This gives rise to a humming sound. So some of the electrical energy is lost in the
form of humming sound.
LC Oscillations :
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When a charged capacitor is allowed to discharge through a non resistive inductor,
electrical oscillations of constant amplitude and frequency are produced. These oscillations
are called LC oscillations.
Qualitative explanation for the production of LC oscillations :
Consider a charged capacitor of capacitance Connected to an inductor of inductance L.
1 q2
Initially the electrical energy stored in the capacitor is U E 
. As there is no current in
2 c
the circuit the energy stored in the magnetic field of inductor is zero
As soon as the capacitor is connected to an inductor the capacitor begins to discharge
through the inductor and hence current starts flowing through the inductor. As a result of this
magnetic field is setup around the inductor. In turn it produces induced emf, which opposes
the growth of current and hence the capacitor takes some finite time to discharge completely.
1
2
When it discharge completely the energy stored in the magnetic field is U B  L I o . Thus
2
electric energy of a capacitor is completely converted into magnetic energy of the inductor.
After the discharge of capacitor is complete, the magnetic flux linked with the inductor
decreases inducing a current in the same direction as the earlier current and charges the
capacitor in opposite direction. The magnetic energy of the inductor is converted into
1 qo2
electrostatic energy of the capacitor. Thus the entire energy is again stored as
in the
2 c
electric field of capacitor
The capacitor is again discharged through the inductor sending current in opposite
direction. The energy is once again stored in the magnetic field in the inductor. Thus the
process repeats in the opposite direction and finally circuit returns to the initial state.
The process then repeats itself indefinitely and the electromagnetic oscillations are
produced. The oscillations are produced due to continuous exchange of electrical energy and
magnetic energy. These oscillations of definite frequency are called LC oscillations.
If the resistance of the circuit is zero, there is no loss of energy and the amplitude of
oscillations remains constant such oscillations are called undamped oscillations. These
oscillations are similar to the mechanical oscillations of a block of mass attached to a spring.
Realistic LC oscillations :
In actual LC circuit the oscillations are not
undamped but they are damped oscillations.
Damped oscillations are those oscillations whose
amplitude decreases with time and finally dies off
Since every inductor has some resistance. Due
to this some energy is dissipated in the form of heat. So amplitude of oscillations goes on
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decreasing. Even if the resistance of inductor is zero, some energy is radiated away in the
form of electromagnetic waves. Radio and TV transmitters based on such radiation.
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