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COM } ON GOOGE
EMI & AC
THEORY
1. MAGNETIC FLUX
Various Methods of Producing induced E.M.F.
(1) The total number of magnetic lines of force passing
normally through an area placed in a magnetic field is equal We have learnt that e.m.f. is induced in a circuit, whenever
to the magnetic flux linked with that area. the amount of magnetic flux linked with the circuit is
changed. As I = BA cos T, the magnetic flux I can be
changed by changing B, A or T. Hence there are three
methods of producing induced e.m.f.
1. By changing the magnitude of magnetic field B,
2. By changing the area A, i.e., by shrinking or stretching or
changing the shape of the coil.
3. By changing angle T between the direction of B and normal
to the surface area A, i.e., changing the relative orientation
of the surface area and the magnetic field.
(2) Net flux through the surface I = ³ B. dA = BA cos T
3. LENZ’S LAW
(T is the angle between area vector and magnetic field
vector) If T = 0o then I= BA, If T = 90o then I = 0 This law gives the direction of induced emf/induced current.
According to this law, the direction of induced emf or current in a
(3) Unit and Dimension : Magnetic flux is a scalar quantity. It’s
circuit is such as to oppose the cause that produces it. This law is
S.I. unit is weber (wb), CGS unit is Maxwell or Gauss × cm2;
based upon law of conservation of energy.
(1wb = 108 Maxwell).
(1) When N-pole of a bar magnet moves towards the coil, the
Num Joule Volt u Coulomb
(4) Other units : Tesla × m 2 flux associated with loop increases and an emf is induced
Amp Amp Amp
in it. Since the circuit of loop is closed, induced current
= Volt × sec = Ohm × Coulomb = Henry × Amp. It’s also flows in it.
dimensional formula [I] = [ML2T–2A–1]
(2) Cause of this induced current, is approach of north pole
2. FARADAY’S LAWS OF EMI and therefore to oppose the cause, i.e., to repel the
approaching north pole, the induced current in loop is in
(1) First law : Whenever the number of magnetic lines of such a direction so that the front face of loop behaves as
force (magnetic flux) passing through a circuit changes an north pole. Therefore induced current as seen by observer
emf is produced in the circuit called induced emf. The O is in anticlockwise direction. (figure)
induced emf persists only as long as there is change or
cutting of flux.
(2) Second law : The induced emf is given by rate of change
dI
of magnetic flux linked with the circuit i.e. e . . For
dt
NdI
N turns e ; Negative sign indicates that induced
dt
emf (e) opposes the change of flux. (3) If the loop is free to move the cause of induced emf in the
coil can also be termed as relative motion. Therefore to
Induced current (i) Induced charge (q) Induced power (P) oppose the cause, the relative motion between the
e N dI e2 N 2 § dI ·
2 approaching magnet and the loop should be opposed.
N
i . dq idt .dI P ¨ ¸ For this, the loop will itself start moving in the direction of
R R dt R R R © dt ¹
motion of the magnet.
Induced charge It depends on (4) It is important to remember that whenever cause of induced
is time indepen- time and resistance emf is relative motion, the new motion is always in the
dent. direction of motion of the cause.
EMI & AC
Table : The various positions of relative motion between the magnet and the coil
Position of magnet
Behaviour of face As a north pole As a south pole As a south pole As a north pole
of the coil
Type of magnetic Repulsive force Attractive force Repulsive force Attractive force
force opposed
Magnetic field linked Cross (×), Increases Cross (×), Decreases Dots () Increases Dots () Decreases
with the coil and it’s
progress as viewed
from left
4. EDDY CURRENT
(i) Dead-beat galvanometer : A dead beat galvanometer
When a changing magnetic flux is applied to a bulk piece of means one whose pointer comes to rest in the final
conducting material then circulating currents called eddy currents equilibrium position immediately without any oscillation
are induced in the material. Because the resistance of the bulk about the equilibrium position when a current is passed
conductor is usually low, eddy currents often have large in its coil.
magnitudes and heat up the conductor.
This is achieved by winding the coil on a metallic
(1) These are circulating currents like eddies in water.
frame the large eddy currents induced in the frame provide
(2) Experimental concept given by Focault hence also named electromagnetic damping.
as “Focault current”.
(ii) Electric-brakes : When the train is running its wheel is
(3) The production of eddy currents in a metallic block leads
moving in air and when the train is to be stopped by
to the loss of electric energy in the form of heat.
electric breaks the wheel is made to move in a field created
(4) By Lamination, slotting processes the resistance path for
by electromagnet. Eddy currents induced in the wheels
circulation of eddy current increases, resulting in to
due to the changing flux oppose the cause and stop
weakening them and also reducing losses causes by them
the train.
(iii) Induction furnace : Joule’s heat causes the melting of a
metal piece placed in a rapidly changing magnetic field.
(iv) Speedometer : In the speedometer of an automobile, a
magnet is geared to the main shaft of the vehicle and it
rotates according to the speed of the vehicle. The magnet
is mounted in an aluminium cylinder with the help of
hair springs. When the magnet rotates, it produces eddy
currents in the drum and drags it through an angle, which
indicates the speed of the vehicle on a calibrated scale.
(v) Energy meter : In energy meters, the armature coil carries
a metallic aluminium disc which rotates between the poles
of a pair of permanent horse shoe magnets. As the
armature rotates, the current induced in the disc tends
(5) Application of eddy currents : Though most of the times to oppose the motion of the armature coil. Due to this
eddy currents are undesirable but they find some useful braking effect, deflection is proportional to the energy
applications as enumerated below consumed.
EMI & AC
1 dI 1
q ³ i dt ³ R dt
dt
R
dI ³ when r < a; E =
r dB
2 dt
; En v r
6. INDUCED ELECTRIC FIELD (1) Consider a conducting rod of length l moving with a
It is non-conservative and non-electrostatic in nature. Its field uniform velocity v perpendicular to a uniform magnetic
lines are concentric circular closed curves.
field B , directed into the plane of the paper. Let the rod be
dB moving to the right as shown in figure. The conducting
A time varying magnetic field always produced induced electrons also move to the right as they are trapped within
dt
the rod.
electric field in all space surrounding it.
Induced electric field (E in) is directly proportional to
dI
From Faraday’s second laws e ..…(ii)
dt
ª Vº
Induced emf e = El = Bvl «E
¬ A »¼
dI dB dB
So ³E in .dA e
dt
A
dt
i.e. E 2 Sr Sa 2
dt
a 2 dB 1
where r t a or E ; E in v (3) Motion of conducting rod on an inclined plane : When
2r dt r conductor start sliding from the top of an inclined plane
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EMI & AC
as shown, it moves perpendicular to it’s length but at an (2) Magnetic force : Conductor PQ experiences a magnetic
angle (90 T ) with the direction of magnetic field. force in opposite direction of it’s motion and
§ BvA · B 2 vA 2
Fm BiA B¨ ¸A
© R ¹ R
dW B 2 vA 2 B2 v 2 A 2
Pmech Pext Fext .v uv
Hence induced emf across the ends of conductor dt R R
e = Bv sin(90 – T)l = Bvl cosT (4) Electrical power : Also electrical power dissipated in
BvA cos T resistance or rate of heat dissipation across resistance is
So induced current i (Directed from Q to P). given as
R
2
The forces acting on the bar are shown in following figure. H § BvA · B2v 2A 2
Pthermal i 2R ¨ ¸ .R ; Pthermal
The rod will move down with constant velocity only if t © R ¹ R
Fm cos T = mg cos (90 – T) = mg sin T (It is clear that Pmech. = Pthermal which is consistent with the
Bil cos T = mg sin T principle of conservation of energy.)
(5) Motion of conductor rod in a vertical plane : If conducting
§ Bv A cos T · mgR sin T
B¨ T ¸A cos T mg sin T vT rod released from rest (at t = 0) as shown in figure then
© R ¹ B 2 A 2 cos 2 T with rise in it’s speed (v), induces emf (e), induced current
(i), magnetic force (Fm) increases but it’s weight remains
8. MOTIONAL EMI IN LOOP BY GENERATED AREA constant.
Rod will achieve a constant maximum (terminal) velocity
If conducting rod moves on two parallel conducting rails
vT if Fm = mg
as shown in following figure then phenomenon of induced
emf can also be understand by the concept of generated B 2 v T2 A 2 mgR
area (The area swept of conductor in magnetic field, during So mg vT
R B2 A 2
it’s motion)
e BvA
i
R R
EMI & AC
Induced emf across the axle of the wheels of the train and it is
across the tips of the wing of the aeroplane is given by e = Bvlv
where l = length of the axle or distance between the tips of the
wings of plane, B v = vertical component of earth’s magnetic field
and v = speed of train or plane.
In time t the area swept by the loop in the field i.e. region II
field as shown below in fig. Flux link with the rotating loop at time t I = BA. Hence induced
1
of emf ‘e’ where e BZr 2
2
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EMI & AC
For uniform rotational motion with Z, the flux linked with coil at (1) Coefficient of self-induction : Number of flux linkages with
any time t the coil is proportional to the current i. i.e. NI v i or
I = NBA cos T = NBA cos Zt NI Li (N is the number of turns in coil and NI – total
i 1 2
U ³0
Lidi
2
Li ;
1 NIi
Also U Li i
2 2
Solenoid
P0 N 2r
L P 0 n 2 AA
A
2 2P 0 N 2 a dI 2 di1
L secondary e 2 N 2 ; e2 M
S dt dt
Coaxial cylinders (iv) Magnetic permeability of medium between the coils (Pr)
P0 r or nature of material on which two coils are wound
L log e 2
2 Sr r1
(v) Distance between two coils (As d increases so M
2.303 r decreases)
P 0 log10 2
2 Sr r1
(vi) Orientation between primary and secondary coil (for 90o
orientation no flux relation M = 0)
(vii) Coupling factor ‘K’ between primary and secondary
coil
EMI & AC
K
Magnetic flux linked in sec ondary
; SP 0 N1N 2 r 2
M
Magnetic flux linked in primary 2R
0dKd1
Two Solenoids
P 0 N1 N 2 A
M
A
Two concentric
(7) The various formulae for M : coplaner square coils
P 0 2 2 N1 N 2 A 2
M
SL
When they are situated close to each other, then net inductance LS = L1 + L2 ± 2M
Mutual induction is absent (k = 0) Mutual induction is present and Mutual induction is present and
favours self inductance of coils opposes self inductance of coils
Leq = L1 + L2
(2) Parallel : If two coils of self-inductances L1 and L2 having When they are situated close to each other, then
mutual inductance are connected in parallel and are far
1 1 1 L1L 2 M 2
from each other, then net inductance L is LP
LP L1 L 2 L1 L 2 r 2M
L1L 2
LP
L1 L 2
EMI & AC
Mutual induction is absent (k = 0) Mutual induction is present and Mutual induction is present and
favours self inductance of coils opposes self inductance of coil
L 1L 2 L1L 2 M 2 L1L 2 M 2
L eq L eq L eq
L1 L 2 L1 L 2 2M L1 L 2 2M
§ E·
i¨ ¸ . Just after closing the switch as i = 0, inductor act
© R¹
as open circuit i.e. broken wires and long after the switch
has been closed as i = i0, the inductor act as a short circuit
i.e. a simple connecting wire.
ª tº
R
E
i i 0 «1 e L » ; where i 0 i max = steady state
«¬ »¼ R
current.
(2) The value of current at any instant of time t after opening
from the steady state condition (i.e. during the decaying
R
t
of current) is given by i i 0e L
L
(3) Time constant (W) : It is given as W ; It’s unit is second.
R
In other words the time interval, during which the current
in an inductive circuit rises to 63% of its maximum value at 14. LC OSCILLATION
make, is defined as time constant or it is the time interval,
during which the current after opening an inductive circuit When a charged capacitor C having an initial charge q0 is
falls to 37% of its maximum value. discharged through an inductance L, the charge and current in the
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EMI & AC
1 rad
Frequency of oscillation is given by Z
LC sec
(3) Working : Force on any arm of the coil is given by Pmechanical Pout e Back e.m.f .
Efficiency K
Psup plied Pin E Supply voltage
F i A u B in fig., force on AB will be perpendicular to
plane of the paper and pointing inwards. Force on CD will (8) Uses of dc motors : They are used in electric locomotives,
be equal and opposite. So coil rotates in clockwise sense electric ears, rolling mills, electric cranes, electric lifts, dc
when viewed from top in fig. The current in AB reverses drills, fans and blowers, centrifugal pumps and air
due to commutation keeping the force on AB and CD in compressors, etc.
such a direction that the coil continues to rotate in the
16. DC GENERATOR
same direction.
If the current produced by the generator is direct current, then the
(4) Back emf in motor : Due to the rotation of armature coil in
generator is called dc generator.
magnetic field a back emf is induced in the circuit. Which
is given by e = E – iR. dc generator consists of (i) Armature (coil) (ii) Magnet (iii)
Commutator (iv) Brushes
Back emf directly depends upon the angular velocity Z of
In dc generator commutator is used in place of slip rings. The
armature and magnetic field B. But for constant magnetic commutator rotates along with the coil so that in every cycle
field B, value of back emf e is given by e v Z or e = kZ when direction of ‘e’ reverses, the commutator also reverses or
(e = NBAZ sinZt) makes contact with the other brush so that in the external load the
current remains in the some direction giving dc
EMI & AC
di
6. A piece of metal and a piece of non-metal are dropped from 15. If main current through a coil increases (in) so will be
dt
the same height near the surface of the earth. The non-
metallic piece will reach the ground first because there will positive (+ve), hence induced emf e will be negative (i.e.
be no induced current in it. opposite emf) Enet = E – e
EMI & AC
L
and it is given by T = 0.693 .
R