MAGNETIC FLUX
Friday, September 4th, 2009
The magnetic flux through any surface placed in a magnetic field is the
number of magnetic lines of force crossing normal to the surface.
It is denoted by the letter
If a uniform magnetic field B passes normally through a plane surface area A,
then the magnetic flux through this area is
ƒÓB=BA
If the magnetic field B makes an angle
drawn to the area A, then the component of the field
normal to this area will be B cos
ƒÓB=BA cosƒÆ
Dimensions of magnetic flux
[ƒÓB]=[ML2A-1T-2]
Unit of magnetic flux is weber (Wb)
CGS unit is maxwell
1 Wb = 108 maxwell
Electromagnetic Induction
The phenomenon of production of induced emf (and hence an induced
current) in a closed circuit as a result of the change in magneti
with it is called electromagnetic induction
Faradayfs Laws of electromagnetic Induction
ƒÓB
Į with the normal
cosĮ, so that
or tesla metre2
total
magnetic flux linked
1. Whenever the magnetic flux linked with a closed circuit changes, an
induced emf and hence an induced current is set up in it and the
induced emf (and current) lasts only so long as the change in flux lasts.
2. The magnitude of induced emf is directly proportional to the time rate
of change of magnetic flux linked with the closed surface.
mathematically |E| = dƒÓB/dt
Lenzfs Law
Lenzfs law states that the direction of induced emf is such that it opposes the
cause which produces it.
Thus, E = . dƒÓB/dt
The negative sign indicates that the direction of induced emf opposes the
change in magnetic flux.
1. Show that Lenzfs Law is in accordance with the law of conservation of
energy.
Consider the north pole of a magnet moved towards a circular coil. According
to Lenzfs law, the emf and current induced in the coil must oppose the
movement of the north pole towards it and therefore in order to repel the
north pole of the magnet a north pole is produced by the flow of induced
current in the anticlockwise direction. This means that work is done in
moving the magnet against the force of repulsion and the work done is
equivalent to the electrical energy produced by electromagnetic induction.
If we consider that Lenzfs law is not valid, then a north pole brought near
may induce a south pole, which in turn attracts the north pole making it move
faster towards the coil. The kinetic energy as well as electrical energy is
gained without spending an equivalent amount of energy or work. This
violates the law of conservation of energy.
Therefore, Lenzfs law is in accordance with the law of conservation of energy.
Determine the direction of induced current in a circular coil when the north pole
of a bar magnet is brought towards it.
Consider the north pole of a magnet moved towards a circular coil. According
to Lenzfs law, the emf and current induced in the coil must oppose the
movement of the north pole towards it and therefore in order to repel the
north pole of the magnet a north pole is produced by the flow of induced
current in the anticlockwise direction.
MOTIONAL EMF
Friday, September 4th, 2009
The emf produced across the ends of a conductor due to its motion in a
magnetic field is called motional emf.
Consider a rectangular frame of conductor
PQRS kept in a uniform magnetic field B
perpendicular to the plane as shown in
diagram. The side PQ is free to slide without
friction on the frame.
Suppose a length x of the loop lies inside the
magnetic field at any instant of time t. Then the
magnetic flux linked with the loop PQRS is
ƒÓ = BA = Blx
According to Faradayfs Law,
The induced emf is
ƒ£ = -dƒÓ/dt = -d(Blx)/dt = -Blx(dx/dt)
Or ģ = Blv
Where dx/dt=-v since the direction of velocity is in the decreasing direction of x.
This induced emf is called motional emf since it is produced by the motion of a
conductor in a magnetic field.
FLEMINGfS RIGHT HAND RULE
If we stretch the thumb, forefinger and the
central finger of right hand in mutually
perpendicular directions and if the thumb points
in the direction of motion and the forefinger
represents the direction of magnetic field, then
the central finger points in the direction of the induced current
Current induced in the loop
I = Blv/R
Force on the movable arm, F = IlB sin 90 = (Blv/R)lB = B2l2v/R
Power delivered by the external source
P = Fv = B2l2v2/R
Power Dissipated as Joule Heating Loss
P = I2R = (Blv/R )2R = B2l2v2/R
Thus, both the powers are equal, indicating that the mechanical energy spent
per second is converted to electrical energy.
Relation between induced Charge and Magnetic Flux
We know, |E|=ƒÓ/t
If R is the resistance,
I = E/R or q/t = (ƒÓ/t) (1/R)
OR
q=ƒÓ/R
The induced charge depends on the net change in magnetic flux and not on
the rate of change of magnetic flux.
Numerical Problems Based on Motional EMF
1. An aircraft with a wingspan of 40 m flies with a speed of 1080 km/h in
the eastward direction at a constant altitude in the northern
hemisphere, where the vertical component of earthfs magnetic field is
1.75 x 10-5T. Find the emf that develops between the tips of the wings.
2. A jet plane is travelling west at 450 m/s. If the horizontal component of
earthfs magnetic field at that place is 4 x 10-4 T and the angle of dip is 30
degree, find the emf induced between the ends of wing having a span of
30 m.
3. A railway track running north . south has twp parallel rails 1.0 m
apart. Calculate the value of induced emf between the rails, when a
train passes at a speed of 90 km/h. The horizontal component of earthfs
magnetic field at that place is 0.3 x 10-4 T and the angle of dip is 60
degree.
4. A conductor of length 1.0 m falls freely under the action of gravity from
a height of 10 m so that it cuts the lines of force of the horizontal
component of earthfs magnetic field of 0.3 x 10-4 T. Find the emf
induced in the conductor.
5. A wheel with 10 metallic spokes each 0.5 m long is rotated with a speed
of 120 rev/minute in a plane normal to the horizontal component of
earthfs magnetic field BH =0.4 G. What is the induced emf between the
axle and rim of the wheel.
METHODS OF GENERATING INDUCED EMF
Friday, September 4th, 2009
An emf is induced in a circuit by changing the magnetic flux linked with the
circuit.
The magnetic flux ƒ³ = BA cos ƒÆ can be changed by
1. Changing the magnetic field B :
Magnetic flux ƒ³ can be changed by changing the magnetic field B and
hence emf can be induced in the circuit
2. Changing area A of the coil :
Magnetic flux ƒ³ can be changed by changing the area of the loop A
which is acted upon by the magnetic field B and hence emf can be
induced in the circuit.
3. Changing the relative orientation of the coil, i.e., the angle Į between B
and A.
ƒ³ = N B A cos ƒÆ
At time t, with angular velocity ƒÖ,
Į = ąt (at t = 0, loop is assumed to be perpendicular to the magnetic field and
ƒÆ = 0‹)
ƒ³ = N B A cos ƒÖt
Differentiating w.r.t. t,
dƒ³ / dt = . NBAƒÖ sin ƒÖt
E = . dƒ³ / dt
E = NBAƒÖ sin ƒÖt
E = E0 sin ƒÖt (where E0 = NBA
The emf changes continuousl
w.r.t. time giving rise to alternating emf.
If initial position of the coil is taken as 0‹, i.e. normal to the coil is at 90‹ with
the magnetic field, then Į
alternating emf and consequently alternating current can be expressed in sin
or cos function.
This method of inducing emf is the basic principle of generators.
Self Induction
Friday, September 4th, 2009
Self Induction is the phenomenon of production of induced emf in a coil when
the strength of current passing through it changes.
The induced emf opposes the growth or decay of current in the coil and hence
delays the current to acquir
Self induction is also called inertia of electricity as it opposes the growth or
decay of current.
Self Inductance:
ƒ³ ƒ¿ I or ƒ³ = LI
t NBAƒÖ is the maximum emf)
continuously in magnitude and periodically in direction
becomes ƒÆ + ƒÎ/2 or ƒÖt + ƒÎ/2 E = E0 cos
ting acquire the maximum value.
y /ƒÖt So,
If I = 1, then L = ƒ³
(where L is the constant of proportionality and is known as Self Inductance or
co-efficient of self induction)
Thus, self inductance is defined as the magnetic flux linked with a coil
when unit current flows through it.
Also, E = . dƒ³ / dt or E = . L (dI / dt)
If dI / dt = 1, then L = E
Thus,
self inductance is defined as the induced emf set up in the coil
through which the rate of change of current is unity.
SI unit of self inductance is henry (H).
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 solenoid:
Magnetic Field due to the solenoid is
B = ƒÊ0nI
Magnetic Flux linked across one turn of the coil is
ƒ³ per turn = B A = ƒÊ0nIA = ƒÊ0NIA / l
Magnetic Flux linked across N turns of the coil is
ƒ³ = ƒÊ0N2IA / l
But, ƒ³ = LI
So, L = ƒÊ0N2A / l = ƒÊ0n2Al
Energy in Inductor:
Small work done dW in establishing a current I in the coil in time dt is dW = .
EI dt
dW = LI dI (since E = -L(dI / dt)
W = ç L I dI = . LI0
AC generator
Friday, September 4th, 2009
The ac generator is a device used for converting mechanical energy into
electrical energy.
Principle
It is based on the principle of electromagnetic induction, according to which
an emf is induced in a coil when it is rotated in a uniform magneti
Essential parts of an AC generator
(i) Armature
Armature is a rectangular coil consisting of a large number of
loops or turns of insulated copper wire wound over a laminated soft
iron core or ring. The soft iron core not only increases the mag
but also serves as a support for the coil
(ii) Field magnets
The necessary magnetic field is provided by permanent magnets in
the case of low power dynamos. For high power dynamos, field is
provided by electro magnet. Armature rotates betwe
poles such that the axis of rotation is perpendicular to the magnetic field.
(iii) Slip rings
The ends of the armature coil are connected to two hollow metallic rings R1
and R2 called slip rings. These rings are fixed to a shaft, to whic
armature is also fixed. When the shaft rotates, the slip rings along with the
armature also rotate.
(iv) Brushes
B1 and B2 are two flexible metallic plates or carbon brushes. They
provide contact with the slip rings by keeping themselves pressed
against the ring. They are used to pass on the current from the
armature to the external power line through
the slip rings.
Working
Whenever, there is a change in
orientation of the coil, the magnetic flux linked
with the coil changes, producing an induced
emf in the coil.
The direction of the induced current is given by
Flemingfs right hand rule.
Suppose the armature ABCD is
, between the magnetic
he magnetic field.
magnetic flux
en which the
initially in the vertical position. It is rotated in
The side AB of the coil moves downwards and the side DC moves
Then according to Flemings right hand rule the current induced in arm AB
flows from B to A and in CD it flows from D to C. Thus the current flows
along DCBA in the coil. In the external circuit the current flows from B1 to
B2.
On further rotation, the arm AB of the coil
moves upwards and DC moves downwards.
Now the current in the coil
In the external circuit the current flows from
B2 to B1. As the rotation of the coil continues,
the induced current in the external circuit
keeps changing its direction for every half a rotation of the coil. Hence the
induced current is alternating in nature. As the armature completes
rotations in one second, alternating current of frequency
produced. The induced emf at any instant is given by
e= Eo sin ƒÖt
The peak value of the emf, Eo = NBA
coil, A is the area enclosed by the coil, B is the magnetic field and
angular velocity of the coil.
EDDY CURRENTS
changes are called Eddy Currents.
Demonstration of Eddy Currents
Take a cylindrical electromagnet fed by an AC
source and place a small metal disc over its top. As
the current is switched on, the magnetic field at the
disc rises from zero to a finite value, setting up eddy
currents which effectively convert it into a small
the anticlockwise direction.
A flows along ABCD.
ƒË cycles per second is
NBAƒÖ where N is the number of turns of the
Friday, September 4th, 2009
Currents induced in soild metallic masses when
the magnetic flux threading through them
rrents upwards.
ƒË
ber ƒÖ is the
magnet. If initially, the top end of the electromagnet acquires N- polarity by
Lenzfs law, the lower side of the disc will also acquire north polarity resulting
in a repulsive force. So, when the electromagnet is switched on, the disc is
thrown off.
Applications of Eddy Currents
1. Induction Furnace
In an induction furnace, high temperature is produced by generating
eddy currents. The material to be melted is placed in a varying
magnetic field of high frequency. Hence a strong eddy current is
developed inside the metal. Due to the heating effect of the current, the
metal melts.
2. Electromagnetic Damping
When current is passed through a galvanometer, the coil oscillates
about its mean position before it comes to rest. To bring the coil to rest
immediately, the coil is wound on a metallic frame. Now, when the coil
oscillates, eddy currents are set up in the metallic frame, which opposes
further oscillations of the coil. This inturn enables the coil to attain its
equilibrium position almost instantly. Since the oscillations of the coil
die out instantaneously, the galvanometer is called dead beat
galvanometer.
3. Electromagnetic Brakes:
A metallic drum is coupled to the wheels of a train. The drum rotates
along with the wheel when the train is in motion.When the brake is
applied, a strong magnetic field is developed and hence, eddy currents
are produced in the drum which oppose the motion of the drum. Hence,
the train comes to rest.
4. Speedometers:
In a speedometer, a magnet rotates according to the speed of the vehicle.
The magnet rotates inside an aluminium cylinder (drum) which is held
in position with the help of hair springs. Eddy currents are produced in
the drum due to the rotation of the magnet and it opposes the motion of
the rotating magnet. The drum inturn experiences a torque and gets
deflected through a certain angle depending on the speed of the vehicle.
A pointer attached to the drum moves over a calibrated scale which
indicates the speed of the vehicle
5. Induction Motor :
Eddy currents are produced in a metallic cylinder called rotor, when it
is placed in a rotating magnetic field. The eddy current initially tries to
decrease the relative motion between the cylinder and the rotating
magnetic field. As the magnetic field continues to rotate, the metallic
cylinder is set into rotation. These motors are used in fans.
Some more Applications
1. Electromagnetic shielding
2. Inductothermy
3. Energy meters
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