Transcript Moving Conductor in a Magnetic Field

Electromagnetic Induction
• emf is induced in a conductor placed in a magnetic
field whenever there is a change in magnetic field.
Moving Conductor in a Magnetic
Field
• Consider a straight conductor moving
with a uniform velocity, v, in a
stationary magnetic field.
• The free charges in the conductor
experience a force which will push
them to one end of the conductor.
• An electric field is built up due to the
electron accumulation.
• An e.m.f. is generated across the
conductor such that
E = Blv.
Induced Current in Wire Loop
• An induced current passes
around the circuit when the
rod is moved along the rail.
• The induced current in the
rod causes a force F = IlB,
which opposes the motion.
• Work done by the applied force to keep the rod moving is
W  Fx  Fv t  IlBv t
• Electrical energy is produced from the work done such that
E = E It = W
E
= Blv
Lenz’s Law
• The direction of the induced current is always so
as to oppose the change which causes the
current.
Magnetic Flux
• The magnetic flux is a measure of the number of
magnetic field lines linking a surface of crosssectional area A.
• The magnetic flux through

a small surface is the
product of the magnetic
flux density normal to the
surface and the area of the
surface.
Unit : weber (Wb)
  B A  BAcos
Faraday’s Law of Electromagnetic
Induction
• The induced e.m.f. in a circuit is equal to the rate
of change of magnetic flux linkage through the
circuit.
d
 
dt
The ‘-’ sign indicates that the induced e.m.f. acts to
oppose the change.
http://physicsstudio.indstate.edu/java/physlets/java/indcur/index.html
Induced Currents Caused by
Changes in Magnetic Flux
• The magnetic flux (number of field lines passing
through the coil) changes as the magnet moves
towards or away from the coil.
http://micro.magnet.fsu.edu/electromag/java/lenzlaw/index.html
Faraday Disk Dynamo
d  vBdr rdrB 
 
R
0
1
2
Brdr  BR
2
Simple a.c. Generator
• According to the Faraday’s law of electromagnetic
induction,
Nd
d
 
  N BA cos t  NBA  sin t
dt
dt
http://www.walter-fendt.de/ph11e/generator_e.htm
Simple d.c. Generator
  NBA  sin t
Eddy Current
• An eddy current is a swirling current set up in a
conductor in response to a changing magnetic
field.
• Production of eddy currents in a rotating wheel
Applications of Eddy Current (1)
• Metal Detector
Applications of Eddy Current (2)
• Eddy current levitator
• Smooth braking device
• Damping of a vibrating system
Back emf in Motors
• When an electric motor is running, its armature
windings are cutting through the magnetic field
of the stator. Thus the motor is acting also as a
generator.
• According to Lenz's Law, the induced voltage in
the armature will oppose the applied voltage in
the stator.
• This induced voltage is called back emf.
Back emf and Power
Armature coils, R
Back emf, Eb
Driving source, V
V   b  IR
Multiplying by I, then
VI   b I  I R
2
b I  VI  I R
2
• So the mechanical power developed in motor
 b I
Variation of current as a motor
is started
I
I max 
V
R
Larger load
Zero load
t
0
• As the coil rotates, the angular speed as well as the
back emf increases and the current decreases until the
motor reaches a steady state.
The need for a starting resistance
in a motor
• When the motor is first switched on,  =0.
• The initial current, Io=V/R, very large if R is small.
• When the motor is running, the back emf increases,
so the current decrease to its working value.
• To prevent the armature burning out under a high
starting current, it is placed in series with a
rheostat, whose resistance is decreases as the
motor gathers speed.
Variation of current with the steady
angular speed of the coil in a motor
I
I max
V

R
V  NBA 
I
R
0
 max
V

NBA

• The maximum speed of the motor occurs when the current
in the motor is zero.
Variation of output power with the
steady angular speed of the coil in a
motor
Po
Pmax
V2

4R
NBA  (V  NBA  )
Po 
R
0
 max
V

NBA

• The output power is maximum when the back emf is ½ V.
Transformer
• A transformer is a device for stepping up or down an
alternating voltage.
• For an ideal transformer,
– (i.e. zero resistance and no flux leakage)
d B
Vp   N p
dt
d B
Vs   N s
dt
Vs N s
 
Vp N p
Transformer Energy Losses
•
Heat Losses
1. Copper losses - Heating effect occurs in the
copper coils by the current in them.
2. Eddy current losses - Induced eddy currents flow
in the soft iron core due to the flux changes in the
metal.
•
Magnetic Losses
1. Hysteresis losses - The core dissipates energy on
repeated magnetization.
2. Flux leakage - Some magnetic flux does not pass
through the iron core.
Designing a transformer to
reduce power losses
• Thick copper wire of low resistance is used to reduce the
heating effect (I2R).
• The iron core is laminated, the high resistance between the
laminations reduces the eddy currents as well as the heat
produced.
• The core is made of very soft iron, which is very easily
magnetized and demagnetized.
• The core is designed for maximum linkage, common method
is to wind the secondary coil on the top of the primary coil and
the iron core must always form a closed loop of iron.
Transmission of Electrical Energy
• Wires must have a low resistance to reduce power loss.
• Electrical power must be transmitted at low currents to reduce
power loss.
• To carry the same power at low current we must use a high
voltage.
• To step up to a high voltage at the beginning of a transmission
line and to step down to a low voltage again at the end we
need transformers.
Direct Current Transmission
• Advantages
– a.c. produces alternating magnetic field which induces
current in nearby wires and so reduce transmitted
power; this is absent in d.c.
– It is possible to transmit d.c. at a higher average
voltage than a.c. since for d.c., the rms value equals
the peak; and breakdown of insulation or of air is
determined by the peak voltage.
• Disadvantage
– Changing voltage with d.c. is more difficult and
expensive.
Self Induction
• When a changing current passes
through a coil or solenoid, a
changing magnetic flux is produced
inside the coil, and this in turn
induces an emf.
• This emf opposes the change in
flux and is called self-induced emf.
• The self-induced emf will be against
the current if it is increasing.
• This phenomenon is called selfinduction.
Definitions of Self-inductance
(1)
• Definition used to find L
The magnetic flux linkage in a coil  the current flowing
through the coil.
  LI
Where L is the constant of proportionality for the coil.
L is numerically equal to the flux linkage of a circuit
when unit current flows through it.
L

I
Unit : Wb A-1 or H (henry)
Definitions of Self-inductance
(2)
• Definition that describes the behaviour of an
inductor in a circuit
d
dI
 
 L
dt
dt
L

dI
dt
L is numerically equal to the emf induced in the circuit
when the current changes at the rate of 1 A in each second.
Inductors
• Coils designed to produce large self-induced
emfs are called inductors (or chokes).
• In d.c. circuit, they are used to slow the growth of
current.
• Circuit symbol
or
Inductance of a Solenoid
• Since the magnetic flux density due to a solenoid is
 o NI
B

• By the Faraday’s law of electromagnetic induction,
d
  N
dt
d
d  o NIA
  N ( BA)   N (
)
dt
dt

2
2

N
A
 o N A dI
o
L



dt
Energy Stored in an Inductor
• The work done against the back emf in bringing
the current from zero to a steady value Io is
Io
W   Idt
0

Io
0
dI
LI dt
dt
Io
  LIdI
0
1 2
 LI o
2
Current growth in an RL circuit
dI
V L
 IR
dt
I 

R
(1  e
 Rt
L
)
• At t = 0, the current is zero.
• So L dIdt  
• As the current grows, the
p.d. across the resistor
increases. So the selfinduced emf ( - IR) falls;
hence the rate of growth of
current falls.
dI
0
• As t
dt
Decay of Current through an Inductor
• Time constant for RL
R
circuit  
L
I  Ioe
 Rt
L
• The time constant is the
time for current to
decrease to 1/e of its
original value.
• The time constant is a
measure of how quickly
the current grows or
decays.
emf across contacts at break
• To prevent sparking at the contacts of a switch in
an inductive circuit, a capacitor is often
connected across the switch.
The energy originally stored
in the magnetic field of the coil
is now stored in the electric
field of the capacitor.
+
-
1 2 1
LI  CV 2
2
2
Switch Design
• An example of using a protection diode with a relay coil.
+
-
• A blocking diode parallel to the inductive coil is used to
reduce the high back emf present across the contacts
when the switch opens.
Non-Inductive Coil
• To minimize the self-inductance, the coils of resistance boxes
are wound so as to set up extremely small magnetic fields.
• The wire is double-back on itself. Each part of the coil is then
travelled by the same current in opposite directions and so
the resultant magnetic field is negligible.