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Electromagnetic Induction

- emf is induced in a conductor placed in a

magnetic field whenever there is a change in

magnetic field.

Faradays work

- Faraday suggested that an e.m.f. is induced in a

conductor when - 1. there is a change in the number of lines

linking it, - 2. it cuts across field lines.

- As shown in the figure, if the coil moves towards

the magnet from X to Y, the number of magnetic

field lines linking it increases from three to

five alternatively we can say it cuts two lines

in moving from X to Y. - Hence, an e.m.f. is induced in the coil.

- The figure shows two stationary coils A and B.
- When a current flowing in coil A increases or

decreases, the magnetic flux linking coil B

increases and decreases respectively. Hence, an

e.m.f. is induced in coil B.

Magnetic flux

- B magnetic flux density (i.e. the number of

magnetic field lines per unit cross-section area)

- A cross-section area,
- magnetic flux linking the area is the product BA

represents the number of field lines linking a

surface of cross-sectional area A. - ? BA
- If B 1 T and A 1 m2, F is defined to be 1

(Tm2) or weber (Wb).

Magnetic flux

- If the surface is not perpendicular to the field

with the normal to the surface making an angle q

to the field, the magnetic flux linking the area

is ? BA cos q. - If F is the flux through the cross-section area A

of a coil of N turns, the total flux through it,

called the flux-linkage, is NF since the same

flux F links each of the N turns.

Example 1

- A circular coil of 20 turns with diameter 10 cm

is placed in a region of uniform magnetic field

of 1.5 T. Find the flux-linkage if the plane of

the coil - (a) is perpendicular to the field,
- Solution

Example 1

- A circular coil of 20 turns with diameter 10 cm

is placed in a region of uniform magnetic field

of 1.5 T. Find the flux-linkage if the plane of

the coil - (b) is along the field, and
- Solution

Example 1

- A circular coil of 20 turns with diameter 10 cm

is placed in a region of uniform magnetic field

of 1.5 T. Find the flux-linkage if the plane of

the coil - (c) makes an angle of 30o to the field.
- Solution

Faradays law

The induced e.m.f. is directly proportional to

the rate of change of flux-linkage or rate of

flux cutting.

- Mathematically, or

. - It is defined that 1 Wb is magnetic flux that

induces in a one-turn coil an e.m.f. of 1 volt

when the flux is reduced to zero in 1 s. - By putting e 1 V, dt 1 s and d(NF) 1 Wb, we

have 1 constant x 1/1. - Hence,

Example 2

- (a) Suppose a 5000-turn coil of cross-section

area 5 cm2 is at right angles to a flux density

of 0.2 T, which is then reduced steadily to zero

in 10 s. Find the e.m.f. induced in the coil. - (b) Find the e.m.f. induced if the normal to the

plane of coil makes an angle of 60o with the

field. - Solution

Lenzs law

Lenz's law

Induced I always flows to oppose the movement

which started it.

In both cases, magnet moves against a force.

Work is done during the motion it is

transferred as electrical energy.

- Lenzs law is incorporated in the mathematic

expression of Faradays law by including a

negative sign to show that current due to the

induced e.m.f. produces an opposing flux change.

So we have

Fleming's right-hand rule

For a wire cutting through a B-field...

motion or force F

magnetic field B

induced current I

Example 3

- State the direction of induced current flowing

through coil B observed by the observer when the

current through coil A increases steadily. - Solution

Calculation of e.m.f.

- Consider a conducting rod of length l moving

sideways with constant velocity v through and at

right angles to a uniform magnetic field of flux

density B. - Area swept out per second by the rod per second

lv - Flux cut per second Blv
- e.m.f induced rate of flux-cutting flux cut

per second - e Blv

Alternative derivation

- Magnetic force Bqv.
- An electric field is built up due to the

accumulation of charges. - Electric force qE
- Finally, equilibrium is reached when magnetic

force acting on electrons is balanced by electric

force. - Hence, qE Bqv ? E Bv
- An e.m.f. e is generated across the conductor

such that - e El Blv.

Example 4

- A metal aircraft of wing span l 32 m is flying

with speed v 190 ms-1 towards the earths

magnetic north pole in a region where the earths

magnetic field BR 4.3 x 10-5 T and the angle of

dip a 65o. - Calculate the e.m.f. induced across its wing

tips. - Solution

Simple a.c. Generator

- According to the Faradays law of electromagnetic

induction,

http//www.walter-fendt.de/ph11e/generator_e.htm

Simple d.c. Generator

Back e.m.f. Sparks appear while opening a switch

- There is current flowing in the coil of the

electromagnet in use. - When the circuit is broken by opening the switch,

the current starts to drop and the flux linkage

through the coil of the electromagnet decreases

suddenly. - By Faradays law, a large induced e.m.f. would

develop across the coil of the electromagnet so

as to oppose the change. - Sparks occur due to the discharge across the

small gap of the switch.

DC motors

- A d.c. motor consists of a coil on an axle,

carrying a d.c. current in a magnetic field. - The coil experiences a couple as in a moving-coil

galvanometer which makes it rotate. - When its plane is perpendicular to the field, a

split-ring commutator reverses the current in the

coil and ensures that the couple continues to act

in the same direction thereby maintaining the

rotation.

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

Multiplying by I, then

- So the mechanical power developed in motor

Variation of current as a motor is started

Larger load

Zero load

- 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, IoV/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

- The maximum speed of the motor occurs when the

current in the motor is zero.

Eddy Current

- An eddy current is a swirling current set up in a

conductor in response to a changing magnetic

field. - When the magnetic flux linkage through a

conductor changes, an e.m.f. is induced in it. - If the conductor is a lump of metal. These are

known as Eddy Currents.

- Eddy currents may be quite large because of the

low resistance of the paths they follow.

Consider a metallic sheet moving away from a

magnetic field.

- By Lenzs law, eddy currents must flow in a

direction to oppose the motion of the sheet. - Hence, eddy currents act as an effective brake to

its motion. - The mechanical work done is converted into

internal energy of the sheet.

Applications 1 Smooth Braking Device

- The eddy currents induced in the copper plate

produce a strong braking effect on the plate

which stops oscillating quickly. - If the copper plate is replaced by one with

slits, the induced eddy currents, which can only

flow within the narrow teeth between the slits,

are greatly reduced. This is because the

resistance of the path which the eddy currents

follow is increased.

Braking effect in moving-coil galvanometer

- As the core swings in a magnetic field, eddy

currents are induced in it. Since the eddy

currents flow in a direction to oppose the

motion, unwanted oscillations are reduced.

- Ideal design of the core is to produce critical

damping in which oscillation is just avoided.

Metal Detector

- A pulsing current is applied to the coil, which

then induces a magnetic field shown in blue. When

the magnetic field of the coil moves across

metal, such as the coin in this illustration, the

field induces electric currents (called eddy

currents) in the coin. - The eddy currents induce their own magnetic

field, shown in red, which generates an opposite

current in the coil, which induces a signal

indicating the presence of metal. - A metal detector can also be used to detect mines

buried underground.

Induction cooker

- The induction cooker uses coils of wire with high

frequency a.c. to produce large eddy currents in

the metal cooking pot placing above. The heating

effect of the eddy current cooks the food. - Moreover, since eddy current is not induced in

its plastic case which is made up of non-metallic

material, the cooker is not hot to touch.

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)

Transformer Energy Losses

- Heat Losses
- Copper losses - Heating effect occurs in the

copper coils by the current in them. - Eddy current losses - Induced eddy currents flow

in the soft iron core due to the flux changes in

the metal. - Magnetic Losses
- Hysteresis losses - The core dissipates energy on

repeated magnetization. - 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 self-induction.

Definitions of Self-inductance (1)

- Definition used to find L

The magnetic flux linkage in a coil ? the current

flowing through the coil.

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.

Unit Wb A-1 or H (henry)

Definitions of Self-inductance (2)

- Definition that describes the behaviour of an

inductor in a circuit

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

- By the Faradays law of electromagnetic induction,

Energy Stored in an Inductor

- The work done against the back emf in bringing

the current from zero to a steady value Io is

Current growth in an RL circuit

- At t 0, the current is zero.
- So

- As the current grows, the p.d. across the

resistor increases. So the self-induced emf (? -

IR) falls hence the rate of growth of current

falls.

- As t??

Decay of Current through an Inductor

- Time constant for RL circuit

- 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.

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.