Magnetism and Electromagnetism Engr. Faheemullah Shaikh.

Magnetism and Electromagnetism

Engr. Faheemullah Shaikh

Wire Coil

• Notice that a current carrying coil of wire will produce a perpendicular field

Magnetic Field: Coil• A series of coils produces a field similar

to a bar magnet – but weaker!

Magnetic Field: Coil

Magnetic Field

Flux Ф can be increased by increasing the current I,

I

Ф I

Magnetic Field

Flux Ф can be increased by increasing the number of turns N,

I

Ф NN

Magnetic Field

Flux Ф can be increased by increasing the cross-section area of coil A,

I

Ф A

N

A

Magnetic Field

Flux Ф can be increased by increasing the cross-section area of coil A,

I

Ф A

N

A

Magnetic Field

Flux Ф is decreased by increasing the length of coil l,

I

Ф

N

A

1

ll

Magnetic Field

Therefore we can write an equation for flux Ф as,

I

Ф

N

A

NIA

ll

or

Ф =μ0 NIA

l

Where μ0 is vacuum or non-magnetic material permeability

μ0 = 4π x 10-7 H/m

Magnetic Field

Ф =μ0 NIA

l

Solenoid

If a coil is wound on a steel rod and connected to a battery, the steel becomes magnetized and behaves like a permanent magnet.

Magnetic Field: Coil• Placing a ferrous material inside

the coil increases the magnetic field

• Acts to concentrate the field also notice field lines are parallel inside ferrous element

• ‘flux density’ has increased

Magnetic Field

By placing a magnetic material inside the coil,

I

N

A

lФ =

μ NIA

l

Where μ is the permeability of the magnetic material (core).

Magnetic Field

By placing a magnetic material inside the coil,

I

N

A

lФ =

μ NIA

l

Where μ is the permeability of the magnetic material (core).

Flux Density

Permeability

• Permeability μ is a measure of the ease by which a magnetic flux can pass through a material (Wb/Am)

• Permeability of free space μo = 4π x 10-7 (Wb/Am)

• Relative permeability:

Reluctance• Reluctance: “resistance” to

flow of magnetic flux

Associated with “magnetic circuit” – flux equivalent to current

• What’s equivalent of voltage?

Magnetomotive Force, F• Coil generates magnetic

field in ferrous torroid• Driving force F needed to

overcome torroid reluctance

• Magnetic equivalent of ohms law

Circuit Analogy

Magnetomotive Force• The MMF is generated by the coil• Strength related to number of turns and

current, measured in Ampere turns (At)

Magnetic Field Intensity

• The longer the magnetic path the greater the MMF required to drive the flux

• Magnetomotive force per unit length is known as the “magnetizing force” H

• Magnetizing force and flux density related by:

Electric circuit:

Emf = V = I x R

Magnetic circuit:

mmf = F = Φ x

= (B x A) x

l

μ A= (B x A) x

l

μ= B x = H x l

= H x l

Magnetic Force On A Current – Carrying Conductor

Magnetic Force On A Current – Carrying Conductor

• The magnetic force (F) the conductor experiences is equal to the product of its length (L) within the field, the current I in the conductor, the external magnetic field B and the sine of the angle between the conductor and the magnetic field. In short

F= BIL (sin)

The force on a current-carrying conductor in a magnetic field:

• When a current-carrying conductor is placed in a magnetic field, there is an interaction between the magnetic field produced by the current and the permanent field, which leads to a force being experienced by the conductor:

• The magnitude of the force on the conductor depends on the magnitude of the current which it carries. The force is a maximum when the current flows perpendicular to the field (as shown in diagram A on the left below), and it is zero when it flows parallel to the field (as in diagram B, on the right):

Fleming's left hand rule shows the direction of the thrust on a conductor carrying a current in a magnetic field.

The left hand is held with the thumb, index finger and middle finger mutually at right angles.

Fleming's left hand rule (for electric motors)

The First finger represents the direction of the Field. The Second finger represents the direction of the Current (in the classical direction, from positive to negative). The Thumb represents the direction of the Thrust or resultant Motion.

Fleming’s left-hand rule

• The directional relationship of I in the conductor, the external magnetic field and the force the conductor experiences

I

F

B

Faraday’s LawFaraday’s Law

Magnetic Field can produce an electric current in a closed loop, if the magnetic flux linking the surface area of the loop changes with time.

The electric Current Produced Induced Current

This mechanism is called “Electromagnetic Induction”



First Experiments

Conducting loop

Sensitive current meter

Since there is no battery or other source of emf included, there is no current in the circuit

Move a bar magnet toward the loop, a current suddenly appears in the circuit

The current disappears when the bar magnet stops

If we then move the bar magnet away, a current again suddenly appears, but now in the opposite direction


Discovering of the First Experiments

1. A current appears only if there is relative motion between the loop and the magnet

3. If moving the magnet’s N-pole towards the loop causes clockwise current, then moving the N-pole away causes counterclockwise.

2. Faster motion produces a greater current

Constant flux, no current is induced in the loop. No current detected by Galvanometer

An Experiment - Situation ASituation A


Constant flux though the loop

DC current I, in coil produces a constant magnetic field, in turn produces a constant flux though the loop

Current in the coil produces a

magnetic field B


An Experiment - Situation B: Situation B: Disconnect battery suddenlyDisconnect battery suddenly

Magnetic field drops to zero

Sudden change of magnetic flux to zero causes a momentarily deflection of Galvanometer needle.

Deflection of Galvanometer

needle

Link: http://micro.magnet.fsu.edu/electromag/java/faraday/index.html


An Experiment - Situation C: Reconnect BatterySituation C: Reconnect Battery

Magnetic field becomes non-zero

Current in the coil produces a

magnetic field B

Sudden change of magnetic flux

through the loop

Deflection of Galvanometer needle in the opposite direction

Conclusions from the experiment

• Current induced in the closed loop when magnetic flux changes, and direction of current depends on whether flux is increasing or decreasing

• If the loop is turned or moved closer or away from the coil, the physical movement changes the magnetic flux linking its surface, produces a current in the loop, even though B has not changed


In Technical TermsTime-varying magnetic field produces an electromotive

force (emf) which establish a current in the closed circuit

3. A combination of the two above, both flux changing and conductor moving simultaneously. A closed path may consists of a conductor, a capacitor or an imaginary line in space, etc.


Electromotive force (emf) can be obtained through the following ways:

1. A time-varying flux linking a stationary closed path. (i.e. Transformer)

2. Relative motion between a steady flux and a close path. (i.e. D.C. Generator)

Faraday summarized this electromagnetic phenomenon into two laws ,which are called the Faraday’s law


Faraday’s First LawFaraday’s First LawWhen the flux magnet linked to a circuit

changes, an electromotive force (emf) will be induced.

Faraday’s Second LawFaraday’s Second LawThe magnetic of emf induced is equal to

the time rate of change of the linked magnetic flux .


Minus Sign Lenz’s Law

Indicates that the emf induced is in such a direction as to produces a current whose flux, if added to the original

flux, would reduce the magnitude of the emf

(volts)



The induced voltage acts to produce an opposing flux







Heinrich F.E. Lenz

• Russian physicist • (1804-1865)• 1834 Lenz’s Law• There is an induced current in

a closed conducting loop if and only if the magnetic flux through the loop is changing.

• Indicates that the emf induced is in such a direction as to produces a current whose flux, if added to the original flux, would reduce the magnitude of the emf

There is an induced current in a closed conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field always opposes the change in the flux.

Right Hand Rule

• If you wrap your fingers around the coil in the direction of the current, your thumb points north.

2 Direction of induced current

In both cases, magnet moves against a force.Work is done during the motion & it is transferred as electrical energy.

Induced I always flows to oppose the movement which started it.

b Lenz's law

Applications of Magnetic Induction• Magnetic Levitation (Maglev) Trains

– Induced surface (“eddy”) currents produce field in opposite direction

Repels magnet Levitates train

– Maglev trains today can travel up to 310 mph Twice the speed of Amtrak’s fastest conventional train!

N

S

rails“eddy” current

Liner induction

0-70 mph in 3 sec

FALLING MAGNET• The copper tube "sees" a

changing magnetic field from the falling magnet. This changing magnetic field induces a current in the copper tube.

• The induced current in the copper tube creates its own magnetic field that opposes the magnetic field that created it.

Fleming Right Hand RuleFleming Right Hand RuleDirection of Induced e.m.f, Magnetic Flux, Conductor Motion

ThumbDirection of

Conductor Motion

Fore FingerDirection of Field Flux

Middle FingerDirection of Induced emf or Current Flow


Fleming's right hand rule shows the direction of induced current flow when a conductor moves in a magnetic field.

The right hand is held with the thumb, first finger and second finger mutually at right angles, as shown in the diagram

Fleming's right hand rule (for generators)

The Thumb represents the direction of Motion of the conductor. The First finger represents the direction of the Field. The Second finger represents the direction of the induced or generated Current (in the classical direction, from positive to negative).

Leakage Flux and Fringing

Leakage flux

fringing

Leakage FluxIt is found that it is impossible to confine all the flux to the iron path only. Some of the flux leaks through air surrounding the iron ring.

Leakage coefficient λ =Total flux producedUseful flux available

Fringing

Spreading of lines of flux at the edges of the air-gap. Reduces the flux density in the air-gap.

Hysteresis loss

Materials before applying m.m.f (H), the polarity of the molecules or structures are in random.

After applying m.m.f (H) , the polarity of the molecules or structures are in one direction, thus the materials become magnetized. The more H applied the more magnetic flux (B )will be produced

When we plot the mmf (H) versus the magnetic flux (B) will produce a curve so called Hysteresis loop

1. OAC – when more H applied, B increased until saturated. At this point no increment of B when we increase the H.

2. CD- when we reduce the H the B also reduce but will not go to zero.

3. DE- a negative value of H has to applied in order to reduce B to zero.

4. EF – when applying more H in the negative direction will increase B in the reverse direction.

5. FGC- when reduce H will reduce B but it will not go to zero. Then by increasing positively the also decrease and certain point it again change the polarity to negative until it reach C.

Hysteresis Loss• Empirical equation

Summary : Hysteresis loss is proportional to f and ABH

Eddy current

metal insulator

When a sinusoidal current enter the coil, the flux also varies sinusoidally according to I. The induced current will flow in the magnetic core. This current is called eddy current. This current introduce the eddy current loss. The losses due to hysteresis and eddy-core totally called core loss. To reduce eddy current we use laminated core

Eddy Current LossEmpirical equation

Core Loss• Core Loss

losscurrenteddyP

losshysteresisPwhere

PPP

e

h

ehc

Inductance• A changing magnetic flux induces an e.m.f. in any

conductor within it• Faraday’s law:

The magnitude of the e.m.f. induced in a circuit is proportional to the rate of change of magnetic flux linking the circuit

• Lenz’s law:The direction of the e.m.f. is such that it tends to produce a current that opposes the change of flux responsible for inducing the e.m.f.

• When a circuit forms a single loop, the e.m.f. induced is given by the rate of change of the flux

• When a circuit contains many loops the resulting e.m.f. is the sum of those produced by each loop

• Therefore, if a coil contains N loops, the induced voltage V is given by

where d/dt is the rate of change of flux in Wb/s

• This property, whereby an e.m.f. is induced as a result of changes in magnetic flux, is known as inductancet

ΦNVdd

TYPES OF INDUCED EMF• Statically induced emf

– Conductor remains stationary and flux linked with it is changed (the current which creates the flux changes i.e increases or decreases)

TYPES– Self induced– Mutually induced

TYPES OF INDUCED EMF

• Dynamically induced emf– Field is stationary and conductors cut across it– Either the coil or the magnet moves.

Self-Inductance

R

Increasing I

Consider a coil connected to resistance Consider a coil connected to resistance R R and voltage and voltage VV. . When switch is closed, the rising current When switch is closed, the rising current I I increases flux, increases flux, producing an internal back emf in the coil.producing an internal back emf in the coil.

Consider a coil connected to resistance Consider a coil connected to resistance R R and voltage and voltage VV. . When switch is closed, the rising current When switch is closed, the rising current I I increases flux, increases flux, producing an internal back emf in the coil.producing an internal back emf in the coil.

R

Decreasing ILenz’s Law:Lenz’s Law: The The back emfback emf (red (red

arrow)arrow) must oppose must oppose change in flux:change in flux:

InductanceThe back emf The back emf EE induced in a coil is proportional to the induced in a coil is proportional to the rate of change of the current rate of change of the current I/I/t.t.

; inductancei

L Lt

E ; inductance

iL L

t

E

An inductance of one henry (H) An inductance of one henry (H) means that current changing at the means that current changing at the rate of one ampere per second will rate of one ampere per second will induce a back emf of one volt.induce a back emf of one volt.

R

Increasing i/t

1 V1 H

1 A/s

Example 1: A coil having 20 turns has an induced emf of 4 mV when the current is changing at the rate of 2 A/s. What is the inductance?

; /

iL L

t i t

E

E

( 0.004 V)

2 A/sL

L = 2.00 mHL = 2.00 mH

Note:Note: We are following the practice of using lower We are following the practice of using lower case case i i for transient or changing current and upper for transient or changing current and upper case I for steady current.case I for steady current.

Note:Note: We are following the practice of using lower We are following the practice of using lower case case i i for transient or changing current and upper for transient or changing current and upper case I for steady current.case I for steady current.

R

i/t = 2 A/s4 mV4 mV

Calculating the InductanceRecall two ways of finding Recall two ways of finding E:E:

iL

t

E

iL

t

EN

t

E N

t

E

Setting these terms equal gives:Setting these terms equal gives:

iN L

t t

Thus, the inductance L can be found from:

Thus, the inductance L can be found from:

NL

I

NL

I

Increasing i/t

R

Inductance L

Inductance of a SolenoidThe The BB-field created by a current -field created by a current II for for

length length l l is:is:

0NIB

and = BA

0 NIA N

LI

Combining the last two equations Combining the last two equations gives:gives:

20N A

L

20N A

L

R

Inductance L

lB

Solenoid

Example 2: A solenoid of area 0.002 m2 and length 30 cm, has 100 turns. If the current increases from 0 to 2 A in 0.1 s, what is the inductance of the solenoid?

First we find the inductance of the solenoid:First we find the inductance of the solenoid:

-7 2 22 T m0 A(4 x 10 )(100) (0.002 m )

0.300 m

N AL

R

lA

L = 8.38 x 10-5 HL = 8.38 x 10-5 H

Note: Note: L L does NOT depend on does NOT depend on current, but on physical current, but on physical parameters of the coil.parameters of the coil.

Note: Note: L L does NOT depend on does NOT depend on current, but on physical current, but on physical parameters of the coil.parameters of the coil.

Example 2 (Cont.): If the current in the 83.8-H solenoid increased from 0 to 2 A in 0.1 s, what is the induced emf?

R

lA

L = 8.38 x 10-5 HL = 8.38 x 10-5 H

iL

t

E

iL

t

E

-5(8.38 x 10 H)(2 A - 0)

0.100 s

E 1.68 mVE 1.68 mVE

Energy Stored in an InductorAt an instant when the current is changing at At an instant when the current is changing at i/i/tt, we have:, we have:

; i i

L P i Lit t

E E

Since the power Since the power PP = Work/t= Work/t, , Work = P Work = P tt. Also the . Also the average value of average value of LiLi is is Li/2Li/2 during rise to the final current during rise to the final current I. I. Thus, the total energy stored is:Thus, the total energy stored is:

Potential energy stored in inductor:

212U Li

R

Example 3: What is the potential energy stored in a 0.3 H inductor if the current rises from 0 to a final value of 2 A?

212U Li

212 (0.3 H)(2 A) 0.600 JU

U = 0.600 J

This energy is equal to the work done in reaching This energy is equal to the work done in reaching the final current the final current II; it is returned when the current ; it is returned when the current decreases to zero.decreases to zero.

L = 0.3 H

I = 2 A

R

The R-L Circuit

R

L

S2

S1

V

E

An inductor An inductor LL and resistor and resistor RR are are connected in series and switch 1 is connected in series and switch 1 is closed:closed:

iiV – V – E E = iR= iR

iL

t

E

iV L iR

t

iV L iR

t

Initially, Initially, i/i/tt is large, making the back emf large and the is large, making the back emf large and the current current ii small. The current rises to its maximum value small. The current rises to its maximum value II when rate of change is zero.when rate of change is zero.

Initially, Initially, i/i/tt is large, making the back emf large and the is large, making the back emf large and the current current ii small. The current rises to its maximum value small. The current rises to its maximum value II when rate of change is zero.when rate of change is zero.

The Rise of Current in L( / )(1 )R L tV

i eR

( / )(1 )R L tV

i eR

At t = 0, I = 0At t = 0, I = 0

At t = At t = , I = V/R, I = V/R

The time constant The time constant L

R

L

R

In an inductor, the current will rise to 63% of its In an inductor, the current will rise to 63% of its maximum value in one time constant maximum value in one time constant = L/R.= L/R.

In an inductor, the current will rise to 63% of its In an inductor, the current will rise to 63% of its maximum value in one time constant maximum value in one time constant = L/R.= L/R.

Time, t

I

i

Current RiseCurrent Rise

0.63 I

The R-L Decay

R

L

S2

S1

V

Now suppose we close Now suppose we close SS22 after energy is after energy is

in inductor:in inductor:

E E = iR= iRi

Lt

E

iL iR

t

iL iR

t

Initially, Initially, i/i/tt is large and the emf is large and the emf E E driving the current is driving the current is at its maximum value at its maximum value II. The current decays to zero when . The current decays to zero when the emf plays out.the emf plays out.

Initially, Initially, i/i/tt is large and the emf is large and the emf E E driving the current is driving the current is at its maximum value at its maximum value II. The current decays to zero when . The current decays to zero when the emf plays out.the emf plays out.

For current decay For current decay in L:in L:

E

ii

The Decay of Current in L

( / )R L tVi e

R( / )R L tV

i eR

At t = 0, At t = 0, ii = V/R = V/R

At t = At t = , , ii = 0 = 0

The time constant The time constant L

R

L

R

In an inductor, the current will decay to 37% of its In an inductor, the current will decay to 37% of its maximum value in one time constant maximum value in one time constant In an inductor, the current will decay to 37% of its In an inductor, the current will decay to 37% of its maximum value in one time constant maximum value in one time constant

Time, t

I

i

Current DecayCurrent Decay

0.37 I

Example 5: The circuit below has a 40-mH inductor connected to a 5- resistor and a 16-V battery. What is the time constant and what is the current after one time constant?

5

L = 0.04 H

16 V

R

0.040 H

5

L

R

Time constant: = 8 msTime constant: = 8 ms

( / )(1 )R L tVi e

R

After time After time

i = 0.63(V/R)i = 0.63(V/R)

16V0.63

5i

i = 2.02 A

Inductors in Series and Parallel

• When several inductors are connected together their effective inductance can be calculated in the same way as for resistors – provided that they are not linked magnetically

• Inductors in Series

• Inductors in Parallel

Mutual Inductance

• When two coils are linked magnetically then a changing current in one will produce a changing magnetic field which will induce a voltage in the other – this is mutual inductance

• When a current I1 in one circuit, induces a voltage V2 in another circuit, then

where M is the mutual inductance between the circuits. The unit of mutual inductance is the Henry (as for self-inductance)

t

IMV

d

d1

2

• The coupling between the coils can be increased by wrapping the two coils around a core– the fraction of the magnetic field that is coupled is

referred to as the coupling coefficient

• Coupling is particularly important in transformers– the arrangements below give a coupling

coefficient that is very close to 1

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Magnetism and Electromagnetism Engr. Faheemullah Shaikh.

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