topic 8 electromagnetic induction and induct ance

7/27/2019 Topic 8 Electromagnetic Induction and Induct Ance

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INTRODUCTION

The discovery of electromagnetic induction by Michael Faraday and JosephHenry in 1831 has changed our understanding of electricity and magnetism. Priorto that, it was known that a battery was a source of electromotive force (emf).However, in electromagnetic induction, the changing magnetic flux through a

TTooppiicc 88

Electromagnetic

Induction and Inductance

LEARNING OUTCOMESBy the end of this topic, you should be able to:

1. Calculate the magnetic flux through a surface;

2. Calculate the induced electromagnetic force (emf) of theelectromagnetic induction process according to faraday’s and lenz’slaws;

3. Calculate the motional emf is created in a conductor moving in auniform magnetic field;

4. Explain how changing magnetic fields induce circulating currents oreddy currents in conducting materials;

5. Describe the principle of an inductor as device to store an electricalenergy in the form of a magnetic field;

6. Define self inductance and mutual inductance and apply them to related problems;

7. Calculate the energy stored in an inductor and to define the magneticenergy density; and

8. Describe the principle and uses of a transformer.



TOPIC 8 ELECTROMAGNETIC INDUCTION AND INDUCTANCE118

circuit induces an emf and an electrical current in the circuit and also in aneighboring circuit. A changing current in a coil induces an emf in adjacent coil.The coupling between the coils is known as the mutual inductance. This is the

principle of a transformer used to step the voltage of an alternating current (ac) upand down. Thus phenomenon of electromagnetic induction is the central principle behind the operation of power-generating stations and transformers.

In this chapter we will discuss how electromagnetic induction occurs in a circuit by changing the magnetic flux, which causes a motion of charged particles in aconductor, thus inducing an emf or a current through a coil of conductor.Faraday’s law and Lenz’s law are the central principles of electromagneticinduction. The effects of electromagnetic inductions are numerous, depending onthe type conductor exposed to the change of magnetic flux. We will also learnabout inductance RL circuits and transformers.

MAGNETIC FLUX

The concept of magnetic flux is vital in order for us to understand the occurrenceof electromagnetic induction. An emf is induced in a coil whenever there is achange in the magnetic flux through it. In Topic 6 we learned how a magneticfield can be described in terms of magnetic lines of force. In this context, themagnetic lines of force help us visualise the magnetic field but provide us with noinformation on the “strength” of the field. In order to know this, we need to utilise

the magnetic flux.

Recall that we identified the electric flux through a surface as the number ofelectric field lines passing through the surface. We will now define the magneticflux in a similar way. Consider a single turn coil of area A. Let the normal to thissurface area make an angle θ with the uniform magnetic field B passing throughit. Figure 8.1. Then, the magnetic flux through the coil is defined as:

cos B BA θ Φ = (8.1)

Notice that cos B θ is just the component of the magnetic field perpendicular to

the plane of the coil.

8.1



TOPIC 8 ELECTROMAGNETIC INDUCTION AND INDUCTANCE 119

Figure 8.1: Definition of magnetic flux

When 90oθ = , the magnetic field lines are parallel to the plane of the loop, then

the flux is zero. See Figure 8.1(a) On the other hand, when 0oθ = , the field is

perpendicular to the plane of the loop and the flux is at the maximum value of BA.

See Figure 8.1(b).

Figure 8.2: The Minimum and maximum flux through a coil

Magnetic flux thus is a measure of the number of magnetic field lines passingthrough the surface of the coil. For a bar magnet, the flux lines are moreconcentrated at the poles, where the magnetic field strength is the greatest.

If the coil contains more than one turn, then the flux through the coil is the sum ofthe total flux through all the individual turns. This is called the flux linkagethrough the whole coil. If the flux contains N turns, then the total flux linkage isgiven by:

cos B NBA θ Φ = (8.2)

The SI unit of magnetic flux is equal to the unit of magnetic field (T) times theunit of area (m

2) or the Weber (Wb). 1 Wb = 1 T. m

2.




Example 8.1

The normal of the surface of a circular loop of radius r = 2 cm makes an angle ofθ = 60° with a magnetic field of magnitude B =0.5 T that passes through this

loop. Calculate the magnetic flux through the loop.

Solution:

Refer to Fig.8.1 with

( )2

2 2 7 2

0.5 , 60 ,

3.14 0.02 10 1.26 10 m

B T

A r

θ

π − −

= =

= = × × = ×

From Eq.8.1,the magnetic flux through the loop is given by7 8

cos 0.5 1.26 10 cos60 3.15 10 Wb B BA θ − −

Φ = = × × × = ×

Example 8.2

1. A loop of area 0.1 2m is placed in a constant magnetic field of 10 Teslaacting downward. What is the flux through the wire loop when the face ofthe loop is:

(a) Perpendicular to the magnetic field,

(b) Parallel to the magnetic field, and

(c) At an angle of 45° to the magnetic field as shown in the figure below?

Solution:

The broken lines indicate the normal to the plane of the loop.For this example 10T, B = A =0.1 2m

(a) 0θ = , 0cos 10 0.1 cos 0 1 Wb B BA θ Φ = = × × =

(b) 90θ = , 0cos 10 0.1 cos90 0 Wb B BA θ Φ = = × × =

(c) 45θ = , 0cos 10 0.1 cos 45 0.71 Wb B BA θ Φ = = × × =

-3 2

-2(11.65×10 kg/m)(9.8m/s )

= = 1.6×107.0A

mg (m/l)g B = =

I l I T




ELECTROMAGNETIC INDUCTION

In Topic 6, we learned about Oersted’s remarkable discovery: an electric current produces a magnetic field. This remarkable discovery inspired Faraday to think

whether it was possible to do the opposite, i.e., to create a current from a magnetic

field. He began his experiment in 1825 and did not succeed until 1831. The

results of Faraday’s work showed that an induced current is produced in a coil by

a changing magnetic field. This phenomenon is known today as electromagnetic

induction.

We can attempt to demonstrate electromagnetic induction by doing a simpleexperiment as shown in Figure 8.3.

Consider a coil of wire connected to a galvanometer. When a magnet is movedtowards the coil as in Figure 8.3(a), the galvanometer needle will deflect in onedirection. If the magnet is moved away from the coil, the galvanometer needlewill deflect in the opposite direction. See Figure 8.3(b). Finally, if the magnet isheld stationary relative to the coil as in Figure 8.3(c), no deflection is observed.

We can also do the experiment by moving a coil relative to the magnet and obtainsimilar results.

From these observations, we can conclude that a current is set up in the circuit aslong as there is relative motion between the magnet and the loop of wire. If thereis no relative motion, electromagnetic induction cannot take place.

Figure 8.3: Electromagnetic induction

8.2




The induced current is a result of the induced emf. In Figure 8.1, if we had used acoil of higher resistance, the induced current will be smaller. This is because theinduced emf is the same, but the resistance is greater.

From experiment, it is also observed that the induced current (and the induced

emf) will be greater if we

(i) Use a coil with a larger surface area

(ii) Use a stronger magnet.

(iii) Increase the number of turns in the coil

(iv) Increase the speed of the magnet

FARADAY’S LAW

Let’s reconsider the experiment shown in Figure 8.3(a) in terms of the magneticflux near the coil. When the magnet was pushed towards the coil, this actioncaused a sudden change in the magnetic flux near the coil. This change was theresult of more magnetic field lines from the magnet entering the coil. As there wasa change in magnetic flux experienced by the coil, an induced emf was set up inthe coil. Since there was an induced emf and a closed circuit, an induced currentflowed in the coil.

What can we conclude from here?

According to Faraday, whenever there is a change in the magnetic flux through acoil, an emf is induced in the coil.

Faraday’s law of electromagnetic induction states that:

the induced emf in a closed loop equals the rate of change of the magnetic flux through the loop.

8.3

ACTIVITY 8.1

An interactive simulation of the experiment above can be found here:

http://micro.magnet.fsu.edu/electromag/java/faraday2/index.html

Observe the movement of the galvanometer as you move the magnet backand forth.




Mathematically, Faraday’s law can be written as

t

ε ΔΦ

= −Δ

B (8.3)

where ε is the induced emf and BΔΦ is the change in the magnetic flux in the

circuit in a time interval of t Δ .

Faraday’s law is used to determine the magnitude of the induced emf obtained inelectromagnetic induction.

If the circuit is a coil consisting of N loops the induced emf is given by

B

N t ε

ΔΦ

= − Δ (8.4)

The negative sign in the equation is a consequence of Lenz’s Law.

LENZ’S LAW

The minus sign in Faraday’s law is a consequence of Lenz’s law. Lenz’s law isalso convenient way of determining the sign or direction of an induced current oremf. Lenz’s law states that:

the induced current in a coil always flows in a direction so that it alwaysopposes the change that caused it.

By “change”, we mean an increase or decrease in the magnetic flux through a coil.

As an example, let us consider the situation in Figure 8.4(a), where a magnet ismoving toward a coil. As the magnet gets closer to the coil, the magnetic flux inthe coil increases. The “change” in the coil is equivalent to an increase in themagnetic flux through the coil. According to Lenz’ law, the induced current in the

coil must now flow in a direction to oppose this change. For this to happen, theinduced current must produce its own magnetic field (called the induced magneticfield) to oppose the increase in the flux created by the magnet.

Notice that the end of the coil facing the approaching magnet has been induced asa north pole as shown in Figure 8.4(b). Thus, this repulsion between the two north poles will oppose the motion of the magnet and consequently reduce the amountof flux entering the coil.

8.4




The right-hand grip rule can be used to determine the direction of the induced

current. Point the fingers in the direction of the induced north. The thumb will

then give the direction of the current, which is counter clockwise in this case.

A current is induced in the

coil as the magnet is moved

towards it

The induced current creates

its own magnetic field in the

coil to oppose the increasing

flux

Application of the right hand

grip rule to determine the

direction of the induced

current

Figure 8.4: Applications of Lenz’s law

Example 8.3

What are the induced emf and current in the coil if its magnetic flux changesaccording to the following? Assume the resistance of the coil is 100 Ω .

(a) From 200 mWb to 800 Wb in 4 s.

(b) From 250 mWb to 20 mWb in 5 s.

ACTIVITY 8.2

An interactive Java applet that illustrates Lenz’s law can be found here:http://micro.magnet.fsu.edu/electromag/java/lenzlaw/

Observe the direction of the induced current in the stationary conductingring as you move the magnet towards or away from it.




Solution:

(a) induced emf equals the rate of change of magnetic flux

(800 – 200)mWb= –150mV

4.0s

2 1Φ -Φ ΔΦε = – = –

Δt Δt = –

induced current –150 mV

= = –1.5mA100Ω

ε I =

R

(b) induced emf equals the rate of change of magnetic flux

(20 – 250) mWb= 46mV

5.0s

2 1Φ – Φ ΔΦ

ε = – = – = – Δt Δt

induced current46mV

= 0.46 mA100Ω

=ε

I = R

Example 8.4

A coil of area20.01m and 100 turns has a total resistance of 0.1Ω . A uniform

magnetic field is applied perpendicular to the plane of the coil. The magnitude ofthis field is decreased from 1 T to zero in 0.5s.

(a) Find the magnitude of the induced emf in the coil

(b) Find the magnitude of the induced current

Solution:

( ) ( )

2

1

2

2 1

0.01m , 100, 0.1 , 0.5

1 0.01 0.01Wb

0 0.01 0 Wb

0 0.01(a) 0.02V

0.50.02

(b) 0.2A0.1

i

f

A N R t s

B A

B A

t t

i R

= = = Ω Δ =

Φ = = × =

Φ = = × =

Φ − Φ −ΔΦ= − = − = − =

Δ Δ

= = =

ε

ε




MOTIONAL ELECTROMAGNETIC FORCE

Consider a straight conductor of length l moving with constant velocity v perpendicular to a uniform magnetic field B directed into the paper as in Fig. 8.2.Using the right-hand rule, the free electrons in the conductor will experience adownward force having a magnitude F qvB= . Here, q represents the charge ofan electron.

The magnetic force causes the free electrons to move to the lower end of the

conductor, thereby creating an excess of positive charges at the upper end of the

conductor. As a result, an electric field E is created that is directed upward. The

upward force E F qE = due to the electric field exactly equals the downward force

m F qvB= due to downward magnetic force.

8.5

EXERCISE 8.1

1. A plane circular loop of area 0.01m2 and possessing 10 turns is placed

in a uniform magnetic field. The direction of the magnetic field makesan angle of 60° with respect to the normal direction to the loop. The

magnetic field strength is now increased from 1 T to 5 T in a time

interval of 5 s. What is the emf generated around the loop during this

time?

2. The magnetic flux in a single loop coil of area 10 cm2 steadily

changes from 10mWb to 15mWb in 1 s. What emf is induced in thecoil?

3. A coil of wire of cross-sectional area 3.0 cm2

and with 500 turns permeter is placed perpendicular to a uniform magnetic field. If the totalresistance of the coil is 50 Ω, at what rate should the magnetic field bechanging to induce a current of 0.24 mA?




Figure 8.5: Motional emf created in a conductor moving in a uniform magnetic field

At the equilibrium, where all free electrons have moved to the lower end of theconductor, we have

E m F F

qE qvB

=

=

Or E = vB (8.5)

But recall from Topic 3 thatV

E d

Δ= , where V Δ is the potential difference

between the upper and lower ends of the conductor. Here d = l ,

This potential difference is also known as the induced emf or the motional emf.Then

V E l vBl ε = Δ = = (8.6)

Example 8.5

A 10 cm long rod in Figure 8.5 is moving at 5 m/s. What is the strength of themagnetic field if a 0.9 V emf is induced in the rod?

Solution10cm = 0.1m, 5m/s, 0.9

0.91.8

5 0.1

l v V

V El vBl

B T vl

= = =

= Δ = =

= = =×

ε

ε

ε




EDDY CURRENTS

In section 8.5, we saw that an induced current is created when a coil of conductor

moves in a uniform magnetic field, and the direction of the current can bedetermined using Lenz’s law. If we replace the coil with a metal piece, and allow

it to oscillate in a magnetic field, induced currents that circulate throughout the

volume of the metal are produced. These circulating currents are called eddy

currents.

Figure 8.6 shows a metal plate oscillating in a magnetic field that is acting into the

plane of the page. As it moves across the field, eddy currents are induced within

the plate in the directions shown.

Figure 8.6: When a plate oscillates in a magnetic field, eddy currents are formed.

8.6

EXERCISE 8.2

A square loop of wire with resistance of 100Ω and length of 4.0 cm ismoved at constant speed of 5.0 m/s across a uniform magnetic field o0.8 T. What are the induced emf and the current in the loop?




By Lenz’ Law, these currents will flow in such a direction so as to oppose the

movement of the plate by creating a resistive force. This resistive force F will

result in the oscillations becoming damped. Such effects can be used to stop the

rotation of a circular saw quickly when the power is turned off. Some electrically- powered trains use eddy-currents as breaking systems. Electromagnets on the train

near the train’s rails are turned on creating eddy currents in the rails.

Another important result of eddy currents is the Joule heating effect through I 2 R,

where R is the resistance through which the current is passing. In induction

furnaces, eddy currents are used to heat materials in a completely sealed container

to avoid any contamination introduced into the materials.

However, heating effects of eddy currents are often undesirable since they

dissipate electrical energy, especially in alternating-current transformers andmotors, To minimise these effects, the core conducting materials are laminated

with thin layers by a non-conducting material to minimise eddy currents formed

and thereby increase the efficiency of this devices.

SELF INDUCTANCE

A magnetic field is formed around a current-carrying coil of conductor andits magnitude can be determined using Ampere law or the Biot-Savart law.

The laws are applied when there is no current change in the coil. However,whenever we change the current in a coil of wire, the magnetic field it produceswill also change. That will in turn change the magnetic flux through the coil.Hence according to Faraday’s law, an induced emf will be produced in the coil.If the coil contains N turns, then from Faraday’s law, the induced emf is:

N t

ε ΔΦ

= −Δ

(8.7)

The relationship between the emf and the current in the coil is known as the self-inductance (or inductance) L.

I L

t ε

Δ= −

Δ (8.8)

The negative sign means that the induced emf opposes the change (an increase ordecrease in the current) causing it.

8.7




From Equation 8.8, we may identify the self-inductance as the magnitude ofinduced emf per unit rate of change of current . From (8.7) and (8.8), the self-inductance L is given by

N Φ L =

I (8.9)

The unit of inductance, L, is the henry (H), measured in volt-second per ampere.One henry is equal to one volt-second per ampere or one ohm-second .

So self-inductance is a measure of a coil's ability to establish an induced voltageas a result of a change in its current. Such a coil is called an inductor . Figure 8.7 isthe two circuit symbols for an inductor. Either one can be used in an electrical

circuit.

Figure 8.7: Two circuit symbols commonly used for an inductor

An inductor is basically a solenoid of length l consisting of n turns per meter of

loops of wire and having the self-inductance of:

20

0

N μ nIA N Φ NBA

L = = = = μ n lA I I I (8.10)

where7-

0 μ = 4π ×10 Wb/A.m .

Example 8.6

What is the self-inductance in a coil of conductor consisting of 1000 turns, whichcarries a steady current of 2.0 A and produces a magnetic flux of 15 mWb?

Solution:

The self-inductance-2

7.5 N Φ (1000)(1.5×10 Wb)

L = = = I 2.0A

H.

Example 8.7

Consider a coil where the current is increased from zero to 2.0 A in 1.0 ms. Ifthe magnitude of the induced voltage across the inductor during this time is 15 V ,what is the inductance of the solenoid?




Solution:

3

3

1.5×10 , 2

From

150.0.375

2

5 10

= 15V, t = s I A

I Lt

L H I

t

ε

ε

ε

−

−

Δ Δ =

Δ=Δ

= = =Δ

Δ ×

MUTUAL INDUCTANCE

We already know that an emf can be induced in a coil by moving a magnet towardit. Figure 8.8 shows another method of inducing an emf by using twoneighbouring coils of wires.

A current flowing in a primary coil 1 produces a magnetic flux 1Φ and hence a

magnetic flux 2Φ through a secondary coil 2. If the current in coil 1, 1 I changes,

the magnetic flux through coil 2 changes as well and an emf 2ε is induced in coil

2. A change in the current in one coil that can induce an emf in an adjacent coil is

known as the mutual inductance.

Figure 8.8: A changing current in the primary coil induces an emf in the secondary coil.

When 1 I changes, the magnetic flux 2Φ through coil 2 also changes, which

induces an emf 2ε in coil 2, given by

2 12 2

I N M

t t ε

ΔΦ Δ= − = −

Δ Δ (8.11)

8.8




where N 2 is the number of turns of coil 2 and M is the mutual inductance of thetwo coils. Since N 2 and M are constant, from Equation 8.11 we can write:

2 2 1 N Φ = MI (8.12)

where 2 2 N Φ is the total flux linkage or the total flux through the secondary coil.

Now, we can define the mutual inductance as:

2 2

1

N Φ M =

I (8.13)

The SI unit of mutual inductance is henry (H). The effect is truly mutual:changing the current in coil 2 can cause a changing magnetic flux 1Φ in coil 1 andhence an emf 1ε in coil 1. Consequently, we can write:

1

2

dI ε = -M

dt and

2

1

dI ε = -M

dt (mutually induced emf’s)

where the mutual inductance can be represented as:

2 2 1 1

1 2

N Φ N Φ M = =

I I (mutual inductance) (8.14)

EXERCISE 8.3

1. A steady current of 2 A in a coil of 400 turns causes a flux o41 10 Wb−

× to pass through the loops of the coil. Calculate:

(a) the average emf induced in the coil if the current is stopped in0.08s.

(b) the inductance in the coil

(c) the energy stored in the coil

2. The mutual inductance between the primary and secondary of atransformer is 0.50 H. Find the induced emf in the secondary whenthe primary current changes at a rate of 5.0 A/s




MAGNETIC-FIELD ENERGY

Because of the inductance, the emf induced in an inductor prevents a powersource or a battery from establishing instantaneous current. Therefore, the powersource has to do work against the inductor to create a steady current I . In anordinary d.c. circuit, part of the energy from the power source is dissipated as heatdue to the resistance R, and the other part is stored as magnetic field in theinductor.

When there is no current, no energy is stored in the inductor. But when the steadycurrent I is passing through the inductor, the total energy is:

2 B

1U = LI 2 (8.15)

This energy is stored in the form of a magnetic field.

In the solenoid with the inductance 2 L = μn Al and the magnetic field of B = μnI , Equation (8.15) may be written as:

2

2

B

1 BU = LI = Al

2 2 μ (8.16)

TRANSFORMERS

A transformer is a device in which two circuits are coupled by a magnetic fieldthat is linked to both. Its function is to transfer electrical energy from the firstcircuit to the other.

The transformer works on the principle of mutual inductance of the primary andsecondary coils winding on an iron core as shown Figure 8.3(a). The circuit

symbol for a transformer with an iron core is shown in Figure 8.3(b).

Figure 8.3: (a) Schematic diagram of a transformer (b) circuit symbol

8.10

8.9




An a.c. power source causes an alternating current in the primary coil, which setsup an alternating magnetic flux in the core and the changing flux induces an emfin the secondary coil. The induced emf in the secondary coil gives rise to an

alternating current in the secondary coil.

1 1

d Φε = -N

dt and

2 2

d Φε = -N

dt (8.17)

or1 1

2 2

ε N =

ε N (8.18)

1 1

2 2

V N =

V N

(8.19)

If a resistance R completes the secondary circuit, then the power delivered to the primary equals the power of the secondary.

1 1 2 2 I V = I V (8.20)

And2 1

1 2

I N =

I N (8.21)

Example 8.8

What turns ratio would be needed for an ideal transformer to provide 12 V (rms)

when connected to 240 V (rms) mains supply.

Solution

According to Equation (8.19)1 1

2 2

V N =

V N or

12 1=

240 20=n

The primary has the greater number of turns by a factor of 20.

In practice, the core of a transformer is not made of a single solid iron block, but consists of iron sheets laminated with non-conducting material.Why?

SELF-CHECK 8.1




• A changing magnetic field or magnetic flux produces an electric current orelectromotive force, which can be explained using Faraday’s law and Lenz’slaw.

• Faraday’ law states that the induced emf in a close loop equals the negative ofthe rate of change of magnetic flux through the loop. In short, the law can be

written ast

ε ΔΦ

= −Δ

B

• Lenz’s law is used to determine the sign or direction of an induced current oremf. Lenz’s law states that the direction of any magnetic induction effect issuch as to oppose the cause of the effect.

• Motional emf is created in a conductor moving in a uniform magnetic field.The induced emf is given by ε = B l v . The direction of emf is reversed whenwe reverse the motion of the conductor.

• Circulating currents are induced in a bulk piece of conductor moving inmagnetic fields or located in changing magnetic fields. These circulatingcurrents are called eddy currents. Eddy currents produce breaking and heatingeffects and can be used in many applications.

• A changing current in a coil causes a changing magnetic flux and induces an

emf in the same coil. The inductance L is the magnitude of induced emf perunit rate of change of current.

• The inductance L is the change of magnetic flux per change of current.

• A changing current in coil 1 causes a changing magnetic flux in coil 2 and anemf is induced in coil 2; likewise a changing current in coil 2 induces an emfin coil 1.

EXERCISE 8.4

A step up transformer is used on a 200 V line to produce 800 V. The primary has 100 turns. Find the number of turns on the secondary.




• A change in the current in one coil that can induce an emf in an adjacent coilis known as mutual inductance.

• An inductor with inductance L carrying current I has energy

1

2

2

BU = LI

• Transformers are employed to step up or down voltage in a.c. powertransmission.

Eddy currents

Electromegnetic induction

Faraday’s law

Lenz’s law

Magnetic flux

Mutual inductance

Self induction

Transformer

1. A loop of diameter 4cm is placed in a magnetic field of magnitude 1 T. The

normal to the loop makes an angle of 45° with the magnetic field. If the fielddecreases to 0.75 T, what is the change in the magnetic flux through the loop?

2. The magnetic field of a TV signal through a circular antenna of radius 5 cm

changes at a rate of 0.1 T/s. Find the induced emf in the antenna if the signal is

normal to the plane of the antenna.

3. 2 coils, A and B are placed at two fixed positions. When coil B has no current

and the current in A increases at a rate of 5A/s, the emf induced in B is 10

mV.

(a) Find the mutual inductance.

(b) At a certain instant, the current in A is zero and the current in B is2.5A.What is the total flux linkage through A ?

topic 8 electromagnetic induction and induct ance

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