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Prepared by Md. Amirul Islam Lecturer Department of Applied Physics & Electronics Bangabandhu Sheikh Mujibur Rahman Science & Technology University, Gopalganj 8100

Magnetism 3

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Page 1: Magnetism 3

Prepared by

Md. Amirul Islam

Lecturer

Department of Applied Physics & Electronics

Bangabandhu Sheikh Mujibur Rahman Science &

Technology University, Gopalganj – 8100

Page 2: Magnetism 3
Page 3: Magnetism 3

Self Induction and Back emf:

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.1, Page – 1015

Consider the circuit consisting of a

switch, a resistor, and a source of

emf, as shown in figure. When the

switch is thrown to its closed

position, the source current does

not immediately jump from zero to

its maximum value Ԑ/R. Faraday’s

law of electromagnetic induction

can be used to describe this effect

as follows: As the source current

increases with time, the magnetic flux through the circuit loop due to this

current also increases with time. This increasing flux creates an induced

emf in the circuit. The direction of the induced emf is such that it would

cause an induced current in the loop, which would establish a magnetic

field that would oppose the change in the source magnetic field.

Page 4: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.1, Page – 1015

Thus, the direction of the induced

emf is opposite the direction of the

source emf; this results in a gradual

rather than instantaneous increase

in the source current to its final

equilibrium value. This effect is

called self-induction because the

changing flux through the circuit

and the resultant induced emf arise from the circuit itself. The emf ԐL

set up in this case is called a self-induced emf. It is also often called a

back emf.

Page 5: Magnetism 3
Page 6: Magnetism 3

Inductor or Solenoid:

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.1, Page – 1015

Consider a coil wound on a cylindrical iron core as shown in figure.

(a) A current in the coil produces a magnetic field directed to the left (Screw

rule) (b) If the current increases, the increasing magnetic flux creates an

induced emf having the polarity shown by the dashed battery (Lenz’s Law) (c)

The polarity of the induced emf reverses if the current decreases.

Page 7: Magnetism 3

Quantitative Analysis:

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.1, Page – 1015

From Faraday’s law we know that the induced emf ԐL is equal to the

negative time rate of change of the magnetic flux (– dФB/dt).

Again, a self-induced emf ԐL is always proportional to the time rate

of change of the source current (dI/dt).

Thus, we can write,

where L is a proportionality constant — called the inductance of the

coil — that depends on the geometry of the circuit and other

physical characteristics. From this expression, we see that the

inductance of a coil containing N turns is,

Page 8: Magnetism 3

Unit of Inductance:

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.1, Page – 1015

From the equation of induced emf, we can write that,

Thus the SI unit of inductance is Henry and can be written as,

Page 9: Magnetism 3
Page 10: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.3, Page – 1021

Because the emf induced in an inductor prevents a battery from

establishing an instantaneous current, the battery must do work

against the inductor to create a current. Part of the energy supplied

by the battery appears as internal energy in the resistor, while the

remaining energy is stored in the magnetic field of the inductor.

Applying KVL, we get,

IԐ is the energy supplied from the source, I2R is the energy

delivered to the resistor and thus, LI(dI/dt) is the energy stored in

the inductor.

Page 11: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.3, Page – 1021

If we let U denote the energy stored in the inductor at any time,

then we can write the rate dU/dt at which energy is stored as,

To find the total energy stored in the inductor, we can rewrite this

expression as dU = LIdI and integrate over the limit 0 to I:

This is the equation of energy stored in an inductor.

Page 12: Magnetism 3
Page 13: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.3, Page – 1021

Very often, the magnetic flux through the area enclosed by a circuit

varies with time because of time-varying currents in nearby circuits.

This condition induces an emf through a process known as mutual

induction, so called because it depends on the interaction of two

circuits.

Consider the two closely wound

coils of wire in cross-sectional view

in figure. The current I1 in coil 1,

which has N1 turns, creates

magnetic field lines, some of which

pass through coil 2, which has N2

turns. The magnetic flux caused by

the current in coil 1 and passing

through coil 2 is represented by

Ф12.

Page 14: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.3, Page – 1021

In analogy to equation L = NФ/I,

we define the mutual inductance

M12 of coil 2 with respect to coil 1:

Induced emf in coil 2 is,

Page 15: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 32.3, Page – 1021

In the preceding discussion, we assumed that the source current is in

coil 1. We can also imagine a source current I2 in coil 2. The

preceding discussion can be repeated to show that there is a mutual

inductance M21 . If the current I2 varies with time, the emf induced

by coil 2 in coil 1 is,

It can be experimentally shown that,

M12 = M21 = M and thus,

The unit of mutual inductance is also Henry.

Page 16: Magnetism 3
Page 17: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 33.8, Page – 1060

When electric power is transmitted over great distances, it is

economical to use a high voltage and a low current to minimize the

I2R loss in the transmission lines. Consequently, 33,000V lines are

common. At the receiving end of such lines, the consumer requires

power at a low voltage. Therefore, a device is required that can

change the alternating voltage and current without causing

appreciable changes in the power delivered. The ac transformer is

that device.

Construction:

The ac transformer consists of two

coils of wire wound around a core of

iron, as illustrated in figure. The coil

on the left, which is connected to the

input alternating voltage source and

has N1 turns, is called the primary

winding (or the primary).

Page 18: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 33.8, Page – 1060

The coil on the right, consisting of N2 turns and connected to a load

resistor R, is called the secondary winding (or the secondary). The

purpose of the iron core is to increase the magnetic flux through the

coil and to provide a medium in which nearly all the flux through

one coil passes through the other coil. Eddy current losses are

reduced by using a laminated core. Iron is used as the core material

because it is a soft ferromagnetic substance and hence reduces

hysteresis losses. Although practical transformer have some power

loss due to the resistance of the coil wire, as we assumed an ideal

transformer, so energy losses in the windings and core are zero.

Working Principle:

According to Faraday’s law the voltage

∆V1 across the primary is,

Page 19: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 33.8, Page – 1060

where ФB is the magnetic flux through each turn. If we assume that

all magnetic field lines remain within the iron core, the flux through

each turn of the primary equals the flux through each turn of the

secondary. Hence, the voltage across the secondary is:

Dividing and then rearranging

these two equations, we get,

When, N2 > N1, then ∆V2 > ∆V1. This setup is referred to as a step-up

transformer. When N2 < N1 , the output voltage is less than the input

voltage, and we have a step-down transformer.

Page 20: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 33.8, Page – 1060

For an ideal transformer, there is no power loss on primary or

secondary winding. Thus,

When a resistive load RL is

connected to the secondary, then

current I2 on the secondary will be,

Furthermore, the current in the primary is,

Where,

Page 21: Magnetism 3

Reference: Physics II by Robert Resnick and David Halliday, Topic – 33.8, Page – 1060

The above equation relates to the input resistance to the output

resistance. Req is the equivalent resistance of the load resistance

when viewed from the primary side. From this analysis we see that a

transformer may be used to match resistances (impedance matching)

between the primary circuit and the load. In this manner, maximum

power transfer can be achieved between a given power source and

the load resistance.

Page 22: Magnetism 3