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• Magnetism

159

Chapter 9

Magnetism

_____________________________________________

9.1 Introduction

Permanent magnets have long been used in navigational compasses. Magnet

always has two poles. Unlike the case where electric charge can be separated

into positive and negative charge, so far no one has found a magnetic monopole.

Any attempt to separate the north and south poles by cutting the magnet fails

because each piece becomes a smaller magnet with its own north and south

poles.

In this chapter, we cover the basic fundamentals of magnetism that

includes magnetic force and electromagnetism, and induction.

9.2 Magnetic Field

Magnetic fields are produced by electric currents, which can be macroscopic

currents in wires or microscopic currents associated with electrons in atomic

orbits. Magnetic field sources are essentially dipolar in nature, having a north

and south magnetic pole. The SI unit for magnetic field is Tesla, which can be

seen from the magnetic part of the Lorentz force law F

= q BxV

to be composed

of (Newton x second)/(Coulomb x meter). A smaller magnetic field unit is the

Gauss (1 Tesla = 10,000 Gauss). Note that Lorentz force law is BxVqEqF !

! .

There are many source of magnetic field. Some illustrations are shown in

Fig. 9.1.

Figure 9.1: Sources of Magnetic field

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160

9.2.1 Magnetic field of the earth

The earth's magnetic field is similar to that of a bar magnet tilted 11 degrees

from the spin axis of the earth. The Curie temperature of iron is about 770 0C,

which lower than the earth's core temperature, whereby it should not have

magnetic field. However, magnetic fields surround electric currents that surmise

circulating electric currents in the Earth's molten metallic core gives rise to

magnetic field.

The earth's magnetic field is attributed to a dynamo effect of circulating

electric current, but it is not constant in direction. Rock specimens of different

age in similar locations have different directions of permanent magnetization.

Evidence for 171 magnetic field reversals during the past 71 million years has

been reported.

Although the details of the dynamo effect are not known in detail, the

rotation of the Earth plays a part in generating the currents, which are presumed

to be the source of the magnetic field. Venus does not have such a magnetic

field although its core has iron content similar to that of the Earth. But the

Venus's rotation period is 243 Earth days that is too slow to produce the dynamo

effect.

9.2.2 Magnetic flux

Magnetic flux "B is the product of the average magnetic field times the perpendicular area that it penetrates i.e. "B = AB

# . It is also equal to BA cos \$,

where \$ is the angle between the magnetic field and the plane of the area. The illustration is shown in Fig. 9.2.

Figure 9.2: Illustration of magnetic flux

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161

For a closed surface, the sum of magnetic flux "B is always equal to zero, which is Gauss' law for magnetism i.e.

(9.1)

No matter how small the volume, the magnetic sources are always dipole

sources so that there are as many magnetic field lines coming in (to the south

pole) as out (from the north pole).

9.2.3 Magnetic field of a current loop

Electric current in a circular loop creates a magnetic field as shown in Fig. 9.3,

which is more concentrated in the center of the loop than outside the loop.

Stacking multiple loops called a solenoid concentrates more fields.

Figure 9.3: Magnetic field of a current loop

9.2.4 Magnetic field contribution of a current element

The Biot-Savart Law relates magnetic fields to the currents, which are their

sources. In a similar manner, Coulomb's law relates electric fields to the point

charges, which are their sources. Biot-Savarts law states that the magnetic field

of a current element 2

0

r4

rxLIdBd

&

'!

, where '0 is equal to . The

illustration is shown in Fig. 9.4.

17 TmA10x4 ((&

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162

Figure 9.4: Magnetic field due a current element

Magnetic field at the center of current loop

The magnetic field at the center of current loop can be derived from Biot-

Savarts law, which states 2

0

r4

rxLIdBd

&

'!

and the illustration shown in Fig. 9.5.

Figure 9.5: Magnetic field at the center of current loop

From Biot-Savarts law, 2

0

r4

rxLIdBd

&

'!

= 2

0

r4

sinIdLdB

&

\$'! . Since the angle between

and Ld

r is 900, therefore sin \$ = sin 900 = 1. Biot-Savarts equation shall be

%&'

! dLr4

IB

2

0 . Therefore the magnetic field at the center of current loop is

rR2

IB 0

'!

(9.2)

Magnetic field on the axis of current loop

The magnetic field at the axis of current loop can be derived from Biot-Savarts

law, which states 2

0

r4

rxLIdBd

&

'!

and the illustration shown in Fig. 9.6.

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163

Figure 9.6: Magnetic field at the axis of current loop

The magnetic field at x-direction cancels each other due to symmetrical

direction. The only component left is the z-component.

The z-component magnetic field \$&

'! cos

r4

rxLIdBd

2

0

z

. r is equal to 22 zR .

Therefore, the z-component magnetic field is 2/322

0

z)Rz(4

kRLIdBd

&

'!

. After

integrating % &! R2dL , the z-component magnetic field

=zB

k)Rz(4

IR22/322

2

0

&

&' (9.3)

9.2.5 Amperes Law

The net electric field due to any distribution of charges follows equation (8.3)

i.e. rr4

E2

0))*

+,,-

.

&/!

q . Similarly the net magnetic field due to any distribution of

current follows Biot-Savarts law 2

0

r4

rxLIdBd

&

'!

. If the distribution has some

symmetry, then Amperes law can be used to find the magnetic field with less

effort. Amperes law states that the sum of the magnetic field along any closed

path is proportional to current that passes through. Mathematically, it is

% % '!\$!# enc0i dscosBsdB

(9.4)

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164

where ds is the length of Amperian loop and Bcos \$ is tangential to the loop as shown in Fig. 9.7.

Figure 9.7: Amperes law applied to an arbitrary Amperian loop encircles two long wires

and excluding a third wire. The dot indicates the current is out of the page and

cross indicates the current is into the page

Applying amperes law for the above Amperian loop, it becomes

% % ('!\$!# )ii( dscosBsdB 210

.

Magnetic field outside a long straight wire with current

Consider the case where a long wire carries current i directly out of the page as

shown in Fig. 9.8. The magnitude of magnetic field B at distance r from the

wire has a cylindrical symmetry. Thus, applying Amperes law, the magnetic

field at the distance r around the wire is enc

0icosBds% '!\$ . Knowing that \$ = 0, ienc = i, and % &! r2ds , the magnetic shall be

B = r2

i0

&

' (9.5)

Figure 9.8: Magnetic field due to long wire carrying current i

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165

Magnetic field inside a long straight wire with current

Consider the case for the magnetic field inside a long straight wire with current

shown in Fig. 9.10.

Figure 9.10: Magnetic field inside a long wire with current

Applying Amperes law enc

0icosBds% '!\$ , where ienc is equal to iRr

i2

2

enc &

&! , the

magnetic field B is equal to

B = rR2

i2

0 )*

+,-

.&

' (9.6)

Magnetic field of solenoid

A tight wound helical coil of wire is called solenoid as shown in Fig. 9.11(a).

The magnetic field at the center of the solenoid that carrying current i and has n

turn per unit length can be calculated using Amperes law.

Figure 9.11: (a) Solenoid and (b) Amperian loop of the solenoid

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166

Consider the Amperian rectangular loop abcd as shown in Fig. 9.12(b). Apply

Amperes law, enc

0icosBds% '!\$ to the Amperian loop, it is % # sdB

=

+ + + = '% #b

a

sdB

% #c

b

sdB

% #d

c

sdB

% #a

d

sdB

0Ni, where N is the number of turn in the

Amperian loop. % = = = 0 since ad and bc are perpendicular to

the magnetic field and outside the solenoid the magnetic field is assumed to be

zero. Thus, the magnetic field at the center of solenoid is

#c

b

sdB

% #d

c

sdB

% #a

d

sdB

B = nih

Ni0

0 '!'

(9.7)

where N/h = n, the number of turn per unit length.

Magnetic field of toroid

A toroid can be considered as a solenoid bent into a shape of hollow doughnut

as shown in Fig. 9.12(a). The magnetic field at the center of the toroid that

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