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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.

    % !# 0AdB

    (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

    carrying current i and has N turn can be calculated using Amperes