Download ECT1026 Field Theory_Chapter 3 Magnetostatics

ECT1026 Field Theory
Chapter 3A Magnetostatics
Chapter 3 Magnetostatics – Part A
3.1 Introduction
Magnetostatics is the study on a phenomenon associated with the steady
motion of charges. Magnetic phenomena were first observed in natural minerals,
readily found near the ancient city of Magnesia (now Manisa, in western Turkey).
These stones are now called magnetite (Fe3O4) and they are examples of permanent
magnets. The phenomenon they exhibit is magnetism. The interactions of permanent
magnets and compass needles were described in terms of magnetic poles. The end of a
bar magnet that points north is called north-pole (N-pole); the other end is called
south-pole (S-pole). Opposite poles attract each other, and like poles repel each other
(Fig. 3-1). The end of a magnet from which the field lines emerge is called the north
pole of the magnet; the other end, where field lines enter magnet is called the south
pole. Magnetic poles always exist in pairs.
F
S
N
N
S
F
F
N
S
S
N
F
S
N
S
N
F F
Figure 3-1: Two bar magnets repel each other when likes poles (N and N, or S and
S) are next to each other. When opposite poles (N and S) are next to
each other, they attract each other.
By analogy to electric interactions (Chapter 2), we describe the interactions in
Fig. 3-1 by saying that a bar magnet sets up a magnetic filed in the space around it and
a second body responds to that field. The magnetic-filed pattern of a bar magnet is
displayed in Fig. 3-2. The magnetic lines encircling a magnet are called magnetic-field

lines and represent the existence of a magnetic field called the magnetic flux density B .
A magnetic field not only exists around permanent magnets but can also be
created by electric current, which was discovered by Hans Christian Oersted in 1819.
He experimentally found out a compass needle was deflected by a current in a wire.
The needle always turned in the direction perpendicular to the current-carrying wire
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ECT1026 Field Theory
Chapter 3A Magnetostatics
and to the radial line connecting the wire to the needle. The current-carrying wire
induced a magnetic field that formed closed circular loops around the wire, as
illustrated in Fig. 3-3.
Figure 3-2: Pattern of magnetic field lines around a bar magnet.
Figure3-3: The magnetic-field lines produced by a current in a long straight
wire form concentric circles around the wire. Here the current is into the page,
as indicated by the .
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ECT1026 Field Theory
Chapter 3A Magnetostatics
Shortly after Oersted’s discovery, André-Marie Ampère announced that
current-carrying wires exert forces on one another and has established Ampère’s law
of force, which describes the law of force between two current elements and is
analogous to Coulombs’s force law in electrostatics. Jean Baptiste Biot and Felix
Savart repeated Oersted’s experiments and put forth a compact law of static magnetic
fields generated by current elements in a circuit, which is known as the Biot-Savart
law.
In this chapter we study the magnetic filed produced by a current-carrying wire.
We introduce two methods for calculating magnetic field: one based on a direct
technique, analogous to Coulomb’s law for the calculation of electric field, and
another based on arguments of symmetry, analogous to Gauss’ law for electric field.
Common Effects of Magnetic Field (on Matter)

When a current-carrying conductor is placed near to a magnetic needle the
needle will deflect

A moving charge particle experiences magnetic force

Any current carrying conductor also experiences a force

Two parallel conductors carrying same direction current are attracted towards
each other, and vice versa.
Characteristics of Magnetic Field or Magnetic Flux Lines
•
The distribution and density of magnetic field is visualized as lines of
magnetic flux. Magnetic flux is the basis used to explain magnetic effects and
magnitudes
•
Direction of magnetic field/lines –
direction of the north-seeking pole of a
compass needle placed in the field
•
Each line of magnetic flux forms a closed-loop
 Lines of magnetic flux never intersect
 Lines of magnetic flux always trying to shorten themselves causing unlike
poles to attract each other
 Lines of magnetic flux (parallel) are in the same direction and repel each other.
They exerted a lateral pressure on one another
 A piece of soft iron can be magnetized thru’ magnetic induction
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Chapter 3A Magnetostatics
 The magnetic field consists of lines of force, which form complete circles
around the conductor. These circles centered around the center of the current
carrying conductor and the circular planes are perpendicular to it
 Right-Hand Screw Rule can be used to relate the direction of current flowing
in a conductor bar and the direction of magnetic field circling it.
 The thumb is pointing along the direction of the current flowing in the
conductor bar, and the fingers point in the direction of the circling magnetic
flux around the conductor
Right-Hand Screw Rule
Thumb: direction of the current flowing in the conductor
Fingers: direction of the circling magnetic flux around the conductor
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Chapter 3A Magnetostatics
parallel and anti-parallel linear currents

Two current carrying conductors are placed in each other parallel; each
will possess a magnetic field of its own, and each will exert a magnetic
field on the other conductor.

A sign “DOT” indicates the current is flowing out along the conductor,
whereas sign “CROSS” indicates the current is flowing in along the
conductor. The magnetic dipole pattern is shown in next slide

A circular conductor with current flowing in it produce magnetic dipole
along the entire circular loop. This circular conductor is called current
loop.
Parallel Linear Current
A sign “DOT” indicates the current is flowing OUT along the conductor
Anti-Parallel Linear Current
A sign “CROSS” indicates the current is flowing IN along the conductor
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ECT1026 Field Theory
Chapter 3A Magnetostatics


3.1.1 Magnetic Flux Density ( B ) and Magnetic Field Intensity ( H )
In Chapter 2 we dealt with static electric fields caused by electric charges at

rest. We saw that electric field intensity E is the only fundamental vector field
quantity required for the study of electrostatics in free space (or vacuum). In a
material medium it is convenient to define a second vector field quantity, the electric

flux density (or electric displacement) D , to account for the effect of polarization.
Two basis equations for the electrostatic are:

  D  v

 E  0
(3-8)
(3-9)


The electrical property of the medium determines the relation between D and E . For


linear and isotropic medium, D  E , where the permittivity  is a scalar quantity.
Stationary charges produce static electric fields, and steady currents produce
magnetic fields For non-time varying, the magnetic fields in a medium with magnetic
permeability  are governed by:

 B  0
 
 H  J
(3-10)
(3-11)


where J is the current density. The magnetic flux density B and the magnetic field

intensity H are related by


B  H
(3-12)
The magnetic permeability  accounts for magnetization properties of a material. 
for a particular material can be defined as
  r 0
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(3-13)
ECT1026 Field Theory
Chapter 3A Magnetostatics
where  r is a dimensionless quantity called the relative magnetic permeability of the
material. For most dielectrics and metals (excluding ferromagnetic materials),  r ~ 1
and thus are characterized by constant magnetic permeability    0  4  10 7 H/m.
The values of  r for some commonly used ferromagnetic materials are given in Table
3-1.
Table 3-1: Relative magnetic permeability  r of
some common ferromagnetic materials
Material
r
Cobalt
250
Nickel
600
Mild steel
2,000
Pure Iron
4,000 – 5,000
Silicon Iron
7,000
Mumetal
~100,000
Purified Iron
~ 200,000
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Chapter 3A Magnetostatics
3.1.2 Current and Current Density
(a) Current
We describe the flow of electric charge by introducing the concepts of current
and current density. Current is a flow of charge, measured by the rate at which charge
passes through any specified surface area, for example, the cross section of a wire.
Current is a scalar quantity. We define the current as the rate at which the charge is
transported past a given point in a conducting medium, that is,
i
dq
dt
(3-1)
where dq is the amount of charge that flows past some given point in time dt . The SI
unit of current is the ampere (A). A current of one ampere corresponds to the
transportation of one coulomb of charge in one second. We used the lowercase letter i
for the current to indicate that it is generally a function of time. We will use the
uppercase letter I to represent a steady current (also called a direct current), that is
current that is constant in time.
(b) Current Density
Since different amount of charges may flow through different parts of a given
cross-sectional area, it is often more appropriate to describe current in term of current

density, denoted as J and having a units of amperes per square meter (A/m2).The

current density J at any given point in a conductor is a vector quantity having the
direction of the flow of positive charge and a magnitude equal to the current per unit
area that is normal to the direction of flow.

Figure 3-2 shows a tube of charge with volume charge density  v .

The charges are moving with a mean velocity u along the axis of the tube.

Over a period t , the chares move a distance   ut .

The amount of charge that crosses the tube’s cross-sectional surface s' in
time t is therefore:
q'   v v   v s'   v uts'
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(3-2)
ECT1026 Field Theory
Chapter 3A Magnetostatics

Figure 3-1: Charges with velocity u moving through a cross section s' .
In the more general case where the charges are flowing through a surface s

whose surface normal is not parallel to u , as shown in Fig. 3-2. In this case, the
amount of charge  q flowing through s is:
or
 
(3-3)
q  vu  s t
 
q  vu  s t q   v ust cos 
(3-4)
and the corresponding current is:
 
q
 vu  s
t
 
 I  J  s


(A/m2)
J  vu
I 
where
(3-5)
(3-6)
(3-7)
is defined as the volume current density or just current density in ampere per square
meter.

Figure 3-2: Charges with velocity u moving through a cross section s' .
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Chapter 3A Magnetostatics
3.1.3 Magnetic Dipoles and Current Loops
A current loop with dimensions much smaller that the distance between the
loop and the observation point is called a magnetic dipole. This is because the pattern
of its magnetic field is similar to that of a permanent magnetic, as well as to the
pattern of the electric field of the electric dipole, as shown in Fig. 3-3.
Figure 3-3: Patterns of (a) the electric field of an electric dipole, (b) the magnetic field
of a magnetic dipole, and (c) the magnetic field of a bar magnet. Far from the sources,
the field patterns are similar in all three cases.

In Chapter 2 we considered the effect of an electric field E on an electric
dipole, which we pictured as two equal and opposite charges separated by a distance.

Defining the electric dipole moment p in a particular way, we found that the electric

field exerted a torque on the electric dipole that tended to rotate the dipole so that p

aligned with E . This statement appears very similar to the effect of a magnetic field
on a current loop: the torque on the loop tends to rotate it so that the normal vector n̂ n

aligns with B . This similarity suggests that we can use equations similar to those for
the electric dipole to analyze the effect of magnetic field on a current loop, which is
referred to as the magnetic dipole.
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Chapter 3A Magnetostatics
3.2 Magnetic Force
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Chapter 3A Magnetostatics
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Chapter 3A Magnetostatics
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Chapter 3A Magnetostatics
3.3 The Biot-Savart Law
The Biot-Savart law describes how the magnetic flux density field at a given

point P is produced an element of length d  of a filamentary wire carrying a steady
current I , as shown in Figure 3.2-1, is:

  0 Id   Rˆ
dB 
4 R 2
(3.3-1)
In this equation,
o

dB is the elemental magnetic flux density in teslas (T), and it direction at
point P is out of the page,
o
o

d  is an element of length in the direction of the current,
o

R̂ is the unit vector pointing from current element d  to point P,


the point P is at a distance R from the current element Id  , where R  RRˆ .
Point P
R

dB
Out of page
I
R̂


Id 


Figure 3.3-1: Magnetic flux density dB generated by a current element Id  .
The direction of the field induced at point P is into the page there as indicates
by  .
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ECT1026 Field Theory
Chapter 3A Magnetostatics

To determine the total magnetic flux density B due to a conductor of finite size, we
need to sum up the contributions due to the current elements making up the conductor.
Hence, the Biot-Savart law becomes:
  I
B 0
4

d   Rˆ
R2

l
(3.3-2)
where l is the line path along which I exists.

When the current sources are specified in terms of volume current density J

(measured in A/m2) over a volume v or surface current density J S (measured in A/m)
over a surface S as showed in Fig. 3.3-2, we can use the equivalence given by
 

Id   Jdv  J S ds
(3.3-3)
to express the Biot-Savart law as follows:
  I
B 0
4
  I
B 0
4


J S  Rˆ
ds
R2
(for a surface current)
(3.3-4)


J  Rˆ
dv
R2
(for a volume current)
(3.3-5)
S
v
(a)
(b)


Figure 3.3-2: (a) Volume current density J and (b) surface current density J S .
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ECT1026 Field Theory
Chapter 3A Magnetostatics
Application of Biot-Savart Law
Example 3.3-1: Magnetic Filed of a Linear Conductor
Using Biot-Savart law find the magnetic flux density at a point P in the plane that
bisects a long straight wire of length 2L carrying a direct current I, as shown in Fig.
3.3-3.
Figure 3.3-3: Long straight wire of
length 2L carrying a direct current I.
Magnetic flux density of
long straight wire
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Example 3.3-2: Magnetic Field of a Pie-Shaped Loop
Determine the magnetic field at the apex O of the pie-shaped loop shown in Fig. 3.2-4.
Figure 3.2-4: Pie-shaped loop of radius a carrying a current I
Solution:
For the straight segments OA and OC, the magnetic field at point O is zero.


As all points along these segments, d  is parallel to R̂ and hence d  Rˆ  0 .


Along segment AC, d  is perpendicular to R̂ , and d   Rˆ  zˆdl  zˆad .
Consequently,
  I
B 0
4
  I
B 0
4

l

d   Rˆ
R2

zˆad
a2

 I
 B  zˆ 0 
4a
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Chapter 3A Magnetostatics
Problem 3.3-1: Magnetic Filed of a Current Loop
A circular loop if radius a carries a steady current I. Determine the magnetic flux
density B at a point P on the axis of the loop, as showed in Fig. 3.2-5. [ B 
z
P (0,0,z)

R

x
a
y


d
I
Figure 3.3-5
3-18
 0 Ia 3
2z 3
].
ECT1026 Field Theory
Chapter 3A Magnetostatics
3.4 Ampère’s Circuital Law (or simply Ampère’s Law)
In electrostatics, problems involving symmetries of one form or another can
often be solved more easily using Gauss’s law than by direct application of
Coulomb’s law. For magnetostatics, the Biot-Savart law and Ampère’s force law are
the point relationships that are analogous to Coulomb’s law, whereas Ampère’s
circuital law (or simply Ampère’s law) serves the same purpose as Gauss’s law.
Ampère’s law is a mathematical consequence of the Biot-Savart law and can simply
be stated as:

C
 
 
H  d    J  ds  I
S
(3.4-1)
Where C is the contour that enclosed the surface S. In words, Ampère’s law states that

the line integral of magnetic filed intensity H around any closed contour is exactly
equal to the total net current passing through the surface enclosed by that contour.

The sign convention for the direction of C is taken so that I and H satisfy the right –
hand rule.
Ampère’s law is useful in solving magnetostatic problems having some degree of
symmetry (usually cylindrical symmetry), when a contour C over which H field is
constant can be identified, in which case the integration on the left-hand side of
Equation (3.3-1) can be readily carried out.
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Chapter 3A Magnetostatics
Application of Ampere’s Law
Example 3.4-1: Magnetic Field of Long Wire
A long (practically infinite) straight wire of radius R0 carries a steady current I which

is uniform over the cross-section of the wire. Find the magnetic flux density B at a
radial distance r from the long axis of the wire for (a) r ≤ R0 , i.e. inside the wire, and
(b) r > R0 , i.e. outside the wire.
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Chapter 3A Magnetostatics
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Example 3.4-2: Magnetic Field of Long Solenoid
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Example 3-4.3: Magnetic Field of a Toroidal Coil
A toroid whose dimension as shown in the figure below has N turns and carries
current I. Determine magnetic flux intensity H inside and outside the toroid.
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