Download Chapter2A 07_08

Chapter 2: Electrostatics - Part A
What we will learn:
 Nature of Electric Charge
 Concept of Electric Field
 Electric Field Intensity
 Electric Flux Density
 Charge and Current Distributions
 Coulomb’s Law
 Gauss’s Law
INTRODUCTION
Field theory lays the foundation of electrical, electronics, telecom and
computer engineering.
Field theory is used in understanding the principle of CRO, Radar,
satellite communication, TV reception, remote sensing, radio
astronomy, microwave devices, fiber optics, electromagnetic
compatibility, instrument landing systems etc.
A number of phenomena, which cannot be adequately explained by
circuit theory, can possibly be explained by field theory.
In circuit theory, voltages and currents are the main system variables,
and they are treated as scalar quantities.
On the other hand, most of the variables in electromagnetic field
theory are vectors having magnitude as well as direction.
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NATURE OF ELECTROSTATIC CHARGE
In general, an atom is electrically neutral (total number of positive
charge of protons = total number of negative charge of electrons).
However, when negatively charged electrons are removed from
atoms of a body, then there will be more positive charges than the
negative charges in the atoms and the body is said to be negatively
charged. On the other hand, if some electrons are added to it, the
number of negative charges is more than positive charges, then the
body is said to be negatively charged.
First Law of ELECTROSTATICS:
“Like charges repel each other, opposite charges attract each
other”
CONCEPT OF ELECTRIC FIELD
Every charged object sets up an electric field in the surrounding
space. A second charge "feels" the presence of this field. The second
charge is either attracted toward the initial charge or repelled from it,
depending on the signs of the charges. Of course, since the second
charge also has an electric field, the first charge feels its presence and
is either attracted or repelled by the second charge, too.
If we place two positively charged particles A and B at a distance as
shown in figure below. There is a repulsive force between these two
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particles. This force is of the action-at-a-distance type and can be felt
without any intermediate medium between A and B.
Let us move particle B away. Point P is the point where particle B
was placed, and is now in the electric field created by particle A.
When particle B is placed back at P, a force F will be exerted on the
particle B by the electric field.
To verify the existence of an electric field at a point, a test charge is
placed at that point. If the test charge feels an electric force, then the
electric field exists at that point. Thus, an electric field is said to exist
at a particular point if an electric force is acting on a charged particle
at that point.
The magnitude and direction of the electric field vector varies from
point to point in the field space. Electric field can also vary with time.
In this subject we are studying on electric field produced by static
electrical charges which is not changing with time.
Electrostatic field - field from electric charge at rest.
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ELECTRIC FIELD INTENSITY
E
A static electric charge sets up an electric field in the region of space
that surrounds it. The quantity we use to measure the strength of the
electric field is its intensity.
The intensity of an electric field (electric field intenity)
E
is the force
exerted by the electric field on a unit test charge.
Imagine an electric charge Q located at the origin of a co-ordinate
system. A test charge q at a distance r from Q experiences a force,
which according to Coulomb’s law is:
F=
1
4 0
Qq
r̂
r2
N, where
r̂
is the unit vector directing radially from Q
to q. Using the above definition the intensity of the electric field
surrounding the charge Q is therefore:
E
=
F
q
=
1
4 0
Q
N
r̂
2
C
r
(or
V
).
m
This result says that an electric charge located at the origin generates
an electric field (E-field) which points radially out from the origin.
The strength of the electric field inversely proportional to the square
(| E | 
1
r2
) of the distance from the origin. Surfaces of constant | E |
form concentric spheres about the origin.
If a few point charges Q1, Q2, Q3, …Qn, at distances r1, r2, r, ... rn
from point P as shown in the figure below, then every point charge
will be exerting force on the test charge Qt at point P, therefore the
net force on Qt will be vector sum of these forces. Thus the electric
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field intensity at point P is the vector sum of each individual electric
field intensity,
E  E1  E 2  E3  ...  En
Electric field intensity
E
at a point is equal to the negative gradient of
electric potential at that point. In other words,
E
is equal to the rate of
fall of potential in the direction of the lines of force.
E
dV
(V / m)
dx
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LINES OF FORCE
Michael Faraday introduces the concept of lines of force or field lines
to represent the electric field. These continuous lines of force
emanate from a positive charge and end on a negative charge. These
lines always leave or enter a conducting surface normally. The lines
indicate the path that a small positive test charge would take if it were
placed in the field. A line tangent to a field line indicates the direction
of the electric field at that point. The arrow shows the direction of the
tangential line. The lines of force do not cross among themselves.
The magnitude of the electric field is represented by the number of
lines of force passing perpendicularly (normally) through a surface.
Thus, there are more lines of force in a strong electric field and less in
a weak field.
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ELECTRIC FLUX
In SI units, one line of electric flux is defined as a tube of lines
(known as Faraday tube) of force emanates from +1 C and terminates
on -1C charge. Since electric flux is numerically equal to the charge,
so unit of flux is measured in coulomb, which is represented by .
So = Q coulombs.
The number of lines of force per unit area is directly proportional to
the electric field intensity at that point. They can be related by the
equation below:
N
 oE
An
For free space.
where N = number of lines of force, An = normal surface area to the
field direction at that point, o the permittivity of the space.
If A1=A2=A and N1>N2, then N1/A1 > N2/A2  0E1>
0E2, and E1>E2
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ELECTRIC FLUX DENSITY D
Flux density is given by the normal flux per unit area, denotes as D. If
a flux of  coulombs passes normally through an area of A m2, then
flux density,
D

A
C/m2. It is related to electric field intensity by the
relation, D   o r E .
Therefore
D
is a vector field similar to which can be represented by
lines of force or lines of displacement (or simply "displacement").
D
is a vector quantity whose direction at every point is the same as
that of
E but
whose magnitude is
 o r  E
. One useful property of D is
that its surface integral over any closed surface equals the enclosed
surface charge.
MAXWELL’S EQUATIONS
Modern electromagnetism is based on a set of four fundamental
relations known as Maxwell’s equations:

 . D = v,
(2.la)


B
 E=(2.lb)
t

 . B = 0,
(2.lc)


 D
 H = J+
,
(2.ld)
t
where E and D are electric field quantities interrelated by D = E,
with  being the electrical permittivity of the material;
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B and H are magnetic field quantities interrelated by B = H, with 
being the magnetic permeability of the material;
v is the electric charge density per unit volume; and
J is the current density per unit area.
 These equations hold in any material, including free space
(vacuum), and at any spatial location (x,y,z).
 His equations, which he deduced from experimental observations
reported by Gauss, Ampere, Faraday, and others, not only
encapsulate the connection between the electric field and electric
charge and between the magnetic field and electric current, but
they also define the bilateral coupling between the electric and
magnetic field quantities. Together with some auxiliary relations,
Maxwell’s equations form the fundamental tenets of
electromagnetic theory.
 In the static case, all charges are permanently fixed in space, or, if
they move, they do so at a steady rate so that v and J are
constant in time. The time derivatives (/t = 0 ) of B and D in
Eqs. (2.l b) and (2.ld) are zero, and Maxwell’s equations reduce to
Electrostatics
 . D = v,
(2.2a)
  E = 0.
(2.2b)
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Magnetostatics
 . B = 0,
(2. 3a)
  H = J.
(2. 3b)
The electric and magnetic fields are no longer interconnected in the
static case. This allows us to study electricity and magnetism as two
distinct and separate phenomena, as long as the spatial distributions
of charge and current flow remain constant in time.
 We refer to the study of electric and magnetic phenomena under
static conditions as electrostatics and magnetostatics,
respectively. Electrostatics is the subject of the present chapter,
and in the Chapter 3 we learn about magnetostatics.
 We study electrostatics not only as a prelude to the study of timevarying fields, but also because it is an important field of study in
its own right. Many electronic devices and systems are based on
the principles of electrostatics. They include x-ray machines,
oscilloscopes, ink-jet electrostatic printers, liquid crystal
displays, copying machines, capacitance keyboards, and many
solid-state control devices. Electrostatics is also used in the design
if medical diagnostic sensors, such as the electrocardiogram (for
recording the heart’s pumping pattern) and the
electroencephalogram (for recording brain activity), as well as in
numerous industrial applications.
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Charge and Current Distributions
 In electromagnetics, we encounter various forms of electric charge
distributions, and if the charges are in motion, they constitute
current distributions. Charge may be distributed by over a volume
of space, across a surface, or along a line.
Charge Densities
 We define the volume charge density v as
q dq
=
v  0 v
dv
v = lim
(C/m3),
(2.4)
where q is the charge contained in an elemental volume v.
 In general, v is defined at a given point in space, specified by (x,
y , z) in a Cartesian coordinate system, and at a given time t. v =
v (x, y, z, t).
 Physically, v represents the average charge per unit volume for a
volume v centered at (x, y, z), with v being large enough to
contain a large number of atoms and yet small enough to be
regarded as a point at the macroscopic scale under consideration.
The variation of v with spatial location is called its spatial
distribution, or simply its distribution.
 The total charge contained in a given volume v is given by
Q = v v dv,
(C),
(2.5)
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 In some cases, particularly when dealing with conductors, electric
charge may be distributed across the surface of a material, in
which case the relevant quantity of interest is the surface charge
density s, defined as
q dq
=
s  0 s
ds
s = lim
(C/m2),
(2.6)
where q is the charge present across an elemental surface area s.
 Similarly, if the charge is distributed along a line, which need not
be straight, we characterize the distribution in terms of the line
charge density l, defined as
q dq
=
l  0 l
dl
l = lim
(C/m),
(2.7)
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Current Density
Figure 2.2
 Consider a tube of charge with volume charge density v, as
shown in Figure (a) above. The charges are moving with a mean
velocity u along the axis of the tube. Over a period t, the charges
move a distance l = u t.
 The amount of charge that crosses the tube’s cross-sectional
surface s in time t is therefore.
q = v v = v l s = vus t,
(2.8)
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 Now consider the more general case where the charges are flowing
through a surface s whose surface normal n is not necessarily
parallel to u, as shown in Figure (b) above. In this case, the
amount of charge q flowing through s is
q = v u (s cos  )t
q = vus t,
(2.9)
and the corresponding current is
I = q/t = vus = Js,
(2.10)
J = vu (A/m2),
(2.11)
where
J is defined as the current density in ampere per square meter. For an
arbitrary surface S, the total current flowing through it is given by
I= S Jds
(A),
(2.12)
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There are two types of current generated by different physical
mechanisms
1) Convection current – generated by actual movement of electrically
charged matter. Do not obey Ohm’s law. J is called the convection
current density.
2) Conduction current – where atoms of the conducting medium do
not move, only the electrons in the outermost shell are moving.
Obeys Ohm’s law.
When the current is generated by the actual movement of electrically
charged matter, it is called a convection current, and J is called the
convection current density.
A wind-driven charged cloud, for example, gives rise to convection
current. In some cases, the charged matter constituting the convection
current consists solely of charged particles, such as the electrons of an
electron beam in a cathode ray tube (the picture tube of the television
and computer monitors).
 Convection current is distinct from conduction current. In a
metal wire, for example, all the positive charges and most of the
negative charges cannot move; only those electrons in the
outermost electronic shells of the atoms can be easily pushed from
one atom to the next if the voltage is applied across the ends of the
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wire. This movement of electrons from atom to atom gives rise
conduction current. The electrons that emerge from the wire are
not necessarily the same electrons that entered the wire at the other
end.
 Because the two types of current are generated by different
physical mechanisms, conduction current obeys Ohm’s law,
whereas convection current does not.
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Coulomb’s Law
The electric field was introduced and defined on the basis of the
results of Coulomb’s experiments on the electrical force between
charged bodies.
1) An isolated charge q induces an electric field E at every point in
space, and at any specific point P, E is given by
E = Rˆ
q
4 R 2
(2.13)
where R̂ is a unit vector pointing from q to P (Figure above), R is the
distance between them, and  is the electrical permittivity of the
medium.
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2) In the presence of an electric field E at a given point in space,
which may be due to a single charge or a distribution of many
charges, the force acting on a test charge q, when the charge is
placed at that point, is given by
q q'
F = q E = Rˆ
(N).
4 R 2
(2.14)
with F measured in newtons (N) and q in coulombs (C), the unit of E
is (N/C).
 For a material with electrical permittivity , the electrical field
quantities D and E are related by
D=E
(2.15)
 = r 0,
(2.16)
with
where
0 =8.85  10-12  (1/36)  10-9 (F/m)
is the electrical permittivity of free space, and r = /0 is called the
relative permittivity (or dielectric constant) of the material.
For most materials and under most conditions,  of the material has a
constant value independent of both the magnitude and direction of E.
If  is independent of the magnitude of E, then the material is said to
be linear because D and E are related linearly, and if it is
independent of direction of E, the material is said to be isotropic.
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Electric Field due to Multiple Point Charges
The expression given by Eq.(2.13) for the field E due to a single
charge can be extended to find the field due to multiple point charges.
 We begin by considering two point charges, q1 and q2, located at
position vectors R1 and R2 from the origin of a given coordinate
system, as shown in Figure below.
z
E
E2
E1
R-R1
q1
R1
P
R-R2
R
q2
R2
y
x
The electric field E is to be evaluated at a point P with position vector
R. At P, the electric field E1 due to q1 alone is given by Eq. (2.13)
with R, the distance between q1 and P, replaced with | R–R1| and the
unit vector R̂ replaced with (R – R1)/ |R – R1|.
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 

q (R  R )
E1 = 1   13
4 R  R
1
(V/m),
(2.17a)
Similarly, the electric field due to q2 alone is
 

q2 ( R  R 2 )
E2 =
 
4 R  R 3
2
(V/m),
(2.17b)
 The electric field obeys the principle of linear superposition.
Consequently, the total electric field E at any point in space is
equal to the vector sum of the electric fields induced by all the
individual charges. In the present case,



E = E1 + E 2
 
 
(R  R1 )
(R  R 2 )
1
=
[ q1   3 + q2   3 ]
4
R  R1
R  R2
(2.18)
 Generalizing the preceding result to the case of N point charges,
the electric field E at position vector R caused by charges q1,
q2,…,qN located at points with position vectors R1, R2, …, RN, is
given by

E=
 
qi (R  R i )
1

  3
4 i 1 R
 Ri
N
(V/m).
(2.19)
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Electric Field due to a Charge Distribution
We now extend the results we obtained for the field caused by
discrete point charges to the case of a continuous charge distribution.
 Consider volume v shown in Figure below. It contains a
distribution of electric charge characterized by a volume charge
density v, whose magnitude may vary with spatial location within
v.
Rq
R
The differential electric field at a point P due to a differential amount
of charge dq = v dv contained in a differential volume dv’ is

ˆ'
dE =R
dq
4 R'
2
(2.20)

where R ' is the vector from the differential volume dv to point P.
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
 
R' = R - Rq
and

R̂' = R ' / R’
Applying the principle of linear superposition, the total electric field

E can be obtained by integrating the fields contributed by all the
charges making up the charge distribution. Thus,


1
E = v d E =
4
 v dv'
ˆ
R
'
v'
R '2
(volume distribution).(2.21a)
 If the charge is distributed across a surface S with surface charge
density s, then dq = s ds. Accordingly,


1
E = s d E =
4
 s ds'
ˆ
(surface distribution) (2.21b)
R
'
s'
2
R'
 If it is distributed along a line l with a line charge density l, then
dq=l dl. Accordingly


1
ˆ ' l dl ' (line distribution)
E = l d E =
R

4 l '
R '2
(2.21c)
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Gauss’s Law

 Gauss’ law states that the total outward flux of D over any closed
surface equals the total charge enclosed Q.
Another words
 
D
S  ds  Q
(2.22)
 
E
S  ds  Q / 
The total outward flux of electric field intensity over any closed
surface equals the total charge enclosed by the surface divided by the
permittivity .
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 The surface S referred to here need not necessarily be a physical
surface, but can be any hypothetical surface, which encloses the
charge. Gauss’ law is useful in helping to determine the electric
field intensity of charge distribution with some symmetry
conditions. The surface S is called a Gaussian Surface.
Using the divergence theorem we can write
 
D
S  ds =



D
dv = Q
v
Or
Q=
v v dv
Thus

  D  v
(2.23)
This is Maxwell’s first equation of electromagnetism. Sometimes this
is called the differential form of Maxwell’s first equation.
 Gauss’s law, as given by Eq. (2.22), provides a convenient method


for determining the electrostatic flux density D , thus E , when the
charge distributions possesses symmetry properties that allow us to
make valid assumptions about the variations of the magnitude and

direction of D as a function of spatial location. Because at every
point on the surface the direction of ds is the outward normal to the

surface, only the normal component of D at the surface
contributes to the integral in Eq.(2.22).
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Example

The integral form of Gauss’s law can be applied to determine D due
to a single isolated charge q by constructing a closed, spherical,
Gaussian surface S of a arbitrary radius R centered at q, as shown in
figure below.
From symmetry considerations, assuming that q is positive, the

direction of D must be radially outward along the unit vector R̂ , and

DR, the magnitude of D , must be the same at all point on the surface,

defined by position vector R .
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 To successfully apply Gauss’s law, the surface S should be chosen

such that, from symmetry considerations, the magnitude of D is
constant and its direction is normal or tangential at every point of
each subsurface of S (the surface of a cube, for example, has six
subsurfaces.
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