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Chapter 3:
Static Electric Fields
3-1. Overview
3-2. Fundamental Postulates of Electrostatics in Free Space
3-3. Coulomb’s Law
3-4. Gauss’ Law and Applications
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3.1 Overview
Electrostatics is the study of the effect of electric
charges at rest, where the electric fields do not change
in time.
Static electric fields in free space;
Conductors and dielectrics in static electric fields
Deductive approach :
Divergence and curl of E Gauss’s law
Coulomb’s law
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3.2 Fundamental Postulates of Electrostatics
in Free Space
Electric field intensity E is defined as the force per unit charge
that a very small stationary test charge experiences when it is
placed in a region where an electric field exists :
(3-1)
According to Eq. (3-1), the force F on a stationary charge q
in an electric field E:
(3-2)
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Two fundamental postulates of electrostatics in free space:
(3-3)
3.2 Fundamental Postulates of Electrostatics
in Free Space
(3-4)
(v : the volume charge density [C/m3])
1.
2.
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Taking the volume integral of both side of Eq. (3-3) over an
arbitrary volume V,
3.2 Fundamental Postulates of Electrostatics
in Free Space
(3-5)
(3-6)
According to the Divergence theorem,
where Q = the total charge contained in volume V bounded by surface S.
“Gauss’s law”
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From Eq. (3-4), i.e., E = 0, integrating E over an open
surface and using Stokes’s theorem :
3.2 Fundamental Postulates of Electrostatics
in Free Space
(3-7)
The scalar line integral of the static electric field intensity
around any closed path vanishes
The scalar product E dl integrated over any path is the
voltage along that path :
“Kirchhoff’s voltage law” : The algebraic sum of voltage drops
around any closed circuit is zero.
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3.2 Fundamental Postulates of Electrostatics
in Free Space
Postulates of Electrostatics in Free Space
Differential Form Integral Form
0
v E0
Q
S dsE
0 E 0C dlE
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The electric field intensity E due to a single point charge q
at a spherical surface with radius R centered at the origin in free
space can be obtained by applying Gauss’s law, i.e., Eq.(3-6) :
3-3 Coulomb’s Law
0
qdsE
SR
S RR aadsE
0
24
q
REdsE RS
R
or
(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.77)
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Eq. (3-8) tells that the electric field intensity of a positive
point charge is in the outward radial direction and has a
magnitude proportional to the charge and inversely
proportional to the square of the distance from the charge.
3-3 Coulomb’s Law
(3-8)
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3-3 Coulomb’s Law
)/(4
)(3
0
mVRR
RRqEP
If the charge q is not at the origin of a coordinate system,
(3-11)
Example 3-1
(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.77)
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When a point charge q2 is placed in the field of another point
charge q1, a force F12 is experienced by q2 due to electric
field intensity E12 of q1 at q2 :
3-3 Coulomb’s Law
(3-13)
The force between two point charges is proportional to the
product of the charges and inversely proportional to the
square of the distance of separation.
“Coulomb’s law”
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3-3.1 Electric Field Due to a System of
Discrete Charges
The total E field at a point is the vector sum of the fields
caused by all the individual charges :
(3-14)
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3-3.2 Electric Field due to a Continuous
Distribution of Charge
The contribution of the charge v dv in a differential volume
element dv to the electric field intensity at the field point P :
(3-15)
(Fundamentals of Engineering Electromagnetics, Addison-Wesley 1993, by David K. Cheng: p.82)
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3-3.2 Electric Field due to a Continuous
Distribution of Charge
(3-16)
(3-17)
(3-18)
(v : the volume charge density [C/m3])
(s : the surface charge density [C/m2])
(s : the line charge density [C/m])
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3-3.2 Electric Field due to a Continuous
Distribution of Charge
Example 3-3 As shown in the solution of example 3-3,
we can solve this problem using Eq.(3-18) in principle.
However, we’d better apply Gauss’s law as shown in
Examples 3-4, 3-5, and 3-6.
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3-4 Gauss’s Law and Applications
Gauss’s law :
(3-24)
The total outward flux of the E field over any closed
surface in free space is equal to the total charge enclosed in
the surface divided by
+ Divergence theorem
Examples 3-4, 3-5, and 3-6