chapter 3: static electric fields -...
TRANSCRIPT
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
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
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)
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.
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”
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.
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
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)
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)
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)
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”
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)
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)
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])
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.
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