This document provides a summary of Lecture 2 on electrostatics. It introduces fundamental concepts such as electric charge, Coulomb's law, electric field, electric potential, and the relationship between electric field and electric potential. Continuous distributions of charge such as volume, surface, and line charges are also discussed. Key equations for calculating electric fields and potentials from these various charge distributions are presented.
1. Lecture 2: Electrostatics
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Lecture 2
from the following blog:
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1
2. Lecture 2
Electromagnetic Fields
• Electromagnetics is the study of the effect
of charges at rest and charges in motion.
• Some special cases of electromagnetics:
– Electrostatics: charges at rest
– Magnetostatics: charges in steady motion (DC)
– Electromagnetic waves: waves excited by
charges in time-varying motion
2
3. Introduction to Electromagnetic
(E)
units = coulombs per square meter (C/m2 = A s /m2)
Lecture 2
Fields
• Fundamental vector field quantities in
electromagnetics:
– Electric field intensity
units = volts per meter (V/m = kg m/A/s3)
– Electric flux density (electric displacement)
– Magnetic field intensity
units = amps per meter (A/m)
– Magnetic flux density
units = teslas = webers per square meter (T =
Wb/ m2 = kg/A/s3)
3
(D)
(H)
(B)
4. Introduction to Electromagnetic Fields
• Universal constants in electromagnetics:
– Velocity of an electromagnetic wave (e.g.,
Lecture 2
light) in free space (perfect vacuum)
c » 3´108 m/s
– Permeability of free space
m = p ´ -
4 10 7 H/m
– Permittivity of free space:
e » ´ -
– Intrinsic impedance of free space:
4
0
8.854 10 12 F/m
0
h »120p W 0
5. Introduction to Electromagnetic Fields
• Relationships involving the universal
constants:
Lecture 2
c = =
5
h m
0
0
0
1
0 0
e
m e
In free space:
B H 0 = m
D E 0 =e
6. Introduction to Electromagnetic
Lecture 2
Fields
6
sources
Ji, Ki
Obtained
• by assumption
• from solution to IE
fields
E, H
Solution to
Maxwell’s equations
Observable
quantities
7. Electrostatics as a Special Case of
Geometric
Optics
Lecture 2
Electromagnetics
7
Maxwell’s
equations
Fundamental laws of
classical
electromagnetics
Special
cases
Electro-statics
Magneto-statics
Electro-magnetic
waves
Kirchoff’s
Laws
¶
t
Statics: º 0
¶
d <<l
Transmission
Line
Theory
Circuit
Theory
Input from
other
disciplines
8. Lecture 2
Electrostatics
• Electrostatics is the branch of
electromagnetics dealing with the effects
of electric charges at rest.
• The fundamental law of electrostatics is
Coulomb’s law.
8
9. Lecture 2
Electric Charge
• Electrical phenomena caused by friction are
part of our everyday lives, and can be
understood in terms of electrical charge.
• The effects of electrical charge can be
observed in the attraction/repulsion of
various objects when “charged.”
• Charge comes in two varieties called
“positive” and “negative.”
9
10. Lecture 2
Electric Charge
• Objects carrying a net positive charge attract
those carrying a net negative charge and repel
those carrying a net positive charge.
• Objects carrying a net negative charge attract
those carrying a net positive charge and repel
those carrying a net negative charge.
• On an atomic scale, electrons are negatively
charged and nuclei are positively charged.
10
11. Lecture 2
Electric Charge
• Electric charge is inherently quantized such
that the charge on any object is an integer
multiple of the smallest unit of charge which is
the magnitude of the electron charge
e = 1.602 ´ 10-19 C.
• On the macroscopic level, we can assume that
charge is “continuous.”
11
12. Lecture 2
Coulomb’s Law
• Coulomb’s law is the “law of action” between
charged bodies.
• Coulomb’s law gives the electric force
between two point charges in an otherwise
empty universe.
• A point charge is a charge that occupies a
region of space which is negligibly small
compared to the distance between the point
charge and any other object.
12
13. F a Q Q R pe
ˆ 12 r
1 2
Lecture 2
Coulomb’s Law
13
2
0 12
=
12 4
Q1
Q2 12 r
12 F
Force due to Q1
acting on Q2
Unit vector in
direction of R12
14. Lecture 2
Coulomb’s Law
• The force on Q1 due to Q2 is equal in
magnitude but opposite in direction to the
force on Q2 due to Q1.
F21 = -F12
14
15. Qt r
Lecture 2
Electric Field
• Consider a point charge
Q placed at the origin of
a coordinate system in
an otherwise empty
universe.
• A test charge Qt brought
near Q experiences a
force:
15
2
F a QQt
Qt r pe
0 4
ˆ
r
=
Q
16. Lecture 2
Electric Field
• The existence of the force on Qt can be
attributed to an electric field produced by Q.
• The electric field produced by Q at a point in
space can be defined as the force per unit
charge acting on a test charge Qt placed at
that point.
E F t
lim
®
Q Q
16
Q
t
t 0
=
17. Lecture 2
Electric Field
• The electric field describes the effect of a
stationary charge on other charges and is an
abstract “action-at-a-distance” concept, very
similar to the concept of a gravity field.
• The basic units of electric field are newtons per
coulomb.
• In practice, we usually use volts per meter.
17
18. Lecture 2
Electric Field
• For a point charge at the origin, the electric
field at any point is given by
E r a Q r pe pe
Qr
( ) = =
3
18
0
2
4 0 4
ˆ
r
r
19. Lecture 2
Electric Field
• For a point charge located at a point P’
described by a position vector
the electric field at P is given by
19
E r QR
( )
=
where
4 3
0 pe
= - ¢
R r r
R r r
R
= - ¢
r¢
Q
P
r R
r¢
O
20. Lecture 2
Electric Field
• In electromagnetics, it is very popular to
describe the source in terms of primed
coordinates, and the observation point in
terms of unprimed coordinates.
• As we shall see, for continuous source
distributions we shall need to integrate over
the source coordinates.
20
21. Lecture 2
Electric Field
• Using the principal of superposition, the
electric field at a point arising from multiple
point charges may be evaluated as
E r Q R
21
( ) å=
=
n
k k
R
k 1
k
3
0 4pe
22. Continuous Distributions of Charge
• Charge can occur as
– point charges (C)
– volume charges (C/m3)
– surface charges (C/m2)
– line charges (C/m)
Lecture 2
22
Ü most general
23. Continuous Distributions of Charge
• Volume charge density
q r Qencl
ev V D ¢
Lecture 2
23
( )
V
¢ =
lim
D ®0
Qencl
r¢ DV’
24. Continuous Distributions of Charge
• Electric field due to volume charge density
¢ ¢
d E r qev r dv R
Lecture 2
24
Qencl r¢ dV’
V’
r P
( ) ( )
3
4 pe
R
0 =
25. Electric Field Due to Volume Charge
E r q r R 3
Lecture 2
Density
( ) ( ) ò
25
¢
¢ ¢
=
V
ev dv
R
1
pe
0 4
26. Continuous Distributions of Charge
• Surface charge density
q r Qencl
es S D ¢
Lecture 2
26
( )
S
¢ =
lim
D ¢®0
Qencl
r¢ D S’
27. Continuous Distributions of Charge
• Electric field due to surface charge density
¢ ¢
d E r qes r ds R
Lecture 2
27
Qencl r¢ dS’
S’
r P
( ) ( )
3
4 pe
R
0 =
28. Electric Field Due to Surface Charge
E r q r R 3
Lecture 2
Density
( ) ( ) ò
28
¢
¢ ¢
=
S
es ds
R
1
pe
0 4
29. Continuous Distributions of Charge
• Line charge density
q r Qencl
el L D ¢
Lecture 2
29
( )
L
¢ =
lim
D ¢®0
Qencl r¢ D L’
30. Continuous Distributions of Charge
• Electric field due to line charge density
¢ ¢
d E r qel r dl R
Lecture 2
Qencl r¢ D L’ r
30
( ) ( )
3
4 pe
R
0 =
P
31. Electric Field Due to Line Charge
E r q r R 3
Lecture 2
Density
( ) ( ) ò
31
¢
¢ ¢
=
L
el dl
R
1
pe
0 4
32. Electrostatic Potential
• An electric field is a force field.
• If a body being acted on by a force is
moved from one point to another, then
work is done.
• The concept of scalar electric potential
provides a measure of the work done in
moving charged bodies in an electrostatic
field.
Lecture 2
32
33. Electrostatic Potential
• The work done in moving a test charge from one
point to another in a region of electric field:
a b W F dl q E dl
Lecture 2
= -ò × = - ò × ®
33
b
a
b
a
a
b
q
F
dl
34. Electrostatic Potential
• In evaluating line integrals, it is customary to take
the dl in the direction of increasing coordinate
value so that the manner in which the path of
integration is traversed is unambiguously
determined by the limits of integration.
W q E aˆ dx a b x
Lecture 2
34
3
= - ò · ®
5
x
b a
3 5
35. Electrostatic Potential
• The electrostatic field is conservative:
– The value of the line integral depends only on
the end points and is independent of the path
taken.
– The value of the line integral around any closed
path is zero.
Lecture 2
35
ò E × dl
= 0
C
C
36. Electrostatic Potential
• The work done per unit charge in moving a
test charge from point a to point b is the
electrostatic potential difference between
the two points:
Lecture 2
º ® = -ò ×
ab E dl
36
b
a
a b
q
V W
electrostatic potential difference
Units are volts.
37. Electrostatic Potential
• Since the electrostatic field is conservative
we can write
ò ò ò
V = - E · dl = - E · dl - E ·
dl
Lecture 2
æ
ò ò
E dl E dl
= - · - - ·
37
V (b) V (a)
a
P
b
P
b
P
P
a
b
a
ab
= -
ö
÷ ÷
ø
ç ç
è
0 0
0
0
38. Electrostatic Potential
• Thus the electrostatic potential V is a scalar
field that is defined at every point in space.
• In particular the value of the electrostatic
potential at any point P is given by
Lecture 2
( ) = -ò ·
V r E dl
38
P
P
0 reference point
39. Electrostatic Potential
• The reference point (P0) is where the potential
is zero (analogous to ground in a circuit).
• Often the reference is taken to be at infinity
so that the potential of a point in space is
defined as
Lecture 2
P
( ) ò
V r = - E ·
dl
¥
39
40. Electrostatic Potential and Electric
W QV Q V b V a
Lecture 2
Field
• The work done in moving a point charge from
point a to point b can be written as
= = - ®
Q E dl
40
{ ( ) ( )}
a b ab
b
= - ò ·
a
41. Electrostatic Potential and Electric
W Q V QE l
Lecture 2
Field
• Along a short path of length Dl we have
D = D = - ×D
V E l
D = - ×D
41
or
42. Electrostatic Potential and Electric
Lecture 2
Field
• Along an incremental path of length dl we
have
dV = -E ×dl
• Recall from the definition of directional
derivative:
dV = ÑV ×dl
42
