Electromagnetic I
EELE 3331
Lecture V
Electrostatic Fields
Dr. Mohamed Ouda
Electrical Engineering Department
Islamic University of Gaza
Coulomb’s Law
Force Due to Multiple Point Charges
Given a point charge Q in the vicinity of a set of N
point charges (Q1,Q2,..., QN), the total vector force
on Q is the vector sum of the individual forces due
to the N point charges.
Electric Field
According to Coulomb’s law, F between two point charges
is directly proportional to the product of the two charges.
Alternatively, we may view each point charge as producing
a force field around it (electric field) which is proportional
to the point charge magnitude.
The force per unit charge experienced by the test charge
Q is defined as the electric field at the point P.
Given our convention of using a positive test charge, the
direction of the vector electric field is the direction of the
force on positive charge.
A convention has been chosen where the source
coordinates (location of the source charge) are defined by
primed coordinates while the field coordinates (location of
the field point) are defined by unprimed coordinates.
Electric Field
Electric Field. Cont’d
The vector electric field intensity E at r (force per
unit charge) is found by dividing the Coulomb
force equation by the test charge Q.
Note:-
• E produced by Q is independent of the magnitude
of the test charge Q.
•
For the special case of a point charge at the
origin (r’ = 0), the electric field reduces to :
Example
Determine the vector electric field at (1,-3,7)m
due to point charges Q’1= 5 nC at (2,0,4)m and
Q’2 =-2 nC at (-3,0,5)m.
Solution
Charges
The electric charge is a fundamental property of
matter. It is measured in Coulombs (C). It was
agreed that the electric current unit Ampere (A)
would be chosen as a basic unit in SI. Thus,
Coulomb is a secondary unit derived as:
i is the electric current in Amperes (A)
Q is the electric charge in Coulombs (C)
t is time
Charges
Atomic nuclei contain particles carrying charge: the
electrons and the protons.
These particles react in opposite way to the influence of
external EM fields. Therefore, they have opposite
charges.
It was agreed that the protons would have the positive
charge, and the electrons would have the negative
charge.
The charge of an electron is equal in magnitude to the
charge of a proton and is
In this course, we shall be concerned with macroscopic
charges, i.e. charge distributions much larger than the
dimensions of the largest atomic nucleus (∝10-15 m).
Charge
Charge is associated with matter. Therefore, it has finite
volume. However, volume charges Q can be always
considered made of even smaller volumes. This is
particularly useful when the volume of charge has
inhomogeneous charge distribution.
Point charge is a charge whose volume can be
considered infinitesimally small (a point) in comparison
with the distance from its center at which its field is
analyzed. It has homogeneous charge distribution
(charge density is constant).
Charge Distribution
Charge Distribution
Charge Distribution
Charge Distribution
Charge Distribution -Summary
Charge Distribution -Summary
Electric Fields Due to Charge Distributions
Electric Fields Due to Charge Distributions
Example
The integrals in the electric field expression may be evaluated
analytically using the following variable transformation:
Example cont’d:- special cases
For the special case of a line charge centered at
the coordinate origin (zA =-a, zB = a) with the
field point P lying in the x-y plane [P = (x,y,0) ],
the electric field expression reduces to
Example cont’d:- special cases
Example cont’d:- special cases
To determine the electric field of an infinite length line
charge, we take the limit of the previous result as a
approaches ∞.
Note:-
The electric field of the infinite-length uniform line charge is
cylindrically symmetric (line source). That is, the electric field is
independent of Φ due to the symmetry of the source.
The electric field of the infinite-length uniform line charge is also
independent of z due to the infinite length of the uniform source.
Example
Evaluate E at a point on the z-axis P=(0,0,h) due to a uniformly
charged disk of radius a lying in the x-y plane and centered
at the coordinate origin.
Example; Cont’d
Example; Cont’d
Electric Scalar Potential
If the force of a field is known, one can determine the work
done in moving a positive test charge from point a to point
b.
The work done per unit charge is called potential
difference (voltage)
From Coulomb’s law, the vector force on a positive point
charge in an electric field is:
F=QE
The amount of work performed in moving this point
charge in the electric field is product of the force and the
distance moved.
The work is positive if the energy of the test charge is
increased. In effect, this means that the work is positive
if the displacement of the test charge is opposite to the
field is force vector.
X
For a closed path (point A = point B), the line integral of the
electric field yields the potential difference between a point
and itself yielding a value of zero
Example
Determine the absolute potential in the electric field of a
point charge Q located at the coordinate origin.
The electric field of a point
charge at the origin is
The potential difference between two points A and B in
the electric field of the point charge is
Potentials of Charge Distributions
Absolute potential for a point charge at the origin)
Absolute potential for a point charge at an arbitrary location)
Absolute potential of a set of point charges
Example
Determine the potential in the x-y plane due to a
uniform line charge of length 2a lying along the
z-axis and centered at the coordinate origin
Example
Determine the potential at the center of a square
loop of side length l which is uniformly charged.
Electric Field as the Gradient of the Potential
Since
Then
Or
Note that the potential derivative is
a maximum when θ=π(when the
direction of the electric field is opposite to the
direction of the path). Thus,
Thus, magnitude of the electric field is equal to the
maximum space rate of change in the potential.
The direction of the electric field
is the direction of the maximum
decrease in the potential (the
electric field always points from
a region of higher potential to a
region of lower potential).
The electric field can be written in terms of the
potential as
The gradient operator
Example (E as the gradient of V)
Electric Flux Density
The electric flux density D in free space is:
The total electric flux
where an is the unit normal to the surface S and
Dn is the component of D normal to S. The
direction chosen for the unit normal (one of two
possible) defines the direction of the total flux.
For a closed surface, the total electric flux is
Gauss’s Law
Gauss’s Law - The total outward electric flux ψ
through any closed surface is equal to the total
charge enclosed by the surface.
where ds = ands and an is the outward pointing unit normal
to S
Example
Given a point charge at the origin, show that Gauss’s law
is valid on a spherical surface (S) of radius ro.
Note
Example
Use Gauss’s law to determine the vector electric field inside
and outside a uniformly charged spherical volume of
radius a.
Divergence Operator / Gauss’s Law (Differential
Form)
The differential form of Gauss’s law is defined in
terms of the divergence operator.
The Divergence Operator
The divergence operator in rectangular
coordinates is
Note
Divergence Theorem
Example
Electric Dipole
An electric dipole is formed by two point charges of
equal magnitude separated by a short distance d.
Energy Density in the Electric Field
Example (3 point charges)
If we reverse the order in which the charges are
assembled
For line, surface or volume charge distributions
Total Energy in Terms of the Electric Field
Using the following vector identity
OR
Example
