1. ENE/EIE 325 Electromagnetic Fields and Waves
Lecture 3
Coulomb’s law, Static Electric Fields and Electric Flux density

2. Outline
Coulomb’s law
Electric field intensity in different charge configurations
Electric flux density

3. Electric charge
A fundamental quantity that is responsible for all electric phenomena.
1e = 1.6x10-19 Coulomb
Law of attraction: positive charge attracts negative charge
Same polarity charges repel one another

4. Coulomb’s Experimental Law+ Q1 Q2 R F
Force of repulsion, F, occurs when charges
have the same sign.
Charges attract when of opposite sign
where

5. Free Space Permittivity
with which the Coulomb force becomes:

6. Coulomb Force with Charges Off-Origin

7. Electric Field Intensity
Consider the force acting on a test charge, Qt , arising from charge Q1:
where a1t is the unit vector directed from Q1 to Qt
The electric field intensity is defined as the force per unit test charge, or
N/C
A more convenient unit for electric field is V/m, as will be shown.

8. Electric Field of a Charge Off-Origin

9. Superposition of Fields From Two Point Charges
For n charges:

10. Ex1 A charge Q is 310-9 C produces the electric field E.
Find E at P, using
First, find the vectors:
Then:

11. Ex1 (continued)
Find E at P, using
Now:
so that:
where

12. Volume Charge Density
Given a charge Q within a volume , the volume charge density is defined as:
….so that the charge contained within a volume is

13. Ex2
Find the charge contained within a 2-cm length of the electron beam shown below,
in which the charge density is

14. Ex2 (continued) Q

15. Electric Field from Volume Charge Distributions
Next, sum all contributions throughout a volume and take the limit as 
approaches zero, to obtain the integral:

16. Line Charge Electric Field
Line charge of constant density L Coul/m lies along the entire z axis.
At point P, the electric field arising from
charge dQ on the z axis is:
where
so that
Therefore
and

17. Line Charge Field (continued)
We have:
By symmetry, only a radial
component is present:

18. Line Charge Field Results

19. Ex3: Off-Axis Line Charge
With the line displaced to (6,8), the field becomes:
where
Finally:

20. Sheet Charge Field or The total field on the x axis is:
Uniform surface charge of density s covers the entire y-z plane.
We begin by writing down the line charge field on the x axis for a strip of
differential width dy´, where we consider the x component of that field:
The total field on the x axis is: or

21. Types of Streamline Sketches of Fields:

22. Methodology of Streamline Construction
The ratio of the y and x field
components gives the slope
of the field plot in the x-y plane

23. Ex4: Line Charge Field Then whose solution is Finally
Begin with the normalized line charge field in cylindrical coordinates:
Convert to rectangular components:
Then
whose solution is
Finally

24. Michael Faraday
Michael Faraday (22 September 1791 – 25 August 1867) was an English scientist who contributed to the fields of electromagnetism and electrochemistry.

25. Faraday Experiment
He started with a pair of metal spheres of different sizes;
the larger one consisted of two hemispheres that could be
assembled around the smaller sphere

26. Faraday Experiment Illustration
Grounded outer
conducting sphere +q +
Insulating or
dielectric material +q +
Charged conducting sphere +q +
Guess what will happen?

27. Observations
Charge transfers from inner to outer sphere without contact.
Charge on outer sphere is of the same magnitude but opposite sign –q.
Charge on outer sphere is the same regardless of the insulating material used.
Charge on outer electrode is the same regardless of electrode’s shape. +q + - –q

28. Faraday Apparatus, Before Grounding+Q
The inner charge, Q, induces
an equal and opposite charge,
-Q, on the inside surface of the
outer sphere, by attracting free
electrons in the outer material toward
the positive charge.
This means that before the outer sphere
is grounded, charge +Q resides on the
outside surface of the outer conductor.

29. Faraday Apparatus, After Grounding
q = 0
ground
attached
Attaching the ground connects the
outer surface to an unlimited supply
of free electrons, which then neutralize
the positive charge layer. The net charge
on the outer sphere is then the charge on
the inner layer, or -Q.

30. Interpretation of the Faraday Experiment
q = 0
Faraday concluded that there occurred
a charge “displacement” from the inner
sphere to the outer sphere.
Displacement involves a flow or flux,  existing
within the dielectric, and whose magnitude
is equivalent to the amount of “displaced” charge.
Specifically:

31. Electric Flux Density q = 0
The density of flux at the inner sphere surface
is equivalent to the density of charge there
(in Coul/m2)

32. Vector Field Description of Flux Density
q = 0
A vector field is established which
points in the direction of the “flow”
or displacement. In this case, the
direction is the outward radial direction
in spherical coordinates. At each surface,
we would have:

33. Radially-Dependent Electric Flux Density
q = 0
D(r)
At a general radius r between
spheres, we would have:
Expressed in units of Coulombs/m2, and
defined over the range (a ≤ r ≤ b) r

34. Point Charge Fields C/m2 (0 < r <∞ ) V/m (0 < r <∞ )
If we now let the inner sphere radius reduce to a point, while maintaining the same charge, and let the
outer sphere radius approach infinity, we have a point charge. The electric flux density is unchanged,
but is defined over all space:
C/m2 (0 < r <∞ )
We compare this to the electric field intensity in free space:
V/m (0 < r <∞ )
..and we see that:

35. Finding E and D from Charge Distributions
We learned in Chapter 2 that:
It now follows that:

36. Displacement Flux Concept
Faraday concluded that there was some sort of "displacement" from the inner sphere to the outer which was independent of the medium. It is called displacement flux (aka electric flux, ).
Displacement flux makes outer sphere charged with –q. 
Inside the outer sphere shell
The inner sphere
Charge in E + - - +q + –q +q + +