Page 67 - Engineering Electromagnetics, 8th Edition

CHAPTER 3 Electric Flux Density, Gauss’s Law, and Divergence 49

for we are restricting our attention to fields in free space until Chapter 6. At that time
we will see that the materials he used will be classified as ideal dielectrics.
His experiment, then, consisted essentially of the following steps:
1. With the equipment dismantled, the inner sphere was given a known positive
charge.
2. The hemispheres were then clamped together around the charged sphere with
about 2 cm of dielectric material between them.
3. The outer sphere was discharged by connecting it momentarily to ground.
4. The outer space was separated carefully, using tools made of insulating material
in order not to disturb the induced charge on it, and the negative induced charge
on each hemisphere was measured.

Faraday found that the total charge on the outer sphere was equal in magnitude to
the original charge placed on the inner sphere and that this was true regardless of the
dielectric material separating the two spheres. He concluded that there was some sort
of “displacement” from the inner sphere to the outer which was independent of the
medium, and we now refer to this flux as displacement, displacement flux, or simply
electric flux.
Faraday’s experiments also showed, of course, that a larger positive charge on the
inner sphere induced a correspondingly larger negative charge on the outer sphere,
leading to a direct proportionality between the electric flux and the charge on the inner
sphere. The constant of proportionality is dependent on the system of units involved,
and we are fortunate in our use of SI units, because the constant is unity. If electric
flux is denoted by (psi) and the total charge on the inner sphere by Q, then for
Faraday’s experiment

= Q

and the electric flux is measured in coulombs.
We can obtain more quantitative information by considering an inner sphere of
radius a and an outer sphere of radius b, with charges of Q and −Q, respectively
(Figure 3.1). The paths of electric flux extending from the inner sphere to the outer
sphere are indicated by the symmetrically distributed streamlines drawn radially from
one sphere to the other.
At the surface of the inner sphere, coulombs of electric flux are produced by the
2
2
charge Q(= )Cs distributed uniformly over a surface having an area of 4πa m .
2
2
2
The density of the flux at this surface is /4πa or Q/4πa C/m , and this is an
important new quantity.
Electric flux density, measured in coulombs per square meter (sometimes de-
scribed as “lines per square meter,” for each line is due to one coulomb), is given
the letter D, which was originally chosen because of the alternate names of displace-
ment flux density or displacement density. Electric flux density is more descriptive,
however, and we will use the term consistently.
The electric flux density D is a vector field and is a member of the “flux density”
class of vector fields, as opposed to the “force fields” class, which includes the electric