Page 74 - Engineering Electromagnetics, 8th Edition

56 ENGINEERING ELECTROMAGNETICS

3.3 APPLICATION OF GAUSS’S LAW: SOME
SYMMETRICAL CHARGE DISTRIBUTIONS
We now consider how we may use Gauss’s law,

Q = D S · dS
S
to determine D S if the charge distribution is known. This is an example of an integral
equation in which the unknown quantity to be determined appears inside the integral.
The solution is easy if we are able to choose a closed surface which satisfies two
conditions:

1. D S is everywhere either normal or tangential to the closed surface, so that
D S · dS becomes either D S dS or zero, respectively.
2. On that portion of the closed surface for which D S · dS is not zero, D S =
constant.
This allows us to replace the dot product with the product of the scalars D S and
dS and then to bring D S outside the integral sign. The remaining integral is then
dS over that portion of the closed surface which D S crosses normally, and this is

S
simply the area of this section of that surface. Only a knowledge of the symmetry of
the problem enables us to choose such a closed surface.
Let us again consider a point charge Q at the origin of a spherical coordinate
system and decide on a suitable closed surface which will meet the two requirements
previously listed. The surface in question is obviously a spherical surface, centered
at the origin and of any radius r. D S is everywhere normal to the surface; D S has the
same value at all points on the surface.
Then we have, in order,

Q = D S · dS = D S dS
S sph
φ=2π θ=π
2
= D S dS = D S r sin θ dθ dφ
sph φ=0 θ=0
2
= 4πr D S
and hence
Q
D S =
4πr 2
Because r may have any value and because D S is directed radially outward,

Q Q
D = a r E = a r
4πr 2 4π 0 r 2