Page 72 - Engineering Electromagnetics, 8th Edition

54 ENGINEERING ELECTROMAGNETICS

or a surface charge,

Q = ρ S dS (not necessarily a closed surface)
S
or a volume charge distribution,

Q = ρ ν dv
vol
The last form is usually used, and we should agree now that it represents any or
all of the other forms. With this understanding, Gauss’s law may be written in terms
of the charge distribution as

ρ ν dv (6)
D S · dS =
S vol
a mathematical statement meaning simply that the total electric flux through any
closed surface is equal to the charge enclosed.

EXAMPLE 3.1
To illustrate the application of Gauss’s law, let us check the results of Faraday’s
experiment by placing a point charge Q at the origin of a spherical coordinate system
(Figure 3.3) and by choosing our closed surface as a sphere of radius a.
Solution. We have, as before,
Q
D = a r
4πr 2
At the surface of the sphere,
Q
D S = a r
4πa 2
The differential element of area on a spherical surface is, in spherical coordinates
from Chapter 1,

2
2
dS = r sin θ dθ dφ = a sin θ dθ dφ
or
2
dS = a sin θ dθ dφ a r
The integrand is
Q Q
2
D S · dS = a sin θ dθ dφa r · a r = sin θ dθ dφ
4πa 2 4π
leading to the closed surface integral

φ=2π θ=π Q
sin θ dθ dφ
φ=0 θ=φ 4π