Page 80 - Engineering Electromagnetics, 8th Edition

62 ENGINEERING ELECTROMAGNETICS

Figure 3.6 A differential-sized gaussian surface about
the point P is used to investigate the space rate of
change of D in the neighborhood of P.

where D x0 is the value of D x at P, and where a partial derivative must be used to
express the rate of change of D x with x,as D x in general also varies with y and z.
This expression could have been obtained more formally by using the constant term
and the term involving the first derivative in the Taylor’s-series expansion for D x in
the neighborhood of P.
We now have

x ∂D x
˙ = D x0 + y z
front 2 ∂x
Consider now the integral over the back surface,

˙ = D back · S back
back
˙ = D back · (− y z a x )
˙ =−D x,back y z
and
x ∂D x
D x,back ˙= D x0 −
2 ∂x
giving

x ∂D x
˙ = −D x0 + y z
back 2 ∂x
If we combine these two integrals, we have

∂D x
+ ˙ = x y z
front back ∂x