Page 85 - Engineering Electromagnetics, 8th Edition

CHAPTER 3 Electric Flux Density, Gauss’s Law, and Divergence 67

and use (14), the expression for divergence in spherical coordinates:
1 ∂ 2 1 ∂ 1 ∂D φ
div D = (r D r ) + (D θ sin θ) +
2
r ∂r r sin θ ∂θ r sin θ ∂φ
Because D θ and D φ are zero, we have
1 d Q
div D = r 2 = 0 (if r = 0)
r dr 4πr 2
2
Thus, ρ ν = 0everywhere except at the origin, where it is infinite.
The divergence operation is not limited to electric flux density; it can be applied
to any vector field. We will apply it to several other electromagnetic fields in the
coming chapters.
D3.8. Determine an expression for the volume charge density associated with
2
4xy 2x 2 2x y
each D field: (a) D = a x + a y − a z ;(b) D = z sin φ a ρ +
z z z 2
z cos φ a φ + ρ sin φ a z ;(c) D = sin θ sin φ a r + cos θ sin φ a θ + cos φ a φ .
4y 2 2
Ans. (x + z ); 0; 0.
z 3

3.6 THE VECTOR OPERATOR ∇
AND THE DIVERGENCE THEOREM
If we remind ourselves again that divergence is an operation on a vector yielding a
scalar result, just as the dot product of two vectors gives a scalar result, it seems possi-
ble that we can find something that may be dotted formally with D to yield the scalar

∂D x ∂D y ∂D z
+ +
∂x ∂y ∂z
Obviously, this cannot be accomplished by using a dot product; the process must be
a dot operation.
With this in mind, we define the del operator ∇ as a vector operator,

∂ ∂ ∂
∇= a x + a y + a z (16)
∂x ∂y ∂z
Similar scalar operators appear in several methods of solving differential equations
2
4
2
2
where we often let D replace d/dx, D replace d /dx , and so forth. We agree on
defining ∇ that it shall be treated in every way as an ordinary vector with the one
important exception that partial derivatives result instead of products of scalars.
Consider ∇ · D, signifying

∂ ∂ ∂
∇ · D = a x + a y + a z · (D x a x + D y a y + D z a z )
∂x ∂y ∂z

4 This scalar operator D, which will not appear again, is not to be confused with the electric flux density.