Page 83 - Engineering Electromagnetics, 8th Edition

CHAPTER 3 Electric Flux Density, Gauss’s Law, and Divergence 65

different regions of the closed surface must be equal. Hence the divergence of this
velocity is zero.
If, however, we consider the velocity of the air in a tire that has just been punc-
tured by a nail, we realize that the air is expanding as the pressure drops, and that
consequently there is a net outflow from any closed surface lying within the tire. The
divergence of this velocity is therefore greater than zero.
A positive divergence for any vector quantity indicates a source of that vector
quantity at that point. Similarly, a negative divergence indicates a sink. Because the
3
divergenceofthewatervelocityaboveiszero,nosourceorsinkexists. Theexpanding
air, however, produces a positive divergence of the velocity, and each interior point
may be considered a source.
Writing (9) with our new term, we have

∂D x ∂D y ∂D z
div D = + + (rectangular) (12)
∂x ∂y ∂z
This expression is again of a form that does not involve the charge density. It is the
result of applying the definition of divergence (11) to a differential volume element
in rectangular coordinates.
2
If a differential volume unit ρ dρ dφ dz in cylindrical coordinates, or r sin θ dr
dθ dφ in spherical coordinates, had been chosen, expressions for divergence involving
the components of the vector in the particular coordinate system and involving partial
derivatives with respect to the variables of that system would have been obtained.
These expressions are obtained in Appendix A and are given here for convenience:

1 ∂ 1 ∂D φ ∂D z
div D = (ρD ρ ) + + (cylindrical) (13)
ρ ∂ρ ρ ∂φ ∂z

1 ∂ 2 1 ∂ 1 ∂D φ
div D = (r D r ) + (sin θ D θ ) + (spherical) (14)
2
r ∂r r sin θ ∂θ r sin θ ∂φ
These relationships are also shown inside the back cover for easy reference.
It should be noted that the divergence is an operation which is performed on a
vector, but that the result is a scalar. We should recall that, in a somewhat similar way,
the dot or scalar product was a multiplication of two vectors which yielded a scalar.
For some reason, it is a common mistake on meeting divergence for the first
time to impart a vector quality to the operation by scattering unit vectors around in

3 Having chosen a differential element of volume within the water, the gradual decrease in water level
with time will eventually cause the volume element to lie above the surface of the water. At the instant
the surface of the water intersects the volume element, the divergence is positive and the small volume
is a source. This complication is avoided above by specifying an integral point.