Page 129 - Engineering Electromagnetics, 8th Edition

CHAPTER 5 Conductors and Dielectrics 111

Figure 5.1 An increment of charge, Q = ρ ν S L, which moves a distance x in
a time t,produces a component of current density in the limit of J x = ρ ν ν x .
2
where ν x represents the x component of the velocity v. In terms of current density,
we find
J x = ρ ν ν x
and in general
J = ρ ν v (3)
This last result shows clearly that charge in motion constitutes a current. We
call this type of current a convection current, and J or ρ ν v is the convection current
density. Note that the convection current density is related linearly to charge density
as well as to velocity. The mass rate of flow of cars (cars per square foot per second)
in the Holland Tunnel could be increased either by raising the density of cars per
cubic foot, or by going to higher speeds, if the drivers were capable of doing so.

2
2
2
D5.1. Given the vector current density J = 10ρ za ρ − 4ρ cos φ a φ mA/m :
(a) find the current density at P(ρ = 3, φ = 30 , z = 2); (b) determine the
◦
total current flowing outward through the circular band ρ = 3, 0 <φ < 2π,
2 < z < 2.8.
2
Ans. 180a ρ − 9a φ mA/m ; 3.26 A
5.2 CONTINUITY OF CURRENT

The introduction of the concept of current is logically followed by a discussion of the
conservation of charge and the continuity equation. The principle of conservation of
charge states simply that charges can be neither created nor destroyed, although equal

2 The lowercase ν is used both for volume and velocity. Note, however, that velocity always appears as
avector v,a component ν x ,ora magnitude |v|, whereas volume appears only in differential form as dν
or ν.