Page 136 - Engineering Electromagnetics, 8th Edition

118 ENGINEERING ELECTROMAGNETICS

when the fields are nonuniform,

a
V ab − b E · dL
R = = (14)
I S σE · dS
The line integral is taken between two equipotential surfaces in the conductor, and
the surface integral is evaluated over the more positive of these two equipotentials.
We cannot solve these nonuniform problems at this time, but we should be able to
solve several of them after reading Chapter 6.

EXAMPLE 5.1
As an example of the determination of the resistance of a cylinder, we find the resis-
tance of a 1-mile length of #16 copper wire, which has a diameter of 0.0508 in.
Solution. The diameter of the wire is 0.0508×0.0254 = 1.291×10 −3 m, the area of
2
−3
2
the cross section is π(1.291×10 /2) = 1.308×10 −6 m , and the length is 1609 m.
7
Using a conductivity of 5.80 × 10 S/m, the resistance of the wire is, therefore,
1609
R = = 21.2
7
(5.80 × 10 )(1.308 × 10 )
−6
This wire can safely carry about 10 A dc, corresponding to a current density of
6
−6
2
2
10/(1.308×10 ) = 7.65×10 A/m ,or 7.65 A/mm .With this current, the potential
difference between the two ends of the wire is 212 V, the electric field intensity is
0.312 V/m, the drift velocity is 0.000 422 m/s, or a little more than one furlong a week,
3
and the free-electron charge density is −1.81 × 10 10 C/m , or about one electron
within a cube two angstroms on a side.
D5.3. Find the magnitude of the current density in a sample of silver for
2
7
which σ = 6.17 × 10 S/m and µ e = 0.0056 m /V · sif(a) the drift velocity
is 1.5 µm/s ; (b) the electric field intensity is 1 mV/m; (c) the sample is a cube
2.5 mm on a side having a voltage of 0.4 mV between opposite faces; (d) the
sample is a cube 2.5 mm on a side carrying a total current of 0.5 A.
2
2
2
Ans. 16.5 kA/m ; 61.7 kA/m ; 9.9 MA/m ; 80.0 kA/m 2
D5.4. A copper conductor has a diameter of 0.6 in. and it is 1200 ft long.
Assume that it carries a total dc current of 50 A. (a) Find the total resistance of
the conductor. (b) What current density exists in it? (c) What is the dc voltage
between the conductor ends? (d)How much power is dissipated in the wire?

2
5
Ans. 0.035 ;2.74 × 10 A/m ; 1.73 V; 86.4 W