Electric Power Generation, Transmission, and Distribution

EP¼

DensityP

«

¼

q

2 p« 0 x

ðÞV=m (13:60)

where DensityP¼electric flux density at pointP
EP¼electric field intensity at pointP
A¼surface of a concentric cylinder with 1-m length and radiusx(m^2 )

«¼« 0 ¼

10 ^9

36 p

¼permittivity of free space assumed for the conductor (F=m)
The potential difference or voltage difference between two outside pointsP 1 andP 2 with correspond-
ing distancesx 1 andx 2 from the conductor center is defined by integrating the electric field intensity
fromx 1 tox 2

V 1  2 ¼

ðx 2

x 1

EP

dx

x

¼

ðx 2

x 1

q

2 p« 0

dx

x

¼

q

2 p« 0

ln

x 2

x 1



ðÞV (13:61)

Then, the capacitance between pointsP 1 andP 2 is evaluated as

C 1  2 ¼

q

V 1  2

¼

2 p« 0

ln

x 2

x 1

ðÞF=m (13:62)

If pointP 1 is located at the conductor surface (x 1 ¼r), and pointP 2 is located at ground surface below
the conductor (x 2 ¼h), then the voltage of the conductor and the capacitance between the conductor
and ground are

Vcond¼

q

2 p« 0

ln

h

r



ðÞV (13:63)

Ccondground¼

q

Vcond

¼

2 p« 0

ln

h

r

ðÞF=m (13:64)

13.5.2 Capacitance of a Single-Phase Line with Two Wires

Consider a two-wire single-phase line with conductors A and B with the same radiusr, separated by
a distanceD>rAandrB. The conductors are energized by a voltage source such that conductor A has
a chargeqþand conductor B a chargeqas shown in Fig. 13.15.
The charge on each conductor generates independent electric fields. Chargeqþon conductor A
generates a voltageVAB–Abetween both conductors. Similarly, chargeqon conductor B generates
a voltageVAB–Bbetween conductors.

l

r (^) A r^ B
D
A B
q (^) A

• −
  q (^) B
  A + rA B rB

D

q+ q

− q+ q−

−

FIGURE 13.15 Electric field produced from a two-wire single-phase system.