EP¼DensityP
«
¼q
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 p¼permittivity 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 ¼ðx 2x 1EP
dx
x¼ðx 2x 1q
2 p« 0dx
x¼
q
2 p« 0ln
x 2
x 1
ðÞV (13:61)Then, the capacitance between pointsP 1 andP 2 is evaluated as
C 1 2 ¼q
V 1 2
¼2 p« 0ln
x 2
x 1ðÞF=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« 0ln
h
r
ðÞV (13:63)Ccondground¼q
Vcond
¼2 p« 0ln
h
rðÞF=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.
lr (^) A r^ B
D
A B
q (^) A
- −
q (^) B
A + rA B rB
Dq+ q
− q+ q−−FIGURE 13.15 Electric field produced from a two-wire single-phase system.