082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

13.3 Definite and indefinite integrals 449
The electric field is a measure of the rate of change of the voltage with position.
Inotherwords,ifthevoltageischangingrapidlywithpositionthenthiscorresponds
to a large magnitude of the electric field. This is illustrated in Figure 13.16. The
magnitude of the electric field at point A is larger than at point B. As a positive
electricfield,E,correspondstoadecreaseinvoltage,V,withpositiontherelationship
betweenE andV, ingeneral, is
E = − dV
(13.10)
dr
This expression can be used to calculate the voltage difference arising as a result of
an electric field. In practice, this is a simplified equation and is only valid provided
r is in the same direction as the electric field. If this is not the case, then a modified
vector formof Equation (13.10) isneeded.
V
A
B
r
Figure13.16
Thegradient ofthe curve, dV , isproportional to the
dr
magnitude ofthe electric field.
In the case of the coaxial cable, E is in the same direction as r and so
Equation (13.10) can be used to calculate the voltage difference between the two
conductors. From Equation (13.10)
dV
dr = −E
Therefore the voltage atanarbitrarypoint,r, isgiven by
∫
V = − E dr
Consequently, the voltage difference between pointsr =bandr =aisgiven by
V b
−V a
=−
∫ b
a
E dr
= − Q ∫ b
1
dr using Equation (13.9)
2πε r
ε 0
r
= − Q
2πε r
ε 0
[lnr] b a
= − Q
2πε r
ε 0
ln( b
a
a
)
Thisgivesthevoltageoftheouterconductorrelativetotheinnerone.Thusthevoltage
of the inner conductor relative tothe outer one is
V a
−V b
=
Q ) b
ln(
2πε r
ε 0
a
➔