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

18.6 Taylor and Maclaurin series 529
y
y(x)
x 0 x 0 +h
x
Figure18.6
Thevaluey(x 0 +h) can be estimated usingvalues ofy
and itsderivatives atx =x 0 .
Ifyand its derivatives are known whenx =x 0
, then the Taylor series can be used to
findyatanearbypoint,wherex =x 0
+h.ThisformofTaylorseriesisusedinnumerical
methods of solving differential equations.
Engineeringapplication18.5
Electricfieldofanelectrostaticdipole
Engineersoftenneedtocalculatetheelectricalfieldduetoseveralstationaryelectric
charges.Theseareknownaselectrostaticproblems.Oneofthesimplestelectrostatic
configurations is that of two charges of opposite polarity separated by a distance,d.
Thisarrangementisknownasanelectrostaticdipole.ItisillustratedinFigure18.7
forcharges +Q and −Q.
The origin of the x axis is located midway between the two charges so that the
charge −Q has coordinate −d/2 and the charge +Qhas coordinated/2.
d
–Q 0 +Q
x
Figure18.7
Electrostatic dipole with two point charges of −Q
and +Qcoulombs respectively.
We wish to calculate the combined electric field of the two charges as a function
of the distance along thexaxis.
The electricfield,E, ofasinglecharge isgiven by
E =
q
4πε 0
r 2
where q is the charge in coulombs, ε 0
is the permittivity of free space (a constant
of approximately 8.85 ×10 −12 F m −1 ), andris the distance from the charge to the
measuring point.
The electricfield atpointxdue tothe left-hand charge is
E LEFT
=
−Q
(
4πε 0
x + d ) 2
2
and forthe right-hand charge the electric field atpointxis
E RIGHT
=
+Q
(
4πε 0
x − d ) 2
2
➔