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

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7.5 The scalar product 245
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X
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Figure7.27
Two charged platessituated in avacuum.
InFigure7.27theelectricfieldvectorEintheregionofspacebetweenthecharged
plateshasadirectionperpendiculartotheplatespointingfromAtoB.Themagnitude
of the electric field vector is constant in this region if end effects are ignored. If
a charged particle is required to move against an electric field, then work must be
donetoachievethis.Forexample,totransportapositivelychargedparticlefromthe
surfaceofplateBtothesurfaceofplateAwouldrequireworktobedone.Thiswould
lead toan increase inpotential of the charged particle.
If s represents the displacement andV the potential it is conventional to write δs
to represent a very small change in displacement, and δV to represent a very small
change inpotential.
Ifaunitpositivechargeismovedasmalldisplacement δsinanelectricfield(Figure
7.27) then the change inpotential δV isgiven by
δV =−E·δs (7.1)
This is an example of the use of a scalar product. Notice that although E and δs are
vector quantities the change inpotential, δV, isascalar.
ConsideragainthechargedplatesofFigure7.27.Ifaunitchargeismovedasmall
displacement along the plane X,then δs isperpendicular toE. So,
δV = −E·δs = −|E‖δs|cosθ
With θ = 90 ◦ ,wefind δV = 0.ThesurfaceXisknownasanequipotentialsurface
because movement of a charged particle along this surface does not give rise to a
change in its potential. Movement of a charge in a direction parallel to the electric
field gives risetothe maximum change inpotential, as forthis case θ = 0 ◦ .
EXERCISES7.5
1 Ifa=3i−7jandb=2i+4jfinda·b,b·a,a·a
andb·b.
2 Ifa=4i+2j−k,b=3i−3j+3kand
c=2i−j−k,find
(a) a·a (b) a·b
(c) a·c (d) b·c
3 Evaluate (−13i −5j) · (−3i +4j).
4 Find the angle between the vectorsp = 7i +3j +2k
andq=2i−j+k.
5 Find the angle between the vectors 7i +j and 4j −k.
6 Findtheanglebetweenthevectors4i −2jand3i −3j.
7 Ifa=7i+8jandb=5ifinda·ˆb.
⎛ ⎞ ⎛ ⎞
3 5
8 Ifr 1 = ⎝1⎠and r 2 = ⎝1⎠ findr 1 ·r 1 ,r 1 ·r 2
2 0
andr 2 ·r 2 .
9 Given thatp = 2qsimplify p ·q, (p +5q) ·q and
(q−p)·p.