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

7.5 The scalar product 241
O
E
Figure7.24
Acharge atOgives riseto an electric fieldE.
Ifasecondchargeisplacedinthisfielditexperiencesaforce.Animportantquantity
is the electric field strength, E. This is a vector field which describes the force
experienced by a unit charge.
A related quantity is the electric displacement, D, also called the electric flux
density, defined as D = εE, where ε is called the permittivity of the medium in
which the field islocated. Note thatDisascalar multiple ofE.
Engineeringapplication7.6
ElectrostaticpotentialV
An important electrostatic field is the electrostatic potentialV. This is an example
of a scalar field. The difference between the potential measured at any two points in
thefieldisequaltotheworkwhichneedstobedonetomoveaunitchargefromone
point to the other. Later, in Chapter 26, we will see that the scalar fieldV is closely
relatedtothe vectorfield E.
7.5 THESCALARPRODUCT
u
b
a
Figure7.25
Two vectorsaandb
separated by
angle θ.
Givenanytwovectorsaandb,therearetwowaysinwhichwecandefinetheirproduct.
Theseareknownasthescalarproductandthevectorproduct.Asthenamessuggest,the
resultoffindingascalarproductisascalarwhereastheresultoffindingavectorproduct
isavector. The scalar product of aandbiswritten as
a · b
Thisnotationgivesrisetothealternativenamedotproduct.Itisdefinedbytheformula
a·b=|a‖b|cosθ
where θ isthe angle between the two vectors asshown inFigure 7.25.