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

252 Chapter 7 Vectors
In equilibrium the magnitude of the electric field, |E H
|, is constant and so we can
write |E H
| = V H
, whereListhe widthof the semiconductor. Hence,
L
IB
A =qp H
0V
L
so that
p 0
= BIL
V H
qA
So, by measuring the value of the Hall voltage, it is possible to calculate the density
of the holes, p 0
, inthe semiconductor.
EXERCISES7.6
1 Evaluate
i jk
(a)
312
∣ 214 ∣
i jk
(c)
010
∣ 104 ∣
(b)
∣
(d)
∣
∣
jk
∣
ij k
−1 2 −3
−4 0 1
i
3 52
−3 −1 4
2 Ifa=i−2j+3kandb=2i−j−k,find
(a) a×b
(b) b×a
3 Ifa=i−2jandb=5i+5kfinda×b.
4 Ifa=i+j−k,b=i−jandc=2i+kfind
(a) (a×b)×c
(b) a×(b×c)
5 Ifp=6i+7j−2kandq=3i−j+4kfind|p|,|q|
and |p ×q|.Deduce the sineofthe anglebetween p
and q.
6 Forarbitraryvectors p andqsimplify
(a) (p+q)×p
(b) (p+q)×(p−q)
7 Ifc=i+jandd=2i+k,findaunitvector
perpendicular to bothcandd.Further, findthe sineof
the anglebetweencandd.
8 A, B,Care pointswith coordinates (1,2,3), (3,2,1)
and (−1,1,0),respectively. Find aunitvector
perpendicular to the planecontaining A, B andC.
9 Ifa=7i−2j−5kandb=5i+j+3k,findavector
perpendicular toaand b.
10 Ifa=7i−j+k,b=3i−j−2kand
c = 9i +j−3k, showthat
a×(b+c)=(a×b)+(a×c)
11 (a) Thearea,A,ofaparallelogramwith baseb
andperpendicular heighthisgiven byA =bh.
Showthat ifthe two non-parallelsidesofthe
parallelogramare representedby the vectorsa
andb,thenthe areais alsogiven by
A=|a×b|.
(b) Findthe area ofthe parallelogramwith sides
representedby2i +3j +k and3i +j−k.
12 Thevolume,V,ofaparallelepipedwithsidesa,band
cisgivenbyV = |a · (b ×c)|.Findthevolumeofthe
parallelepiped with sides 3i +2j +k,2i +j+k and
i+2j+4k.
13 Suppose a forceFactsthrough the pointPwith
position vector r.Themoment aboutthe origin, M,
ofthe force isameasureofthe turningeffect ofthe
force andis given by M = r ×F.Aforce of4Nacts
in the directioni +j+k,andthrough the point with
coordinates (7,1,3).Findthe momentofthe force
about the origin.
14 Inthe theory ofelectromagnetic waves an important
quantityassociatedwith the flow ofelectromagnetic
energy isthe Poyntingvector S.This isdefined as
S = E ×Hwhere Eis the electric field strength and
Hthe magnetic field strength. Suppose thatin a plane
electromagnetic wave
( ) 2πz
E=E 0 cos
λ − ωt j