Chapter 1 VECTOR ALGEBRA
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1 Chpter 1 VECTOR LGEBR
2 INTRODUCTION: Electromgnetics (EM) m be regrded s the stud of the interctions between electric chrges t rest nd in motion. Electromgnetics is brnch of phsics or electricl engineering in which electric nd mgnetic phenomen re studied.
3 PPLICTION EM principles find ppliction in vrious disciplines such s; micowves,ntenns electric mchines stellite communictios Bioelectromgnetics plsms,nucler reserch,fiberoptics
4 Sclr nd Vectors sclr is quntit tht hs onl mgnitude. Quntities such s time,mss,distnce,temperture,entrop h,electric potentil.
5 VECTORS vector is quntit tht hs both mgnitude nd direction. Vector quntities include velocit,force,displcement nd electric field intensit.
6 UNIT VECTOR vector hs both mgnitude nd direction. The mgnitude of is sclr written s or unit vector long is defined s vector whose mgnitude is unit nd its direction is long, tht is;
7 UNIT VECTOR n
8 vector in Crte coordintes m be represented s; (,, ) + +
9 VECTOR DDITION & SUBTRCTION Two vectors nd B cn be dded together to give nother vector C C + B Let (,, ) nd B(B, B,B ) C ( + B ) + ( + B ) + ( + B )
10 Vector subtrction is similrl crried out s; D B + (-B) D ( -B ) + ( -B ) + ( -B )
11 Vector lgebr
12 Vector lgebr ddition ssocitive lw +(B+C) (+B)+C commuttive lw +B B+ multipliction b sclr B B distributive lw ( B+C) B + C
13 POSITION ND DISTNCE VECTOR point P in Crte coordintes m be represented b (,,). The position vector r p (rdius vector)of point P is s the directed distnce from the origin O to P; i.e., r p OP + +
14 P (3,4,5) 0 Z5 X3 Y4 POSITION VECTOR OP
15 Distnce Vector P r PQ r p Q 0 r Q r PQ r Q -r p
16 Emples: If nd B2 +. Find: () The components of long (b) The mgnitude of 3-B (c) unit vector long + 2B
17 Solution: () The component of long is -4 (b) 3-B 3(10,-4,6)-(2,1,0) (30,-12,18)-(2,1,0) (28,-13,18) Hence;
18 3 B ( ) ( 18)
19 ( c ) Let C + 2B (10,-4,6) + (4,2,0) (14,-2,6) unit vector long C is c C C 14 2 ( 14, 2,6 ) + ( )
20 Emples:2 Points P nd Q re locted t (0,2,4) nd (-3,1,5).Clculte () The position vector P (b) The distnce vector from P to Q (c) The distnce between P nd Q (d) vector prllel to PQ with mgnitud of 10
21 Solution 2 () r p (b) r PQ r Q r P (-3,1,5)-(0,2,4)(-3,- 1,1) (c) Since r PQ is the distnce vector from P to Q,the distnce between P nd Q is the mgnitude of this vector;tht is, d r PQ
22 (d) Let the required vector be,then Where 10 is the mgnitude of.since is prllel to PQ, it must hve the sme unit vector s r PQ or r QP. Hence,
23
24 VECTOR MULTIPLICTION When two vector nd B re multiplied,the result is either sclr or vector depending on how the re multiplied. Thus there re two tpes of vector multipliction: 1. Sclr (or dot) product:.b 2. Vector(or Cross)product:XB
25 DOT PRODUCT The dot product of two vector nd B,written.B is defined geometricll s the product of the mgnitudes of nd B nd the ine of the ngle between them..b B B
26 The Vector Field Emple The Dot product.b B B B in the direction of You need to normlie before the dot product.
27 B Where is the smller ngle between nd B. The result of.b is clled either the sclr product. Let (,, ) nd B (B,B,B ).B B + B + B
28 Two vectors nd B re sid to be orthogonl (or perpendiculr) with ech other if.b0 Note tht dot product obes the following; (i) Commuttive Lw;.BB. (ii) Distributive lw :.(B+C).B +.C. 2
29 (iii) lso note tht
30 Cross Product The cross product of two vector nd B,written X B, is vector quntit whose mgnitude is the re of the prllelopiped formed b nd B B B
31 The Cross Product B N B B B Emple B B B 2 : 3 B 1 : B
32 The cross product hs the bsic following properties: (i) It is not commuttive: B B (ii) It is not ssocitive; ( B C )( B ) C (iii) It is distributive (B+C)XB +XC
33 (iv) 0 lso note tht;,,
34 Cross Product ug clic permuttion
35 Emple:1 Determine the dot product nd cross product of the following vectors B
36 SOLUTION The dot product is.b (2)(-1) + (3) (-5) + (-4)(6) -41 The Cross Product B is B [ (3)(6) (-4)(-5)] + [(-4)(-1) (2)(6)] + [(2)(-5) (3)(-1)]
37 lterntivel B - B B
38 Sclr Triple Product Given three vector,b nd C.(BC)B.(C)C.(B) If (,, ), B(B,B,B ) nd C(C,C,C ).(BC) B B B Z C C C
39 Vector Triple product For vectors,b, nd C, we define the vector triple product s (BC)B(.C)-C(.B) It should be noted tht (.B)C(B.C) But (.B)CC(.B)
40 Coordinte Sstems nd Trnsformtion. Crte Coordintes (,,) Circulr Clindricl Coordintes ( ρ,, ) Sphericl Coordintes ( r,, )
41 Crte Coordinte Consists of three mutull orthogonl es,,, nd point in spce is denoted s P( 1, 1, 1 ). The loction of the point is defined b the intersection of three plnes.
42 Z P( 1, 1, 1 ) Z
43 The Crte Coordinte Sstem
44 The differentil surfces in crte coordintes Differentil Surfces ds dd ds dd ds dd ds dd d d d ds dd ds dd
45 Crte Coordintes (,,) The Rnges of the coordinte vribles,, re;
46 Vector Components nd Unit Vectors
47 The Clindricl Coordintes Sstem. Form b three surfces One is plne for constnt, 1 The net surfce is clinder centered on the is of rdius ρ The third surfce is plne perpendiculr to the plne nd rotte bout the is b ngle Unit vector ρ,, point in the direction of increg coordinte vlue.
48 Circulr Clindricl Coordinte Sstem ρ dρ ρ d d 1 - Unit Vector vr with The coordinte Since direction chnges 2 Dot Product ρ d d ρ dρ d d
49 Clindricl coordinte sstem tn ρ ϕ ρ ρ ρ ρ
50
51
52 Z ρ S C Q D B d ρd dρ ρ
53 ρ d ρ ρ d
54 Differentil norml res in clindricl coordintes. ρd ρ d d ρ d d ρ ρ d
55 In Clindricl Coordintes,differentil elements cn be found s follows; (1) Differentil displcement is given b di dρ + ρd + ρ d (2) Differentil norml re is given b ds ρ d d ρ dρd ρddρ
56 (3) Differentil volume is given b dv ρ d ρ d d
57 Circulr Clindricl Coordinte Sstem ρ () ρ () Dot Product ρ ρ 0 tn + + ρ ρ + + ρ ρ ρ ( + + ) ρ ρ + ρ ρ () ρ ( ) 1 ( + + ) + () () ( + + ) ρ 0
58 Emple: () Determine the volume enclosed b clinder of rdius ρ nd the length L s well s the surfce re of tht volume. (b) The surfce re?
59 Solution: To determine the volume enclosed we integrte; V v dv ρ ρ 0 πρ 2 2π L L 0 0 ρdrdd dv
60 2π L 2π ρ 2π ρ S 0 0 ρ d d Sides + 0ρ 0 rd dr bottom + 0ρ 0 rd top dr πρL + 2πρ 2
61 Sphericl Coordintes point P cn be represented s ( r,, ) in this coordinte sstem vector in sphericl coordintes is written s or ( r,, ) + + r r Where, nd r re unit vectors long -direction. r,, nd
62 The Sphericl Coordinte Sstem ( ) r ( ) ( ()) r r ( ) r r tn
63 r tn Sphericl coordinte sstem r ϕ r r r
64 The Sphericl Coordinte Sstem ( ) r ( ) ( ()) r r ( ) rdr d ( ) r ( ) r 2 dr d d d r 2 ( ) dr d d
65 The rnges of vribles re; 0 r 0 π 0 2 π
66 r d dr rd r d d
67 Z r d r rd Differentil rc length for constnt
68 Z r d r r d rd Differentil rc length for constnt
69 Differentil norml res in sphericl coordintes; r d r r d rd dr rd () (b) (C) dr
70 (1) The differentil displcement is dl dr + rd + r r d (2) The Sphericl Coordinte Sstem ds r 2 r rdrd d drd d r
71 The differentil volume is dv r 2 drd d
72 Emple: Determine the volume enclosed b sphere of rdius r
73 Solution: v dv v r r 0 π 0 2π 0 r 2 drd d 4 3 π r 2
74 The reltionships between the vribles (,,) of the Crte Coordinte Sstem. P (,,)P ( ρ,, ) ρ ρ ρ
75 The mgnitude of is; 2 2 ρ The reltionship between,, nd re obtined geometricll from, ρ, ρ + ρ
76 + + Let ρ ρ convert to crte coordinte ( + + ) ρ ρ ρ ( ) ( ) + + ) ρ (
77 ρ ρ ρ π 2 + ρ
78 . ( + + ) ρ ρ ( ) ( ) + + ) ρ ρ ( + ρ Z Z
79 r
80 Performing dot-product ρ ρ
81 Clinder to Crte Crte to Clinder ρ ρ ρ tn 1
82 EXMPLES:4 Given point P(-2,6,3) nd Vector + (+),epress P nd in clindricl coordintes. Evlute t P in Crte nd Clindricl sstems.
83 Solution: t Point P: -2, 6, 3 Hence, ρ tn tn Z3
84 Thus, P(-2,6,3)P(6.32, ,3) In the crte stem, t P is 6 + For vector,, + 0, Hence the clindricl sstem ρ
85 ρ + ( + ) + ( + ) But ρ, ρ
86 ρ ρ ( ) [ ] ( ) [ ] ρ ρ ρ ρ ρ ρ ),, ( , 2 6 tn 40 ρ Hence P t
87 Crte Coordintes to Sphericl Coordintes Sstem Z ρ r P(,, ) P( r,, ) P( ρ,, ) r r ρ ρ ρ
88 r
89 In Mtri form,the ),, ( ),, ( r r r r r r 0 ),, ( ),, ( 0 r r tion Trnsform Inverse The
90 Dot Product of unit vectors in sphericl nd crte coordintes sstem. r 0
91 2 2 r tn tn 1
92 r r r
93 Emple: Evlute t P in the sphericl sstem s in Emple 4
94 In the sphericl sstem ) ( ) ( ) ( or r r
95 END
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