Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra


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1 Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to mgnitude (nd possibl phse). Emples: velocit, ccelertion, force Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle Unit Vectors We cn write relvlued vector s where: mgnitude of the vector (dimensionless) unit vector in the direction of Two vectors re sid to be equl if (nd onl if) the hve the sme mgnitude nd direction. Vector ddition Two vectors, not in the sme or opposite directions, determine plne. The sum of two vectors is nother vector in the sme plne. Vector ddition cn be represented grphicll using either the prllelogrm rule or the hedtotil rule. Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle
2 Vector Subtrction ules of Vector ddition Vector subtrction is rell just vector ddition, i.e., D ( ) where ( ) Commuttive lw: ssocitive lw: ( C) ( ) C Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle Product of Sclr nd Vector k ( sgn( k) )( k) For k > 0, is in the sme direction s For k < 0, is in the opposite direction s Position Vector The position vector of point in spce is the directed distnce from the origin to tht point. P The position vector is forml mthemticl w to stte the coordintes of point in spce. Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle
3 Copright 00 b Jmes T. berle Distnce Vector The distnce vector is the directed distnce from one point in spce to nother. P P The distnce between the points is the mgnitude of the distnce vector! Vector Multipliction: Sclr (Dot) Product The sclr (dot) product of two vectors is sclr tht is denoted b It is defined s cosθ θ Note tht Copright 00 b Jmes T. berle θ is the smller of the two ngles between nd, i.e., 0 θ π. Vector Multipliction: Sclr (Dot) Product (Cont d) The sclr (dot) product obes the following rules: Commuttive lw: Distributive lw: ( C) C No ssocitive lw since is meningless ( C) Vector Multipliction: Vector (Cross) Product The vector (cross) product of two vectors is vector tht is denoted b It is defined s sinθ n unit vector in the direction determined b the right  hnd rule (nd thus perpendiculr to the plnecontining nd ). θ θ is the smller of the two ngles between nd, i.e., 0 θ π. Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle
4 Vector Multipliction: Vector (Cross) Product (Cont d) The cross product obes the distributive lw: ( C) C The cross product is not commuttive: The cross product is not ssocitive: ( C) ( ) C Products of Three Vectors: Sclr Triple Product nd Vector Triple Product The sclr triple product produces sclr ( C) ( C ) C ( ) The vector triple product produces vector ( C) ( C) C( ) CC rule sclrs There is no such thing s division b vector! Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle Wh Do We Need Coordinte Sstems? b: OTHOGONL COODINTE SYSTEMS: Crtesin Coordinte Sstem The lws of electromgnetics (like ll the lws of phsics) re independent of prticulr coordinte sstem. However, ppliction of these lws to the solution of prticulr problem imposes the need to use suitble coordinte sstem. It is the shpe of the boundr of the problem tht determines the most suitble coordinte sstem to use in its solution. Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle
5 Orthogonl ighthnded Coordinte Sstems coordinte sstem defines set of three reference directions t ech nd ever point in spce. The origin of the coordinte sstem is the reference point reltive to which we locte ever other point in spce. position vector defines the position of point in spce reltive to the origin. These three reference directions re referred to s coordinte directions, nd re usull tken to be mutull perpendiculr (orthogonl). Orthogonl ighthnded Coordinte Sstems Unit vectors long the coordinte directions re referred to s bse vectors. In n of the orthogonl coordinte sstems, n rbitrr vector cn be epressed in terms of superposition of the three bse vectors. Consider bse vectors such tht â â3 â Such coordinte sstem is clled righthnded Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle Orthogonl ighthnded Coordinte Sstems Note tht the bse vectors cn, in generl, point in different directions t different points in spce. Obviousl, if the re to serve s references, then their directions must be known priori for ech nd ever point in spce. Coordinte Sstems Used in EEE 340 In EEE 340, we shll solve problems using three orthogonl righthnded coordinte sstems: Crtesin clindricl sphericl (,, ) ( r,φ, ) (,θ,φ ) Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle
6 Copright 00 b Jmes T. berle Crtesin Coordintes The point P(,, ) is locted s the intersection of three mutull perpendiculr plnes:,,. The bse vectors re The bse vectors stisf the following reltions:,, â â â Copright 00 b Jmes T. berle Crtesin Coordintes (Cont d) lso, In contrst to clindricl nd sphericl coordinte sstems, the bse vectors in Crtesin coordintes re independent of position. 0, 0, 0, q p pq q p,,, ; δ Copright 00 b Jmes T. berle Crtesin Coordintes (Cont d) The position vector to the point P(,, ) is given b P(,, ) O Copright 00 b Jmes T. berle Crtesin Coordintes (Cont d) The distnce vector to the point Q(,, ) from the point P(,, ) is given b ( ) ( ) ( ) P Q ( ) ( ) ( )
7 Crtesin Coordintes (Cont d) Consider n rbitrr vector in Crtesin coordintes: Sclr (dot) product: Crtesin Coordintes (Cont d) Consider nother rbitrr vector: Vector (cross) product: ( ) ( ) ( ) Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle Crtesin Coordintes (Cont d) The differentil length vector is the distnce vector from the point P(,,) to the djcent point Q(d,d,d). C dl P d d d dl Note tht the differentil lengths d, d nd d re not independent but depend on the specific pth long which P nd Q lie. Crtesin Coordintes (Cont d) differentil surfce vector t point on coordinte equl to constnt surfce is defined s the cross product of the differentil length vectors in the other two coordinte directions with the order of the vectors chosen such tht the differentil surfce vector points in the direction of incresing coordinte. constnt, d S d d d d constnt, constnt, d S d d d d d S d d d d Copright 00 b Jmes T. berle Copright 00 b Jmes T. berle
8 Crtesin Coordintes (Cont d) The differentil volume element t point within region is defined s the sclr triple product of the differentil length chnges in ech of the three coordinte directions with the order of the vectors chosen such tht the differentil volume element is positive. dv d ( d d) d d d Copright 00 b Jmes T. berle
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