Bent 3133 Lect1

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Copyright © 2007 Oxford University Press

DEFINITION

A quantity can be either a scalar or a vector.

A scalar is a quantity that has only magnitude.Examples are time, mass, distance, temperature,entropy, electric potential etc…

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DEFINATION

A vector is a quantity that has both magnitudeand direction.

Examples are velocity, force, displacement,electric field intensity, magnetic field intensity

etc……

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DEFINATION

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DEFINATION

Electromagnetic (EM) heory is essentially astudy of some particular fields.

A field is a function that specifies a particularquantity every!here in a region.

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DEFINATION

"f the quantity is scalar, the field is scalar field.Examples of scalar field # temperature distributionin a building, sound intensity in a theater, electric

potential in a region, etc…."f the field is vector, the field is vector field.Examples of vector field # gravitational force on a

body in space, the velocity of raindrops in theatmosphere, Electric $ield and Magnetic $ield,intensities,etc……

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UNIT VECTOR


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UNIT VECTOR

hus !e may !rite vector A as

A % A aA

"n cartesian coordinates vector A may be

represented asA % Ax ax &Ay ay & A' a'

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UNIT VECTOR

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UNIT VECTOR

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Figure 1.1 (a) nit vectors a x, a y, and a z , (b) components of A along a x, a y, and a z .

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Vector addition

!o vectors A and can be added together togive vector C,

C % A &

% ( Ax ax &Ay ay & A' a' ) &

( x ax & y ay & ' a' ) % (Ax ! x " ax & (Ay & y " ay & (A' & ' ) a'


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Figure 1.# *ector addition C = A + # $a" parallelogram rule, $b" head+to+tail rule.

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Vector subtraction

*ector subtraction is similarly carried out as

D % A % A & (+ ) % (Ax % x " ax & (Ay + y " ay & (A' + ' ) a'

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Figure 1.& *ector subtraction D = A − # $a" parallelogram rule, $b" head+to+tail rule.

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T'ree basics la(s of algebra

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T'ree basics la(s of algebra)


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*osition and distance vectors

he +osition vector r- (or radius vector) of point -(x,y,') is defined as the the directed distance fromthe origin to -,

r - % - % xax &yay & 'a'

he position vector of point - is useful in definingits position in space.

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$or example if - is at (/,0,1) in cartesian

coordinates then its position vector

r- % O* % /ax &0ay & 1a'

his is illustrated in $ig. 2.0

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Figure 1., "llustration of position vector r P = /a x + 0a y

+ 1a z .

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he distance vector is the displacement fromone point to another.

"f - and 3 are given by (x-,

y-

, '-

) and (x3,

y3

,'3 ), the distance vector (or separation vector)is,

r-3 % r3 + r- % (x3 x-)ax & (y3 y-)ay & ('3 '-)a'

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Figure 1.- 4istance vector r PQ.

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Ea/+le 1

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Ea/+le 1

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Ea/+le 1


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Ea/+le #

-oints - and 3 are located at (5,6,0) and (+/,2,1).7alculatea)he position vector - b)he distance vector from - to 3c)he distance bet!een - and 3

d)A vector parallel to -3 !ith magnitude of 25

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Eercise 1

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Eercise #

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Eercise &

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Figure 1.0 $or Example 2./.

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VECTOR U2TI*2ICATION

8hen t!o vectors A and are multiplied, theresult is either a scalar or a vector depending onho! they are multiplied. !o types of vectormultiplication

9:calar (or dot ) product # A99*ector (or cross ) product # A x

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VECTOR U2TI*2ICATION

Multiplication of three vectors A,, and C canresult in either

9:calar triple product # A 9 ( x C)9*ector triple product # A x ( x C)

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Dot *roduct

he dot +roduct of t!o vectors A and ,!ritten as A9, is defined geometrically as the product of the magnitude of A and and thecosine of the angle bet!een them.

A9 3 A cos ;A)

8here ;A) is the smaller angle bet!een A and.

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Dot *roduct

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Dot *roduct

he dot product obeys the follo!ing#

2.7ommutative la!< A9 % 9A6.4istributive la!<

A9 ( ! C) % A9 ! A9C/. A9A % =A=6 % A6

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Dot *roduct

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Cross *roduct

he cross +roduct of t!o vectors A and ,!ritten as A x , is a vector quantity !hosemagnitude is the area of the parallelogram

formed by A and , and is in the direction ofadvance of a right handed scre! as A is turnedinto .

A x % A sin ;A) an

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Cross *roduct

!here an is a unit vector normal to the planecontaining A and . he direction of an ista>en as the direction of the right thumb !henthe fingers of the right hand rotate from A to.

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Copyright © 2007 Oxford University PressElements of Electromagnetics Fourth Edition adi!u #(

Figure 1.4 he cross product of A and is a vector !ith magnitude equal to the area of the parallelogram and direction, as indicated.

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Figure 1.5 4irection of A × and an using $a" the right+hand rule and $b" the right+handed+scre! rule.

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Cross *roduct

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Cross *roduct

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Cross *roduct

/. "t is distributive< A x ( & C) % A & A x C

0. A x A % 5

?ote# ax x ay % a'

ay x a' % axa' x ax % ay

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Cross +roduct

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Figure 1.6 7ross product using cyclic permutation $a" Moving cloc>!ise leads to positive results. $b" Moving countercloc>!ise leads to negative results.

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7calar Tri+le *roduct

@iven vector A, , and C, !e define the scalartriple product as,

A 9 ( x C) % 9 (C x A) % C 9 (A x )

he result is a scalar.

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7calar Tri+le *roduct

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Vector Tri+le *roduct

$or a vector A, , and C, !e define the vectortriple product as

A x ( x C) % $A 9 C) C(A 9 )

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Co/+onents of a vector

@iven a vector A, !e define the scalarcomponent A) of A along a vector as,

A) % A cos ;A) %=A==a)=cos ;A)

r A) % A 9 a)

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Co/+onents of a vector

?ote the vector component A) of A along issimply the scalar component multiplied by aunit vector along ,

A) % A)a) % (A 9 a))a)


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Copyright © 2007 Oxford University PressElements of Electromagnetics Fourth Edition adi!u $"

Figure 1.18 7omponents of A along # $a" scalar component A B, $b" vector component A B .

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Ea/+le &

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Ea/+le &

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Ea/+le &

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Ea/+le ,


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Ea/+le ,

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Ea/+le -

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Ea/+le 0

:ho! that points -2 (1,6,+0), -6 (2,2,6), and -/ (+/,5,B) all lie on a straight line. 4etermine theshortest distance bet!een the line and point -0

(/,+2,5).

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Eercise ,

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E i -

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Eercise -

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E i 0

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Eercise 0

7onsider a rigid body rotating !ith a constantangular velocity D radians per second about a fixedaxis through 5 as in figure belo!. et r be the

distance vector from 5 to -, the position of a particlein the body. he magnitude of the velocity u of the body at - is=u=% dD % =r= sin ; = 9= or u % 9 x r. "f the rigid body is rotating at / radFs about an

axis parallel to ax 6ay & 6a' and passing through point (6,+/,2), determine the velocity of the body at(2,/,0)

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Figure 1.1& $or -roblem 2.B.

Bent 3133 Lect1

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