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CHAPTER 1:

Vector Fields

ELECTROMAGNETIC FIELDS

THEORY

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Scalar

Scalar:A quantity that has only magnitude.

For example time, mass, distance,

temperature and population are scalars. Scalar is represented by a lettere.g., A, B

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Vector

Vector: A quantity that has both magnitudeand

direction.

Example: Velocity, force, displacement and electric

field intensity. Vector is represent by a letter such asA, B, or

It can also be written as

where A is which is the magnitude and is unitvector

aAA

A

a

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Unit Vector

A unit vector along A is defined as a vector whosemagnitude is unity (i.e., 1) and its direction is alongA.

It can be written as aAor

Thus

a

A

A

aAA

A

A

A

Aa

||

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Vector Subtraction

Vector subtraction is similarly carried out as

D=AB=A+ (-B)

A

B

BA

B

Figure (a)

Figure (b)

Figure (c)

Figure (c) shows that vector D is a vector

that is must be added to Bto give vectorASo if vectorA and Bare placed tail to tail

then vector Dis a vector that runs from the

tip of BtoA.

A

B

BA

ABA

B

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Vector multiplication

Scalar (dot ) product (AB)

Vector (cross) product (A X B)

Scalar triple productA (B X C)

Vector triple productA X (B X C)

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Multiplication of a vector by a scalar

Multiplication of a scalar kto a vectorAgives a vectorthat points in the same direction asAand magnitude equal

to |kA|

The division of a vector by a scalar quantity is a

multiplication of the vector by the reciprocal of the scalar

quantity.

AkA

Ak1|| k

1|| k

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Scalar Product

The dot product of two vectors and , written asis defined as the product of the magnitude of and , and

the projection of onto (or vice versa).

Thus ;

Where is the angle between and . The result of dotproduct is a scalar quantity.

BA

cos|||| BABA

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Vector Product

The cross (or vector) product of two vectors A and B,written as is defined as

where; a unit vector perpendicular to the plane thatcontains the two vectors. The direction of is taken asthe direction of the right thumb (using right-hand rule)

The product of cross product is avector

nBABA AB sin||||

nn

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Right-hand Rule

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Dot product

If and then

which is obtained by multiplying A and B

component by component.

It follows that modulus of a vector is

zzyyxx BABABABA

),,( zyx BBBB

),,( zyx AAAA

222|| zyx AAAAAA

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Cross Product IfA=(Ax, Ay, Az), B=(Bx, By, Bz) then

zyx

yx

yxz

xz

xzy

zy

zyx

zyx

zyx

aBB

AAa

BB

AAa

BB

AA

BBB

AAA

aaa

BA

zxyyxyzxxzxyzzy aBABAaBABAaBABA )()()(

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Cross Product

Cross product of the unit vectors yield:

yaz

xzy

zyx

aaa

aaa

aaa

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Example 1

Given three vectors P=

Q=

R=

Determine

a) (P+Q) X (P-Q)

b) Q(R X P)

c) P(Q X R)

d) e) P X( Q X R)

f) A unit vector perpendicular to both Q and R

zx

aa 2zyx aaa 22

zyx aaa 32

QRsin

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Solution (cont)

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To find the determinant of a 3 X 3 matrix, we repeat the first two rows andcross multiply; when the cross multiplication is from right to left, the result

should be negated as shown below. This technique of finding a determinant

applies only to a 3 X 3 matrix. Hence

Solution (cont)

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Solution (cont)

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Solution (cont)

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Solution (cont)

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Cylindrical Coordinates

Very convenient when dealing with problems havingcylindrical symmetry.

A point P in cylindrical coordinates is represented as (,, z) where

: is the radius of the cylinder; radial displacement from the z-axis

: azimuthalangle or the angular displacement from x-axis

z: vertical displacement z from the origin (as in thecartesian system).

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a

a

za


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The range of the variables are

0 < , 0 < 2, - < z <

vector in cylindrical coordinates can be writtenas (A,A, Az) or Aa+ Aa+ Azaz

The magnitude of is

222|| zAAAA

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Relationships Between Variables

The relationships between the variables (x,y,z) of the

Cartesian coordinate system and the cylindrical system (,

, z) are obtained as

So a point P (3, 4, 5) in Cartesian coordinate is the same

as?

zz

xy

yx

/tan 1

22

zz

y

x

sin

cos

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Relationships Between Variables

So a point P (3, 4, 5) in Cartesian coordinate is the same asP ( 5, 0.927,5) in cylindrical coordinate)

5

927.03/4tan

543

1

22

z

rad

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Spherical Coordinates (r,,)

The spherical coordinate system is useddealing with problems having a degree ofspherical symmetry.

Point P represented as (r,,) where r : the distance from the origin,

: called the colatitude is the angle between z-axis andvector of P,

: azimuthal angle or the angular displacement fromx-axis (the same azimuthal angle in cylindricalcoordinates).

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Spherical Coordinates

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The range of the variables are

0 r< , 0 < , 0 < < 2

A vectorAin spherical coordinates written as(Ar,A,A) or Arar+ Aa+ Aa

The magnitude of A is

222|| AAAA r

Spherical Coordinates (r,,)

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Relation to Cartesian coordinates system

x

y

zyx

zyxr

1

221

222

tan

)(tan

cos

sinsin

cossin

rz

ry

rx

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Relationship between cylinder and spherical

coordinate system

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Point transformation between cylinder and spherical

coordinate is given by

or

Point transformation

22 zr z 1tan

sinr cosrz

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Express vector B=

in Cartesian and cylindrical coordinates. Find Bat (-3,

4 0) and at (5, /2, -2)

Example

aaa cos10 rr

r

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Differential Elements

In vector calculus the differential elementsin length,

area and volumeare useful.

They are defined in the Cartesian, cylindrical and

spherical coordinate

Di

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Cartesian Coordinates

P Q

S R

dy

dz

dx

A B

D C

xx

y yax ay

az

z

Differential displacement : zyx dzadyadxald

fferentialelements

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z

y

x

dxdySd

dxdzSd

dydzSd

a

a

a

Differential normal area:


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Differentialdisplacement

Differential

normal area

Differentialvolume

zyx dzdydxld aaa

z

y

x

dxdySd

dxdzSd

dydzSd

a

a

a

dxdydzdv


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Differential displacement :zdzddld aaa

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zaddSd

dzadSd

dzadSd


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Differential normal

area

Differential volume

zdzaadadld

zaddSd

dzadSd

dzadSd

dzdddv


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drd sin

Differential displacement : adrarddrald r sin

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a

a

a

rdrdSd

drdrSd

ddrSdr

sin

sin2



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Differential

normal area

Differentialvolume

adrarddrald r sin

ardrdSd

adrdrSd

addrSdr

sin

sin2

ddrdrdv sin2


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1. The gradient of a scalar V, written as V

2. The divergence of a vectorA, written as

3. The curl of a vectorA, written as

4. The Laplacian of a scalar V, written as

Written as is the vector differential operator. Alsoknown as the gradient operator. The operator in useful indefining:

Del Operator

A

A

V2

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Example :

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Divergence

In Cartesian coordinates,

In cylindrical coordinates,

In spherical coordinate,

z

z

y

y

x

x AAA

A

z

AAA

z

1)(

1A

A

rA

rAr

rrA r

sin

1)sin(

sin

1)(

1 22

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Example

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zyx

zyx

AAAzyx

A

aaa

Curl of a Vector

z

z

AAA

z

aaa

1A

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Curl of a Vector

In spherical coordinates,

ArrAA

r

arraa

r

r

r

)sin(

)sin(

sin

12

A

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Examples on Curl Calculation

)(cossin 2

xzxye

zyx

aaa

xy

zyx

A

y

A

x

A

x

A

z

A

z

A

y

A xyzxyzzyx aaaxA

xyxexyyxzxzz cossincos2000 zyx aaaxA

xyxexyyxzz cos2sin zy aaxA

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Laplacian of a scalar

The Laplacian of a scalar field V, written as 2V is defined

as the divergence of the gradient of V.


2

2

2

2

2

22

z

V

y

V

x

VV

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In spherical coordinates,

2

2

222

2

2

2 V

sinr

1Vsin

sinr

1

r

Vr

rr

1V

Laplacian of a scalar

2

2

2

2

2

2 11

z

VVVV

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References:

Richard S . Muller and Theodore I. KaminsDeviceElectronics for Integrated Circuits

Jacob Milman, Christos C. Halkias

IntegratedElectronics

Specialization is an art of knowing more and

more about less and less and finally knowingeverything about nothing - Zentill [ ]

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L1 Chapter -I