Fundamentals of Electromagnetics

Introductory ConceptsCoordinate Systems & Transformations

Vector Analysis

RICHU JOSE CYRIACNIT CALICUT

Why Study Electromagnetics? Electromagnetics is everywhere!!!It’s around you ..hmm…but you cannot detect

It’s the basic from which circuit theory is developed

Maxwell’s Equations Kirchoff’s laws

Its essential for a communication, signals are sent as Electromagnetic waves

What is the basic of electromagnetics? CHARGEElectromagnetics is the study of CHARGES

Electromagnetic study can be divided into three

o Electrostatics : charges are at restoMagnetostatics : charges are at steady motiono Electrodynamics : charges are in time varying motion(gives rise to wave that propagate and carry energy and information)

What is Electromagnetics?

rest motion

Signals Amplification

Modulation

Antennas

“This part of the program is sponsored by” EM waves……….!!!!

Where Electromagnetic waves?

•Maxwell’s equationEntire subject in one slide…!!!!!!

vD

0 B

t

BE

t

DJH

Maxwell’s equation

Electrostatics(Only E-field)

Magnetostatics(Only H-field)

Electromagnetic waves

(both E&H field )

Fundamental laws of electromagnetics

0t

» Is it difficult to study?

No…No…Never…..It’s interesting…..

NB: Maths is just a powerful tool…..Physical interpretation of the mathematical result is the key…

»We will acquire the tool for so called “OPERATION EMT” first….OK?

VECTOR ANALYSIS

How to study?????

VECTOR ANALYSIS

Co-ordinate sytems CartesianCylindricalSpherical

Transformations of co-ordinate systemsVector Calculus

Orthogonal and non-orthogonalNon orthogal is hard to work with….so we will discuss only

orthogonal co-ordinate systems

Co-ordinate systems

Orthogonal Coordinate Systems: (coordinates mutually perpendicular)

Spherical Coordinates

Cylindrical Coordinates

Cartesian Coordinates

P (x,y,z)

P (r, θ, Φ)

P (r, θ , z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Selection of co-ordintes depends upon the symmetry of problem

Engineers always choose the easy way

x

z

y

yaxa

za

Unit Vector Representation for

Rectangular Coordinate System

xaThe Unit Vectors imply :

ya

za

Points in the direction of increasing x

Points in the direction of increasing y

Points in the direction of increasing z

Cartesian Coordinate System

VECTOR REPRESENTATION: UNIT VECTORS

zzyyxx aAaAaAA ˆˆˆ

r

f

z

P

x

z

y

Cylindrical Coordinate System

za

a

ra

The Unit Vectors imply :

za

Points in the direction of increasing r

Points in the direction of increasing j

Points in the direction of increasing z

ra

a

VECTOR REPRESENTATION: UNIT VECTORS

zzrr aAaAaAA ˆˆˆ

Spherical Coordinate System

r

f

P

x

z

y

q

a

a

ra

The Unit Vectors imply :Points in the direction of increasing r

Points in the direction of increasing j

Points in the direction of increasing q

ra

aa

VECTOR REPRESENTATION: UNIT VECTORS

aAaAaAA rr ˆˆˆ

zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ

CARTESIAN CYLINDRICAL SPHERICAL

ORDER Sradhikkanam…!!!

r,f, z r, q ,f

Summary

VECTOR REPRESENTATION: UNIT VECTORS

Cartesian Coordinates

Differential quantities:Length:

Area:

Volume:

dzzdyydxxld ˆˆˆ

dxdyzsd

dxdzysd

dydzxsd

z

y

x

ˆ

ˆ

ˆ

dxdydzdv

Cylindrical Coordinates

Differential quantities:

Length:

Area:

Volume:

dzzrddrrld ˆˆˆ

rdrdzsd

drdzsd

dzrdrsd

z

r

ˆ

ˆ

ˆ

dzrdrddv

Distance = r sinq df

x

y

dfr sinq

r

f

P

x

z

y

q

Spherical Coordinates

PLEASE NOTE Bhaiyajiii….!!!.....its important

Spherical Coordinates

Differential quantities:

Length:

Area:

Volume:

dRRddRR

dldldlRld R

sinˆˆˆ

ˆˆˆ

RdRddldlsd

dRdRdldlsd

ddRRdldlRsd

R

R

R

ˆˆ

sinˆˆ

sinˆˆ 2

ddRdRdv sin2

dRdl

Rddl

dRdlR

sin

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ) Cartesian CoordinatesP(x,y,z)

Spherical CoordinatesP(r, θ, Φ)

Cylindrical CoordinatesP(r, θ, z)

TRANSFORMATIONS

TRANSFORMATIONS: CYLINDRICAL CARTESIAN

θr

P(r, θ, z)

xy

z

POINT TRANSFORMATION: If u are given a point in one co-ordinate system and to convert it to another

zzryrx sincos

zzx

yyxr 122 tan

zz

yx

yxr

ˆˆ

cosˆsinˆˆ

sinˆcosˆˆ

VECTOR TRANSFORMATION: If u are given a vector in one co-ordinate system and to convert it to another

zz

ry

rx

ˆˆ

cosˆsinˆˆ

sinˆcosˆˆ

TRANSFORMATIONS: SPHERICAL CYLINDRICAL

AREA INTEGRALS

• integration over 2 “delta” distances

dx

dy

Example:

x

y

2

6

3 7

AREA = 7

3

6

2

dxdy = 16

Note that: z = constant

VECTOR ANALYSIS

SCALAR FIELDS

VECTOR FIELDS

Example: Velocity vector field - jet stream

SOME TOOLS FROM OUR MATHS TOOL KIT

DEL operator- An operator that we are going to do operations on scalars and vectors

zk

yj

xi

Does not have any significance its own, but have significance when it OPERATES

DEL OPERATOR-PHYSICAL INTERPRETATION

‘ T ’ be a scalar and we are operating on T

RESULT OF OPERATION?????

T is a vector in the direction of the most rapid change of T,

and its magnitude is equal to this rate of change

If u substitute a point we get a direction in which a maximum variation from that point occurs

jxiyz

xyk

y

xyj

x

xyiT

)()()(xyxyzTLet )(

‘ T ’ be a vector and we are operating on T

RESULT OF OPERATION????? T is the net flux of T per unit volume at the point considered, countingvectors into the volume as negative, and vectors out of the volume as positive.

DEL OPERATOR-PHYSICAL INTERPRETATION

z

T

y

T

x

T

kTjTiTz

ky

jx

iT

zyx

zyx

)(

T Known as Gradient of a scalar

A Known as Divergence of a vector