EMT01

Fundamentals of

Electromagnetics

Introductory Concepts, Vector Fields and Coordinate Systems

Outline

• Introductory Concepts • Vector Fields

Research Areas of Electromagnetics

• Antenas

• Microwaves

• Computational Electromagnetics • Electromagnetic Scattering

• Electromagnetic Propagation • Radars

• Optics • etc …

What is Electromagnetics?

• Electromagnetics is the study of the effect of

charges

-at rest or in motion.

• Some special cases of electromagnetics:

– Electrostatics : charges at rest

– Magnetostatics : charges in steady motion (DC)

– Electrodynamics : waves excited by charges in

time-varying motion (car energy and information)

Scalar and Vector Fields

• A scalar field is a function that gives us a single value of some variable for every point in space.

• Examples: voltage, current, energy, temperature

• A vector is a quantity which has both a magnitude and a direction in space.

18

Scalar Fields

e.g. Temperature: Every location has associated value (number with units)

19

Scalar Fields - Contours

• Colors represent surface temperature

20

Fields are 3D

•T = T(x,y,z)

•Hard to visualize

 Work in 2D

21

Vector Fields

Vector (magnitude, direction) at every point in space

VECTOR REPRESENTATION

3 PRIMARY COORDINATE SYSTEMS: • RECTANGULAR

• CYLINDRICAL • SPHERICAL

Choice is based on symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL Spheres - SPHERICAL

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 z P(x,y,z) θ z r x y z

P(r, θ, z)

θ Φ r z y x

P(r, θ, Φ)

Page 108

-Parabolic Cylindrical Coordinates (u,v,z) -Paraboloidal Coordinates (u, v, Φ)

-Elliptic Cylindrical Coordinates (u, v, z) -Prolate Spheroidal Coordinates (ξ, η, φ) -Oblate Spheroidal Coordinates (ξ, η, φ) -Bipolar Coordinates (u,v,z)

-Toroidal Coordinates (u, v, Φ) -Conical Coordinates (λ, μ, ν)

-Confocal Ellipsoidal Coordinate (λ, μ, ν) -Confocal Paraboloidal Coordinate (λ, μ, ν)

Cartesian Coordinates P(x,y,z)

Spherical Coordinates P(r, θ, Φ)

Cylindrical Coordinates P(r, θ, z)

x

y z

P(x,y,z)

θ z

r

x y

z

P(r, θ, z)

θ

Φ r

z

y x

VECTOR NOTATION

xa A a A a

A

A  ˆ  ˆ  ˆ

Rectangular or Cartesian Coordinate System x z y z z y y x

xB A B A B

A B

A      Dot Product

z y x z y x z y x B B B A A A a a a B A ˆ ˆ ˆ     Cross Product





1 2 2 2 z y

x A A

A

A    Magnitude of vector

(SCALAR)

Cartesian Coordinates

z y

x yA zA

A x

A  ˆ  ˆ  ˆ

Page 109

x

y

z

Z plane

y plan e

x plane



 _{   }

 2 2 2

z y

x A A

A A

A A  

xˆ yˆ

zˆ

x₁

y₁

z₁

A_x Ay

A_z

( x, y, z)

Vector representation

Magnitude of A

Position vector A

) , ,

(x₁ y₁ z₁ A

1 1

1 ˆ ˆ

ˆx yy zz

x  

Base vector properties

0 ˆ ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ             x z z y y x z z y y x x y x z x z y z y x ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ      

x

y

z

A_x Ay

A_z A

 B Dot product: z z y y x

xB A B A B

A B

A     

Cross product: z y x z y x B B B A A A z y x B A ˆ ˆ ˆ     Back Cartesian Coordinates Page 108

VECTOR REPRESENTATION: CYLINDRICAL COORDINATES

Cylindrical representation uses: r ,, z

z z r

ra A a A a

A

A  ˆ   ˆ  ˆ

z z r

r

B

A

B

A

B

A

B

A













UNIT VECTORS:





Dot Product

(SCALAR)

r



z

P

x

z

VECTOR REPRESENTATION: SPHERICAL COORDINATES r  P x z y 

Spherical representation uses: r ,,  UNIT VECTORS:







 a

a

aˆ_r ˆ ˆ

  

 a A a

A a

A

A  _r ˆ_r  ˆ  ˆ

 





B

A

B

A

B

A

B

x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

y

aˆ

x

aˆ

z

aˆ Unit Vector Representation for Rectangular

Coordinate System

x

aˆ

The Unit Vectors imply :

y

aˆ

z

aˆ

Points in the direction of increasing x Points in the direction of increasing y Points in the direction of increasing z

r



z

P

x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

Cylindrical Coordinate System z

aˆ

 aˆ

r

aˆ

The Unit Vectors imply :

z

aˆ

Points in the direction of increasing r Points in the direction of increasing 

Points in the direction of increasing z

r

aˆ

 aˆ

VECTOR REPRESENTATION: UNIT VECTORS

Spherical Coordinate System

r



P

x

z

y



 aˆ

 aˆ

r

aˆ

The Unit Vectors imply :

Points in the direction of increasing r

Points in the direction of increasing 

Points in the direction of increasing 

r

aˆ

 aˆ

 aˆ













RECTANGULAR Coordinate

Systems

CYLINDRICAL Coordinate

Systems

SPHERICAL Coordinate

Systems

NOTE THE ORDER!

r,, z r,,

Note: We do not emphasize transformations between coordinate systems

VECTOR REPRESENTATION: UNIT VECTORS

METRIC COEFFICIENTS

1. Rectangular Coordinates:

When you move a small amount in x-direction, the distance is dx

In a similar fashion, you generate dy and dz

Cartesian Coordinates Differential quantities: Length: Area: Volume: dz z dy y dx x l

d  ˆ  ˆ  ˆ

dxdy z s d dxdz y s d dydz x s d z y x ˆ ˆ ˆ       dxdydz dv  Page 109

METRIC COEFFICIENTS

2. Cylindrical Coordinates:

Distance = r d

x y

d

r

Differential Distances:

3. Spherical Coordinates:

Distance = r sin d

x y

d

r sin

Differential Distances:

( dr, rd, r sind )

r



P

x

z

y



METRIC COEFFICIENTS

Representation of differential length dl in coordinate systems:

z y

x dy a dz a

a dx

l

d   ˆ   ˆ   ˆ

z

r r d a dz a

a dr

l

d   ˆ     ˆ   ˆ



  

 a r d a

rd a

dr l

d   ˆr   ˆ  sin  ˆ

rectangular

cylindrical

spherical

AREA INTEGRALS

• integration over 2 “delta” distances

dx

dy

Example:

x y

2 6

3 7

AREA =   

7

3 6

2

dx

dy _{= 16}

Note that: z = constant

In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant

Representation of differential surface element:

z

a

dy

dx

s

d









ˆ

Vector is NORMAL to surface

DIFFERENTIALS FOR INTEGRALS

Example of Line differentials

or or

Example of Surface differentials

z

a dy

dx s

d    ˆ or ds  rd  dz  aˆ_r Example of Volume differentials dv  dx  dy  dz

x

a dx

l

Base Vectors

A₁

r radial distance in x-y plane

Φ azimuth angle measured from the positive x-axis Z    r 0  2 0 

     z Cylindrical Coordinates          ˆ ˆ ˆ , ˆ ˆ ˆ , ˆ ˆ ˆ r z r z z r z

r A zA

A r A

a

A  ˆ   ˆ  ˆ _  ˆ

Pages 109-112

Back ( r, θ, z)

Vector representation





 _{   }

 2 2 2

z

r A A

A A

A A  

Magnitude of A

Position vector A

Base vector properties

1

1

ˆ

ˆ

_r

_z

_z

Dot product:

z z r

rB A B A B

A B

A     _{ } 

Cross product: z r z r B B B A A A z r B A  

ˆ ˆ ˆ     B A Back Cylindrical Coordinates Pages 109-111

Cylindrical Coordinates Differential quantities: Length: Area: Volume: dz z rd dr r l

d  ˆ  ˆ   ˆ

       rdrd z s d drdz s d dz rd r s d z r ˆ ˆ ˆ    dz rdrd

dv  

          

 ˆ ˆ , ˆ ˆ ˆ , ˆ ˆ ˆ

ˆ _{R R}

R

Spherical Coordinates

Pages 113-115

Back (R, θ, Φ) 

 

A A

A R

A  ˆ _R  ˆ  ˆ

Vector representation



 _{   }

 2 2 2

  A A A A A

A   _R

Magnitude of A

Position vector A

1

ˆ

_R

R

Dot product:

  

 B A B

A B

A B

A    _{R R}  

Cross product:       B B B A A A R B A R R ˆ ˆ ˆ     Back B A Spherical Coordinates Pages 113-114

Spherical Coordinates Differential quantities: Length: Area: Volume:              _{ } d R Rd dR R dl dl dl R l d _R sin ˆ ˆ ˆ ˆ ˆ ˆ                        RdRd dl dl s d dRd R dl dl s d d d R R dl dl R s d R R R ˆ ˆ sin ˆ ˆ sin ˆ ˆ 2      

 R dRd d

dv 2 sin

        d R dl Rd dl dR dl_R sin Pages 113-115 Back

z z y x y x r ˆ ˆ cos ˆ sin ˆ ˆ sin ˆ cos ˆ ˆ            z z y x y x r A A A A A A A A          

 sin cos

sin cos

Back

Cartesian to Cylindrical Transformation

z z x y y x r       ) / ( tan 1 2 2  Page 115