# lecture 1 - introduction€¦ · 5 chapters 1, 2 and 3 in your text book! ab x x y y z z x x y y z...

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Electromagnetics EE 340 Dr. Mohammad S. Sharawi EE, KFUPM 2012 1 Lecture 1 - Introduction

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Electromagnetics

EE 340

20121

Lecture 1 - Introduction

Why Study EM?

2

Because EM phenomena is in all electrical/electronic based equipment ….

Electrical/electronic components/equipment are almost everywhere …

Thus, EM is everywhere these days …

Computers, Cell Phones, Car Controllers, Power Lines, etc.

Wireless communications is based on EM wave propagation ...

High speed digital design is based on EM wave propagation …

Fields around power lines are EM fields …

3

4

?

Vector Algebra Quick Review (MATH302!)

5

Chapters 1, 2 and 3 in your text book!

zzyyxxAB

zzyyxx

zyx

zzyyxxA

zyx

zzyyxxA

BABABAABaAaAaA

AAA

aAaAaAA

a

AAAA

aAaAaAaA

++==•

•=•=•=

++

++===

++=

++==

θcos

ˆ , ˆ , ˆctor,another veor axisan on vector a of projection

ˆˆˆˆ

)(cartesian ˆˆˆˆ

222

222

BAAAA

AAA

A

6

( ) ( ) ( ) zyx azzayyaxxRRR

ˆˆˆ 121212

1212

−+−+−=−=

Position Vector,

Example 1.1:Given a vector

find the magnitude of A, its unit vector and the angle that it makes with the z-axis? [On the Board.]

zyx aaa ˆ2ˆ2ˆ −+−=A

7

Cross Product,

Scalar Triple product,ABBA

BA

×−=×

==×

yxz

xzy

zyx

zyx

zyx

zyx

nAB

aaaaaaaaa

BBBAAAaaa

aAB

ˆˆˆ

ˆˆˆ

ˆˆˆ

ˆˆˆˆsinθ

( ) ( ) ( )

zyx

zyx

zyx

CCCBBBAAA

=

×•=×•=×• BACACBCBA

8

Vector Triple Product,

( ) ( ) ( )BACACBCBA ••−••=××

A) Cartesian Coordinates (x,y,z)

9

z

y

x

zyx

==

==

++=

ˆ

ˆ ˆ

ˆˆˆ

sl

B) Cylindrical Coordinates (ρ,φ,z)

10

zz , siny , cos

, tan ,

ˆˆˆˆˆˆˆˆˆ

ˆˆˆ

122

222

===

=

=+=

++=

++=

φρφρ

φρ

φρ

ρφ

φρ

φρ

φφρρ

x

zzxyyx

aaaaaaaaa

AAAA

aAaAaA

z

z

z

z

zzA

- useful for problems with cylindrical symmetry

11

Transformation matrices

−=

−=

zz

y

x

z

y

x

z

AAA

AAA

AAA

AAA

φ

ρ

φ

ρ

φφϕφ

φφϕφ

1000cossin0sincos

1000cossin0sincos

12

z

z

φρρφρρ

ρ

φρ

φρρ

φ

ρ

φρ

==

=

=

++=

ˆ

ˆ

ˆˆˆˆ

sl

C) Spherical Coordinate System (r,ϴ,φ)

13

used in problems with spherical symmetry.

θφθφθ

φθ

θφ

φθ

φθ

φθρ

φφθθ

cosz , sinsiny , cossin

tan , tan ,

ˆˆˆˆˆˆˆˆˆ

ˆˆˆ

122

1-222

222

rrrx

xy

zyx

zyxr

aaaaaaaaa

AAAA

aAaAaA

r

r

r

rr

===

=

+=++=

++=

++=

A

14

Transformation Matrices,

−=

−−=

φ

θ

φ

θ

θθφφθφθφϕθφθ

φφθφθφθθϕθφθ

AAA

AAA

AAA

AAA

r

z

y

x

z

y

xr

0sincoscossincossinsinsincoscoscossin

0cossinsinsincoscoscos

cossinsincoscos

15

φθθ

φφθφθθ

φθθθ

φ

θ

φθ

ddrdrdv

r

r

sin

ˆ ˆsin

ˆsin

ˆsinˆˆ

2

2

=

===

++=

s

l

16

A vector (magnitude and direction) that represents the maximum space rate of increase of a function A.

zyx azAa

yAa

xAA ˆˆˆ

∂∂

+∂∂

+∂∂

=∇

Divergence of a Vector and Divergence Theorem

17

The divergence of A at a point P is the outward flux per unit volume as the volume shrinks about P.

The divergence theorem states that the total outward flux of a vector field A through the closed surface S is the same as the volume integral of the divergence of A.

zA

yA

xAA zyx

∂∂

+∂

∂+

∂∂

=•∇

∫∫ •∇=•VS

dvAsdA

Curl of a Vector and Stokes’s Theorem

18

The curl of A is a rotational vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is normal to the direction of the area when oriented to make maximum circulation.

Stokes’s Theorem: The circulation of a vector field A around a closed path is equal to the surface integral of the curl of A over the open surface bounded by the path.

( ) AAcurl ×∇=

( )∫∫ •×∇=•SL

sdAldA