22
Dr.-Ing. R. Marklein - EFT I - SS 05 1 Elektromagnetische Feldtheorie I (EFT I) / Electromagnetic Field Theory I (EFT I) 3rd Lecture / 3. Vorlesung University of Kassel Dept. Electrical Engineering / Computer Science (FB 16) Electromagnetic Field Theory (FG TET) Wilhelmshöher Allee 71 Office: Room 2113 / 2115 D-34121 Kassel Universität Kassel Fachbereich Elektrotechnik / Informatik (FB 16) Fachgebiet Theoretische Elektrotechnik (FG TET) Wilhelmshöher Allee 71 Büro: Raum 2113 / 2115 D-34121 Kassel Dr.-Ing. René Marklein [email protected] http://www.tet.e-technik.uni-kassel.de http://www.uni-kassel.de/fb16/tet/marklein/index.html

Elektromagnetische Feldtheorie I (EFT I) / Electromagnetic ... · Elektromagnetische Feldtheorie I (EFT I) / Electromagnetic Field Theory I (EFT I) 3rd Lecture / 3. Vorlesung University

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Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

1

Elek

trom

agne

tisch

e Fe

ldth

eorie

I (E

FT I)

/El

ectr

omag

netic

Fie

ld T

heor

y I (

EFT

I)

3rd

Lect

ure

/ 3.

Vor

lesu

ng

Univ

ersi

ty o

f Kas

sel

Dep

t. El

ectr

ical

Eng

inee

ring

/ Co

mpu

ter

Scie

nce

(FB

16)

Elec

trom

agne

tic F

ield

The

ory

(FG

TET

)W

ilhel

msh

öher

Alle

e 71

Off

ice:

Roo

m 2

113

/ 21

15D

-341

21 K

asse

l

Univ

ersi

tät K

asse

lFa

chbe

reic

h El

ektr

otec

hnik

/ In

form

atik

(F

B 16

)Fa

chge

biet

The

oret

isch

e El

ektr

otec

hnik

(F

G T

ET)

Wilh

elm

shöh

er A

llee

71Bü

ro: R

aum

211

3 /

2115

D-3

4121

Kas

selD

r.-I

ng. R

ené

Mar

klei

nm

arkl

ein@

uni-

kass

el.d

eht

tp:/

/ww

w.te

t.e-t

echn

ik.u

ni-k

asse

l.de

http

://w

ww

.uni

-kas

sel.d

e/fb

16/t

et/m

arkl

ein/

inde

x.ht

ml

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

2

Diff

eren

t Coo

rdin

ate

Syst

ems

/ Ve

rsch

iede

ne K

oord

inat

ensy

stem

e

●Ca

rtes

ian

(Rec

tang

ular

) Coo

rdin

ate

Syst

em/

Kart

esis

ches

Koo

rdin

aten

syst

em

●Cy

lindr

ical

Coo

rdin

ate

Syst

em/

Zylin

derk

oord

inat

ensy

stem

●Sp

heric

al C

oord

inat

e Sy

stem

/ Ku

gelk

oord

inat

ensy

stem

Wha

t is

the

bene

fit o

f the

Use

of a

Pro

blem

Mat

ched

Co

ordi

nate

Sys

tem

s?

/ W

as is

t der

Nut

zen

der

Verw

endu

ng e

ines

pro

blem

ange

pass

ten

Koor

dina

tens

yste

men

?

(Eas

ier)

Sol

utio

n of

the

Prob

lem

und

er C

once

rn!/

(E

infa

cher

e) L

ösun

g de

s be

trac

htet

en P

robl

ems?

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

3

Posi

tion

Vect

or /

Ort

svek

tor

(Pos

ition

svek

tor)

()

()

()

=

(

)(

)(

)x

yz

xy

zx

yz

xy

z

RR

R

xy

z

=+

+

++

=+

+

RRR

RR

RR

Re

Re

Re

ee

e

Cart

esia

n Co

ordi

nate

Sys

tem

/ Ka

rtes

isch

es K

oord

inat

ensy

stem

Vect

oria

l Vec

tor

Com

pone

nts

/ Ve

ktor

ielle

Vek

tork

ompo

nent

en(

)(

,,

)(

)(

,,

)

()

(,

,)

xx

xx

yy

yy

zz

zz

Rxyz

xRxyz

y

Rxyz

z

==

==

==

RR

ee

RR

ee

RR

ee

Scal

ar V

ecto

r Co

mpo

nent

s/

Skal

are

Vekt

orko

mpo

nent

en(

,,

)(

,,

)

(,

,)

x y zRxyz

xRxyz

y

Rxyz

z

= = =

Ort

hono

rmal

Uni

t Vec

tors

/

Ort

hono

rmal

e Ei

nhei

tsve

ktor

en,

,

||

||

||

1x

yz

xy

zx

yz

⊥⊥

==

=

eee

ee

ee

ee

Coor

dina

tes

/ Ko

ordi

nate

n,

,;

,

,xyz

xyz

−∞<

<∞

y

z

x

xxe

yye

zze

R

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

4

Fiel

d Ve

ctor

/ Fe

ldve

ktor

Cart

esia

n Co

ordi

nate

Sys

tem

/ Ka

rtes

isch

es K

oord

inat

ensy

stem

x

Ort

hono

rmal

Uni

t Vec

tors

/

Ort

hono

rmal

e Ei

nhei

tsve

ktor

en,

,

||

||

||

1

xy

z

xy

z

xy

z

⊥⊥

==

=

eee

ee

e

ee

e

Coor

dina

tes

/ Ko

ordi

nate

n,

,xyz

()

()

()

()

= A

(,

,)

A(

,,

)A

(,

,)

xy

z

xy

zx

yz

xyz

xyz

xyz

=+

+

++

AR

AR

AR

AR

ee

e

y

z

xxe

yye

zze

R

()

AR

xeye

zex y z

−∞<

<∞

−∞<

<∞

−∞<

<∞

Lim

its/

Gre

nzen

Arbi

trar

y Ve

ctor

Fie

ld/

Belie

bige

s Ve

ktor

feld

: Per

pend

icul

ar /

Senk

rech

t⊥

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

5

Not

atio

n an

d Fi

eld

Qua

ntiti

es /

Not

atio

n un

d Fe

ldgr

ößen

()

()

()

()

3

12

31

12

3

3 V

ecto

r Com

pone

nts /

3 V

ekto

rkom

pone

nten

,,

,,

= E

(,

,,)

E(

,,

,)E

(,

,,)

= E

(,

,,)

= E

(,

,,)

ii

ii

xy

z

xy

zx

yz

xx

i

xx

tt

tt

xyzt

xyzt

xyzt

xxxt

xxxt

=

=+

+

++

ER

ER

ER

ER

ee

e

e

e

()

()

()

()

()

()

()

()

()

()

9 D

yadi

c Com

pone

nts /

9 dy

adisc

he K

ompo

nent

en

,,

,,

,,

,

,,

,

(,

,,)

(,

,,)

(,

,,)

+(

,,

,)(

xxxy

xz

yxyy

yz

zxzy

zz

xxxy

xzxx

xy

xz

yxyy

yx

tt

tt

tt

t

tt

t

xyzt

xyzt

xyzt

xyzt

x

εε

εε

εε

ε

εε

ε

εε

ε

εε

=+

+

++

+

++

+

=+

+

+

RR

RR

RR

R

RR

R

ee

ee

ee

ee

33

12

31

j1 1

23

,,

,)(

,,

,)

+(

,,

,)(

,,

,)(

,,

,)

(

,,

,)

(

,,

,)

ij

ij

ij

ij

yzyy

yz

zxzy

zzzx

zy

zz

xx

xx

i xx

xx

yzt

xyzt

xyzt

xyzt

xyzt

xxxt

xxxt

ε

εε

ε

ε

ε==

+

++

= =

∑∑

ee

ee

ee

ee

ee

ee

ee

Vect

or /

Vek

tor:

El

ectr

ic F

ield

Str

engt

h/

Elek

tris

che

Feld

stär

keD

yad

/ D

yade

: Pe

rmitt

ivity

Dya

d/

Perm

ittiv

itäts

dyad

e

with

Ein

stei

n’s

Sum

mat

ion

Conv

entio

n /

mit

Eins

tein

sche

r Su

mm

atio

nsko

nven

tion

Eins

tein

‘s S

umm

atio

n Co

nven

tion:

If a

inde

x ap

pear

s tw

o tim

es a

t one

sid

e of

an

equa

tion

(and

not

at t

he o

ther

sid

e), t

he in

dex

is a

utom

atic

ally

sum

med

ove

r 1

to 3

. /

Eins

tein

sche

Sum

men

konv

entio

n: W

enn

ein

Inde

x au

f ein

er S

eite

ein

er G

leic

hung

zw

eim

al v

orko

mm

t (un

d au

f der

and

eren

nic

ht),

wird

dar

über

von

1 b

is 3

sum

mie

rt.

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

6

Posi

tion

Vect

or /

Ort

svek

tor

(Pos

ition

svek

tor)

()

()

()

=

(

)(

)(

)(

)(

)

()

rz

rz

rz

rz

RR

R

rzϕ

ϕϕ

ϕϕ

ϕ

=+

+

++

=+

RRR

RR

RR

Re

Re

Re

ee

Cylin

dric

al C

oord

inat

e Sy

stem

/ Z

ylin

derk

oord

inat

ensy

stem

Vect

oria

l Vec

tor

Com

pone

nts

/ Ve

ktor

ielle

Vek

tork

ompo

nent

en(

)(

)(

)(

)(

)

()

()

rr

rr

zz

zz

Rr

r

Rz

ϕϕ

==

= ==

RR

ee

RR

0

RR

ee

Scal

ar V

ecto

r Co

mpo

nent

s/

Skal

are

Vekt

orko

mpo

nent

en(

,,

)(

)(

,,

)0

(,

,)

rr

zz

Rr

zr

Rr

z

Rr

zz

ϕ

ϕϕ

ϕ ϕ

= = =

e e

Ort

hono

rmal

Uni

t Vec

tors

/

Ort

hono

rmal

e Ei

nhei

tsve

ktor

en(

),(

),

()

()

|(

)||

()|

||

1r

z

rz

rz

ϕ

ϕϕ

ϕϕ

ϕϕ

ϕϕ

⊥⊥

==

=

ee

e

ee

ee

ee

Coor

dina

tes

/ Ko

ordi

nate

n,

,;

0

,02

,r

zr

ϕπ

≤<∞

≤<

−∞<

<∞

y

z

x

()

rr

ϕe

zze

R

ϕ

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

7

Fiel

d Ve

ctor

/ F

eldv

ekto

rCy

lindr

ical

Coo

rdin

ate

Syst

em/

Zylin

derk

oord

inat

ensy

stem

()

()

()

()

()

()

= A

(,

,)

(,

,)

(,

,)

rz

rz

rz

rz

Ar

zAr

ϕϕ

ϕϕ

ϕϕ

ϕ

=+

+ ++

AR

AR

AR

AR

ee

e

Ort

hono

rmal

Uni

t Vec

tors

/

Ort

hono

rmal

e Ei

nhei

tsve

ktor

en(

),(

),

()

()

()

()

1

rz

rz

rz

ϕ

ϕ ϕ

ϕϕ

ϕϕ

ϕϕ

⊥⊥

==

=

ee

e

ee

e

ee

e

Coor

dina

tes

/ Ko

ordi

nate

n,

,r

y

z

x

()

rr

ϕe

zze

R

ϕ

()

AR

()

e

ze

()

ϕϕ

e

0 02

r

π≤

<∞

≤<

−∞<

<∞

Lim

its/

Gre

nzen

Arbi

trar

y Ve

ctor

Fie

ld/

Belie

bige

s Ve

ktor

feld

: Per

pend

icul

ar /

Senk

rech

t⊥

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

8

Posi

tion

Vect

or /

Ort

svek

tor

(Pos

ition

svek

tor)

()

()

()

=

(

)(

,)

()

(,

)

(

)(

)

(,

)

R RR

R

RR

R R

ϑϕ

ϑϑ

ϕϕϑ

ϕϑϕ

ϕ

ϑϕ

=+

+ +

+

=

RR

RR

RR

R

Re

Re

Re

e

Sphe

rical

Coo

rdin

ate

Syst

em/

Kuge

lkoo

rdin

aten

syst

em

Vect

oria

l Vec

tor

Com

pone

nts

/ Ve

ktor

ielle

Vek

tork

ompo

nent

en(

)(

,,

)(

,)

(,

)(

)(

,,

)(

,)

()

(,

,)

()

RR

RR

RR

RRR

RR

ϑϑ

ϑ

ϕϕ

ϕ

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϕ

==

==

==

RR

ee

RR

e0

RR

e0

Scal

ar V

ecto

r Co

mpo

nent

s/

Skal

are

Vekt

orko

mpo

nent

en(

,,

),(

,,

),(

,,

)RRR

RR

RR

ϑϕ

ϑϕ

ϑϕ

ϑϕ

Ort

hono

rmal

Uni

t Vec

tors

/

Ort

hono

rmal

e Ei

nhei

tsve

ktor

en,

,

||

||

||

1R R

R

ϑϕ

ϑϕ

ϑϕ

⊥⊥

==

=

eee

ee

ee

ee

Coor

dina

tes

/ Ko

ordi

nate

n,

,;

0

,0;0

2R

Rϑϕ

ϑπ

ϕπ

≤<∞

≤≤

≤<

y

z

x

ϕ

()

,R

ϕe

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

9

Fiel

d Ve

ctor

/ Fe

ldve

ktor

Sphe

rical

Coo

rdin

ate

Syst

em/

Kuge

lkoo

rdin

aten

syst

em

()

()

()

()

()

()

()

,

= A

(,

,)

,(

,,

),

(,

,)

R RR

t

RAR

AR

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϕ

=+

+

++

AR

AR

AR

AR

ee

e

Ort

hono

rmal

Uni

t Vec

tors

/

Ort

hono

rmal

e Ei

nhei

tsve

ktor

en(

)(

)(

)(

)(

)(

)(

)(

)(

)

,,

,,

,,

|,

||

,|

||

1

R R R

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϑϕ

ϕ

ϑϕ

ϑϕ

ϕ

ϑϕ

ϑϕ

ϕ

⊥⊥

==

=

ee

e

ee

e

ee

e

Coor

dina

tes

/Ko

ordi

nate

n,

,Rϑϕ

y

z

x

ϕ

()

,R

ϕe

()

AR ()

ϕϕ

e

()

,R

ϑϕ

e

()

ϑϕ

e

Arbi

trar

y Ve

ctor

Fie

ld/

Belie

bige

s Ve

ktor

feld

Lim

its/

Gre

nzen

: Per

pend

icul

ar /

Senk

rech

t⊥

0 0 02

R ϑπ

ϕπ

≤<∞

≤≤

≤<

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

10

Sphe

rical

Coo

rdin

ates

/Ku

gelk

oord

inat

enCy

lindr

ical

Coor

dina

tes

/Zy

linde

rkoo

rdin

aten

Cart

esia

nCo

ordi

nate

s/

Kart

esis

che

Koor

dina

ten

x y z

cos

sin

r rz

ϕ ϕ

sin

cos

sin

sin

cos

R R R

ϑϕ

ϑϕ

ϑ

22

arct

an

xy y x

z+r zϕ

sin

cos

R R

ϑϕ

ϑ

22

2

22

arct

an

arct

an

xy

z

xy

zy x

++

+

22

arct

an

rz r z

ϕ

+R ϑ ϕ

Tran

sfor

mat

ion

Tabl

e /

Umre

chnu

ngst

abel

le

z

y

x

ϕ

Coor

dina

tes

of D

iffer

ent

Coor

dina

te S

yste

ms

/Ko

ordi

nate

n ve

rsch

iede

nen

Koor

dina

tens

yste

men

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

11

cos

sin

cos

xr

ϑϕ

==

1.Fo

rmul

ate

xas

a fu

nctio

n of

the

cylin

der

and

sphe

rical

coo

rdin

ates

./

Form

ulie

re x

als

Funk

tion

der

Zylin

der-

und

Kuge

lkoo

rdin

aten

.

2.Fo

rmul

ate

ras

a fu

nctio

n of

the

Cart

esia

n an

d sp

heric

al c

oord

inat

es.

/ Fo

rmul

iere

ral

s Fu

nktio

n de

r Ka

rtes

isch

en u

nd K

ugel

koor

dina

ten.

3.Fo

rmul

ate

as

a fu

nctio

n of

the

cylin

der

coor

dina

tes.

/ Fo

rmul

iere

als

Fun

ktio

n de

r Zy

linde

rkoo

rdin

aten

.

22

sin

rx

yR

ϑ=

+=

22

22

22

1

(co

s)

(si

n)

cos

sin

xy

rr

rr

ϕϕ

ϕϕ

=

+=

+=

+=

22

xy

+ 22

xy

+

Exam

ples

/ B

eisp

iele

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

12

Sphe

rical

Coo

rdin

ates

/Ku

gelk

oord

inat

enCy

lindr

ical

Coo

rdin

ates

/Zy

linde

rkoo

rdin

aten

Cart

esia

n Co

ordi

nate

s/

Kart

esis

che

Koor

dina

ten

xy

zx

yz

AA

A=

++

Ae

ee

rz

rz

AA

ϕ=

++

Ae

ee

RRA

AA

ϑϕ

ϑϕ

++

A=

ee

e

x y zA A A

cos

sin

sin

cos

r r

z

AA

AA A

ϕ ϕ

ϕϕ

ϕϕ

− +

sin

cos

cos

cos

sin

sin

sin

cos

sin

cos

cos

sin

R R

R

AA

AA

AA

AA

ϑϕ

ϑϕ

ϑ

ϑϕ

ϑϕ

ϕ

ϑϕ

ϑϕ

ϕ

ϑϑ

+−

++

cos

sin

sin

cos

xy

xy

z

AA

AA

A

ϕϕ

ϕϕ

+

−+

r zA A Aϕ

sin

cos

cos

sin

R RAA

AA

ϕ

ϑ

ϑϑ

ϑϑ

+ −

sin

cos

sin

sin

cos

cos

cos

cos

sin

sin

sin

cos

xy

z

xy

z

xy

AA

AA

AA

AA

ϑϕ

ϑϕ

ϑ

ϑϕ

ϑϕ

ϑ

ϕϕ

++

+−

−+

sin

cos

cos

sin

rz

rz

AA

AA

A ϕ

ϑϑ

ϑϑ

+ −RA A Aϑ ϕ

Tran

sfor

mat

ion

Tabl

e /

Umre

chnu

ngst

abel

le

Scal

ar V

ecto

r Co

mpo

nent

s in

Diff

eren

t Coo

rdin

ate

Syst

ems

/Sk

alar

e Ve

ktor

kom

pone

nten

in v

ersc

hied

enen

Koo

rdin

aten

syst

emen

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

13

Exam

ple:

Coo

rdin

ate

Tran

sfor

mat

ion

of th

e Po

sitio

n Ve

ctor

/ Be

ispi

el: K

oord

inat

entr

ansf

orm

atio

n de

s O

rtsv

ekto

r

()

()

()

,,

,,

,,

zx

y

xy

zRxyz

Rxyz

Rxyz

xy

z=

++

Re

ee

Posi

tion

Vect

or in

the

Cart

esia

n Co

ordi

nate

Sys

tem

/

Ort

svek

tor

im K

arte

sisc

hen

Koor

dina

tens

yste

m

(,

,,

,,

)

cos

sin

(,

,,

,,

)si

nco

s(

,,

,,

,)

rx

yz

xy

xy

zx

y

zx

yz

z

Rr

zRRR

RR

Rr

zRRR

RR

Rr

zRRR

ϕϕ

ϕϕ

ϕϕ

ϕ

=+

=−

+=

(,

,)

(,

,)

cos

(,

,)

(,

,)

sin(

,,

)(

,,

)

x y zRr

zxr

zr

Rr

zyr

zr

Rr

zzr

zz

ϕϕ

ϕϕ

ϕϕ

ϕϕ

==

==

==

()

()

()

,,

,,

,,

cos

sinz

xy

xy

zRr

zRr

zRr

z

rr

z ϕϕ

ϕ

ϕϕ

=+

+R

ee

e

Tran

sfor

mat

ion

of th

e Co

ordi

nate

s /

Tran

sfor

mat

ion

der

Koor

dina

ten

Posi

tion

Vect

or in

the

Cart

esia

n Co

ordi

nate

Sys

tem

as

a Fu

nctio

n of

Cyl

inde

r Co

ordi

nate

s /

Ort

svek

tor

im K

arte

sisc

hen

Koor

dina

tens

yste

m a

ls F

unkt

ion

der

Zylin

derk

oord

inat

en

Tran

sfor

mat

ion

of th

e Sc

alar

Vec

tor

Com

pone

nts

/ Tr

ansf

orm

atio

n de

r ska

lare

n Ve

ktor

kom

pone

nten

22

1

co

sco

ssi

nsi

n

(c

ossi

n)

cos

sin

sin

cos

0

r zz

Rr

rr

r

Rr

r

RR

ϕ

ϕϕ

ϕϕ

ϕϕ

ϕϕ

ϕϕ

=

=+

=+

=

=−

+= =

()

rz

rz

RR

rz

ϕ=

+R

ee

Posi

tion

Vect

or in

the

Cylin

der

Coor

dina

te S

yste

m /

O

rtsv

ekto

r in

dem

Zyl

inde

rkoo

rdin

aten

syst

em

()

()

()

(,

,,

,,

)

,,

()

,,

()

,,

rz

rz

rz

ryRRR

Rr

yR

ry

Rr

ϕϕ

ϕ

ϕϕ

ϕϕ

ϕ=

++

R

ee

e?Po

sitio

n Ve

ctor

in th

e Cy

linde

r Co

ordi

nate

Sys

tem

/

Ort

svek

tor

im Z

ylin

derk

oord

inat

ensy

stem

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

14

Fara

day‘

s In

duct

ion

Law

in In

tegr

al F

orm

/Fa

rada

ysch

es In

dukt

ions

gese

tz in

Inte

gral

form

(1)

()

()

()

St

Ct

St

=∂

m(

)(

)(

)(

)d

(,

)(

,)

(,

)d

Ct

St

St

St

tt

tt

=∂

=−

−∫

∫∫∫∫

ER

dRBR

dSJ

RdS

ii

i

Fara

day‘

s In

duct

ion

Law

/ Fa

rada

ysch

es In

dukt

ions

gese

tz

Tim

e D

epen

dent

Sur

face

/Ze

itabh

ängi

ge F

läch

eTi

me

Dep

ende

nt C

onto

ur /

Zeita

bhän

gige

Kon

tur

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

15

Fara

day‘

s In

duct

ion

Law

in In

tegr

al F

orm

/Fa

rada

ysch

es In

dukt

ions

gese

tz in

Inte

gral

form

(2)

Fara

day‘

s In

duct

ion

Law

/ Fa

rada

ysch

es In

dukt

ions

gese

tz

[](

)(

)Ct

St

=∂∫

dRi

(,)t

ER dR

(,)t

ER

dRiSc

alar

Pro

duct

of E

and

dR

= ta

ngen

tial p

roje

ctio

n of

E o

nto

dR /

Sk

alar

prod

ukt v

on E

auf

dR

= T

ange

ntia

lpro

jekt

ion

von

E au

f dR

[V]

Vect

oria

l Diff

eren

tial L

ine

Elem

ent /

Vek

torie

lles

diff

eren

tielle

s Li

nien

elem

ent

[m]

Elec

tric

Fie

ld S

tren

gth

/ El

ektr

isch

e Fe

ldst

ärke

[V/m

]

Clos

ed C

onto

ur In

tegr

al /

Ges

chlo

ssen

es K

urve

nint

egra

l[m

]

dR=

dRs

Vect

oria

l Diff

eren

tial L

ine

Elem

ent /

Ve

ktor

ielle

sdi

ffer

entie

lles

Lini

enel

emen

t

Tang

entia

l Uni

t Vec

tor

/ Ta

ngen

tiale

r Ei

nhei

tsve

ktor

Scal

ar D

iffer

entia

l Lin

e El

emen

t/ S

kala

res

diff

eren

tielle

s Li

nien

elem

ent

m(

)(

)(

)(

)d

(,

)(

,)

(,

)d

Ct

St

St

St

tt

tt

=∂=−

−∫

∫∫∫∫

ER

dRBR

dSJ

RdS

ii

i

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

16

Diff

eren

t Pro

duct

s /

Vers

chie

dene

Pro

dukt

e

C=ABi

Scal

ar P

rodu

ct /

Ska

larp

rodu

kt

=C

AB

=C

A×B

Vect

or P

rodu

ct /

Vek

torp

rodu

kt

Dya

dic

Prod

uct /

Dya

disc

hes

Prod

ukt

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

17

Scal

ar P

rodu

ct (D

ot o

r In

ner

Prod

uct)

/ Sk

alar

prod

ukt (

Punk

tpro

dukt

ode

r in

nere

s Pr

oduk

t) (1

)

cos

(,

)

cos

AB

ABAB

φ

φ

=∠

=

AB

AB

AB

i

cosAB

=

ABφ

A

B

cosAB

=

ABφEn

clos

ed A

ngle

/

Eing

esch

loss

ener

Win

kel

cos

cosBA AB

BA ABφ φ

= = =

AB

BA

ii

()

()

cos

cos

ABAB

φφ

=−

cos

arcc

os

AB ABφ φ

=

=

AB

AB

AB

AB

i

i

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

18

Scal

ar P

rodu

ct (D

ot o

r In

ner

Prod

uct)

/ Sk

alar

prod

ukt (

Punk

tpro

dukt

ode

r in

nere

s Pr

oduk

t) (2

)

10

0

01

0 10

0

()

()

+

+

xy

zx

yz

xy

zx

yz

xx

xy

xz

xx

xy

xz

yx

yy

yz

yx

yy

yz

zx

zy

zz

zx

zy

zz

AA

AB

BB

AB

AB

AB

AB

AB

AB

AB

AB

AB

A

==

=

==

= ==

=

=+

++

+

=+

+

++

++

=

AB

ee

ee

ee

ee

ee

ee

ee

ee

ee

ee

ee

ee

ii

ii

i

ii

i

ii

i

xx

yy

zz

BAB

AB

++

12

31

23

12

31

23

11

22

33

3 1

()

()

()

()

i

i

xy

zx

yz

xy

zx

yz

xx

yy

zz

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

iAA

AB

BB

AB

AB

AB

AA

AB

BB

AB

AB

AB

AB

=

=+

++

+

=+

+

=+

++

+

=+

+

=∑

AB

ee

ee

ee

ee

ee

ee

ii

i

xy

z⊥

⊥e

ee

Ort

hono

rmal

Uni

t Vec

tors

/

Ort

hono

rmal

e Ei

nhei

tsve

ktor

en

1 0 0

xx

xy

xz

= = =

ee

ee

eei i i

0 1 0

yx

yy

yz

= = =

ee

ee

eei i i

0 0 1

zx

zy

zz

= = =

ee

ee

eei i i

1 2 3

xx

yx

zx

= = =

Cart

esia

n Co

ordi

nate

s /

Kart

esis

che

Koor

dina

ten

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

19

Scal

ar P

rodu

ct (D

ot o

r In

ner

Prod

uct)

/ Sk

alar

prod

ukt (

Punk

tpro

dukt

ode

r in

nere

s Pr

oduk

t) (3

)

33

11

33

11

33

11

()

()

or/o

der

ij

ij

ij

ij

ij

ij

ij

ij

ij

ij

ij x i

xy

zx

yz

xy

zx

yz

xx

xx

ij

xx

xx

ij

xx

xx

ij

xx

xx

xx

ij

B

AA

AB

BB

AB

AB

AB

AB

AB

δ

δ

δ

==

==

==

=

=

=

=+

++

+

= = = = =

∑∑

∑∑

∑∑

AB

ee

ee

ee

ee

ee

ee

ee

ii

i

i

i

i

ij

x j xx

jj

ii

xij

x

A AB

xx

AB

AB

δ= =

=

1 0ij

ij

ij

δ=

=

Kron

ecke

r D

elta

/

Kron

ecke

r-D

elta

with

Ein

stei

n’s

Sum

mat

ion

Conv

entio

n /

mit

Eins

tein

sche

r Su

mm

atio

nsko

nven

tion

Eins

tein

‘s S

umm

atio

n Co

nven

tion:

If a

inde

x ap

pear

s tw

o tim

es a

t one

sid

e of

an

equa

tion

(and

not

at t

he o

ther

sid

e),

the

inde

x is

aut

omat

ical

ly s

umm

ed o

ver

1 to

3. /

Ei

nste

insc

he S

umm

enko

nven

tion:

Wen

n ei

n In

dex

auf e

iner

Se

ite e

iner

Gle

ichu

ng z

wei

mal

vor

kom

mt (

und

auf d

er

ande

ren

nich

t), w

ird d

arüb

er v

on 1

bis

3 s

umm

iert

.

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

20

Mag

nitu

de o

f a V

ecto

r /

Betr

ag e

ines

Vek

tors

10

0

01

0 10

0

(AA

A)

(AA

A)

A

AA

AA

A

+

AA

AA

AA

+ A

AA

AA

A

xy

zx

yz

xy

zx

yz

xx

xy

xz

xx

xy

xz

yx

yy

yz

yx

yy

yz

zx

zy

zz

zx

zy

zz

==

=

==

= ==

=

= =+

++

+

=

++

+

+

+

+

AAA e

ee

ee

e

ee

ee

ee

ee

ee

ee

ee

ee

ee

i

i

ii

i

ii

i

ii

i

1 2

22

2

A

AA

AA

A

AA

A

A

xx

yy

zz

xy

z

=+

+

=+

+

=

33

11

2

ij

ij

ij

ij

ij

ij

ij

i

xx

xx

ij

xx

xx

xx

xx

x

AB

AA

AA

A

δ

==

=

= = = = =

∑∑

AAA

ee

ee

ee

i

i

i

i

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

21

Exam

ple:

Pos

ition

Vec

tor

and

Elec

tric

Fie

ld S

tren

gth

Vect

or /

Beis

piel

: Ort

svek

tor

und

elek

tris

cher

Fel

dstä

rkev

ekto

r

(,

,)

R(

,,

)R

(,

,)

R(

,,

)

xy

zx

yz

xy

z

xyz

xyz

xyz

xyz

xy

z=

++

=+

+R

ee

ee

eeCa

rtes

ian

Coor

dina

te S

yste

m/

Kart

esis

ches

Koo

rdin

aten

syst

em

(,

)(

,,

,)

E(

,,

,)

E(

,,

,)

E(

,,

,)

xy

zx

yz

txyzt

xyzt

xyzt

xyzt

= =+

+ER

Ee

ee

Elec

tric

Fie

ld S

tren

gth

Vect

or /

El

ektr

isch

e Fe

ldst

ärke

vekt

or

22

2

(,

,)

ˆ (,

,)

(,

,)

x

yz

xyz

xyz

xyz

xy

z

xy

z

=

++

=+

+

RR

R ee

e

()(

)2

22

(,

,)

(,

,)

(,

,)

x

yz

xy

z

xyz

xyz

xyz

xy

zx

yz

xy

z

= =+

++

+

=+

+

RR

R

ee

ee

ee

i

i

22

2

(,

,)

ˆ (,

,)

(,

,)

EE

E

EE

E

xy

zx

yz

xy

z

xyz

xyz

xyz

=

++

=+

+

EE

Ee

ee

()(

)2

22

(,

,)

(,

,)

(,

,)

EE

EE

EE

E

EE

xy

zx

yz

xy

zx

yz

xy

z

xyz

xyz

xyz

= =+

++

+

=+

+

EE

E

ee

ee

ee

i

i

Posi

tion

Vect

or /

O

rtsv

ekto

r

Mag

nitu

de o

f the

Pos

ition

Vec

tor

(Dis

tanc

e) /

Be

trag

des

Ort

svek

tor

(Abs

tand

)

Mag

nitu

de o

f the

Ele

ctric

Fie

ld S

tren

gth

Vect

or

(Str

engt

h) /

Bet

rag

des

elek

tris

che

Feld

stär

keve

ktor

s (S

tärk

e)

Posi

tion

Unit

Vect

or (D

irect

ion)

/

Ort

sein

heits

vekt

or(R

icht

ung)

Elec

tric

Fie

ld S

tren

gth

Unit

Vect

or (D

irect

ion)

/

Elek

tris

che

Feld

stär

keei

nhei

tsve

ktor

(Ric

htun

g)

Dr.-

Ing.

R. M

arkl

ein

-EFT

I -S

S 05

22

End

of L

ectu

re 3

/En

de d

er 3

. Vor

lesu

ng