ELEC TROMAGNETICS (EE 262)

Suygated Text baokK

alectn maynehes by M-N.D Sodiku
2.
ENgincainy aacomaguehes
Macahan Hl Pubicahon.
by Htayt and Buck,

- In this core, we wll deal wth Physcs dt Eucto

magnehic wawes ich is comtosed o Ebetne ad
magnethie ielda. As, edsctne feld and mamehie fields
ane vectos auanhy we wlltat discus Vector alaaba.
Then, we wl discus caleulath en sthc elchic tields
and magnetie ields d to Vaioscharye distibten,
arious ans ike Grauss la, Biot Sawantlw
and Ane cinuital law
Then we wll discuss Phenomama t wawe
Ahouyh dibt medium like tae apae, dieltnc
amd cenduets.
Aaat we wl disoUss wawe
dungnnt kund ot comni cation chanel ike
amd Trans mis)on enes.
 UNIT- 1

is a byaneh 4 Physies in which

Eletnc amd Magnetie phenemona and thu abpiahiot
ar studied
Eleetie and maymete phenoen a can be summanged
in Mawell's equatiens

V= the vecor diaomtialoaby

eletne dunuity (efa)
- elschc hed intnity (v/m)
magnehc ld ntonaity
f volme change denty
wsent denity
So, e wll be wth (ma) vectos
Quantthes. So, ne wl veotor
as t is a math emati cal trel herine d
Aeuamt can ethen be salen or a vetr
: A Scalan s a usntty that has
oly magitda
t : TiHe, maA dis tamce, Temteahe
cuwent, vo tape ete
vector auanti: both a Hhas has
mantude and diechen
E:g: velocty i Dis piacmons, Forca,
Electie jld inkenity , mapnehe
ted nenity ete
Avector auntty is reneened by a leter
wth an arrow en tot t s h au A D
by a
a bold tace ltter seh as A
Aheld is a tunctin that pes
pani u an
1p the is scalan, the iolol is aaid to
bo salan ald.
E9. Tenerahne disti buti on in a puidig
Sound untemaiy n a Theate
Elactni otenti al n a gn.
Te the suantty s vectoY, the held
feld is said
to Be vec os held
atmohhne.
UNIT VECTOR :
Aveethr A has both mapnitud and dischon
The magitude oh Äis a Scaln writen as
A or
Aurit vector along A(a) issduhined as

vechoy wohgse majnu s and

A. Thws

A = A aA

tems 4 b mapitude Aand

dinecio aA
A yetr A w Carteaian co- ovdi nae
be repreaonted
 Azag
A

Aya

co-0rinat
A Vector A Un carei an
A

X=(A, Ay, Az) oY Ay aa t Aay t Az az

Called
h ,A ,Ay and Az ane

Comþorona and
Z dineti n

. A = A= A2+ Ay-+ A
A + Azaz
A
Vector addihon and subtscthi en:
A

(Ara yay ttaz

+ Bzaz

=A+ B
(An t Bx) a +(Ayt By) ay t
(Azt Bz) az
A -B =
+(-B)
(A- Ba) a t (Ay- Bylay t
A

(Az- Bz) a2

Panolalgen

Head to Taul rule

A
Siday tpr veabr Aubtr ach on . ¢- B

Parallalagom rule
).

Head to Tail sule

AtB = B-+A

Acsociatie : A+ (B+E) -(A B)+2

Disti butie : KCR+ B) = KÃ+ Kî
hne Kis a sealanaawnty
ht
Position and distance veetor

o,0,)
A point Pin crteaian co-ornate
be sepresonted by (9,). than

Positon Vector? Apotion vector (op) or

Tadis vctr h bount p elorens
Pos on a pount P in space
S in
elaion to an anbid ay ogmmce. A
) = Op = Xay +

lot P(2,3, ) ,then sp= 2a + 3ay +

Pistanee veebr: Tt is the dispecamont en
ene Boint to another.
P

).

(6
(ya
an d non untam Vecto
contnt or
held is Baid to be
A vectos
untorm t does not dopond on
vaniabe , 9, Z
For o : A= 34,

2y
m vector

Same nyotne
Vanies fin pot -to
 A

4ay +Gaz

almy ay
B
unt vocts A+ 2 B

b. 3À-8 =(28, -13, 13)

J2s)+(3)+ (03)

35.74
SÄ+ 28= (14,-3, )

+ O39

(as expectd)
 Pots P and a ne ocated at (o, 2, 4) and
(-3, , s). alulate:
(o) The pouti en vectr p
(5) The distamee Vectoy p to &
c) The distnce bowen Pand Q
(d) AVeetos panale to Pa wth
Selution
magitudsdo.
(a)
2)
()
= (3,1,5) - ( o,2,4)
(-9,.-, )
Cc) d =(dpo= )9+)+l = 3.37
a) (ot
et the aeAhed cetor be A ,thon

A is ) P8 t muat hawe dame

unit vee as res Thus,
- t (-3,-l, )
3.31
A= (-9.045 ay- 3.0|5ay +3. D1S
NECTOR MULTIPLICATIDN &
hon t vestoss A and B ane mui liedy
the es! s ethon aa Bcalan os a Veetr
ci) Scalan (os) Dot foduet (A"B)
i) Vectoy (or) che procuct CÄXB)
Mutli cathin thre vectoss A.B and
ci) Scalan Trise faaduct ( (BXC))
Gi) vechy Teitlu faaduct (7x(Bx)
DOT PRODUCT

AB =

Dot oduot batweom tuo Voctoss is

dotyicd geemhicaly as the oduct
A and B nd cosne
A smaen angle bahen thonn, when
ane dhawn tail to tal
A
 y ay
a
a < a
||=A =
Ay+
Az Ay+ - AA: (in)
2
(8+) = Abutwe: Doti )
Cemtahue:
ohe
with
each foandiuler) ovthogenal
(or
Àal
ba to e and8 vectoss
A Two
NOTE&
BzAz t By y +ByAy = BA.
z Bz +
az Az +.
ay A= if
CROSS PRODUT

Sun

an: nt veety nomal to the

ard

The dineasen o an is taken as

(a) dineehen thumb shen
tngos te i t bamd sote
tom Rto B. (Rigtt hamd Rute)

OAB

Rigld hamd Srew rule

Advance hando seemw as
 ay <
Ax Ay Az
Bx Bz

Not commutti
’
It is anti commutehie

Not associahic :
()

(i) Dstibutiue :
®x (B+) = (XxB) +(#xe)
(h) AX A = 0

X a

bw ducts ane ob}anct

homuahon.
Mov ng counter deae loads t -e
Fesuts.
Scalan Tslo odut 3
B
(X#) =

A Ay Az

Bx By
Cx

Vector T e poduct :
Ax(Bxe) - B(AE) -¿(*-3)
bac cab' lo

# Compomonts a Vectv :
The Puyhion or comporunt 4 a yecor
9isen dneh en can Be Saalan or vector.
A
Scalan comtenont (AB)
hetos 8 as
 Ag =

The vechs comonent (As) A alny

sinbly sealn componod (Ao) muhlied by
unit vectoy aleng

Xcan be hesoved into two orthogenal

Compenne ene combonant As is fanalle
anothen
 3a7
B= 2dy - 54
Calculale am betwen Aand B
Sol
A:B
Cos Da. B) =

3
O. lD92
J26x29

(0.1092)
83.73°

a= 2a - a t 2a
R= 2a
Calculate :
ta) F+a)x (P-8)
(t) unit vecy to
P: & X R oth Q and R

(3) ) The compoment

() P CqxR)
- Fx(P-K) + EX(P- t)

2. az
2

= 2a 4|2ay +4a
ay
& (RxF) - (2,) 2
ay
3
2 T

p3) Scalan Trie otuet :

2

2.

2
d)

Ia||&
JS =0B9T6
Ji4

Px(axa) - a(7.)-R (?.8)

=(2,-, 2)3 -
(2-9) 2

=(2,5, 4)
A

s Aunidt voctoy pntendiuln to both

amdR
gen

=t (o45,0. 293, - DS9¬)

SO
(P.

(P. &

(P. a)a

2 (2,-l, 2)

- 0 444ay- O222ay +t
 COORDINA Tt S YSTEMS

Thhee beat Knn co-or din ade sysBems ane

(i) arteaj an
( cal

i) sphni cat
Be
A ceniudnable amount tine mat
bwed
aved
by chosng a cocrdi nae
that best Juen prblom.
Rizht homdd and o sttogonal
CARTE SIAN CoORDINA TE S 3

yz)

The co-o7dnate
Vau able x, and z
 tuommeesured
s tandangle, mthal azi caled is
hom tnce disTadial the fi
dt
asrupraen P
is point Any
hawing probloms
dealing
with ane he
conwenient ystem
s coosdnate This
TES3 DLNA cDoRCyLINRICAL
az) Az + yAy
A:) Ay, (An, A=
wen
?as
beodinates
Can co- cartesi
an Un A
g: adis oybindn pas ing hieuyh Pe)
tadial distanee hom Z- ax s

azi muthal nge

Sane as in antesian co-0di nate

< 271

Avetos A

A= Ap a, t Ag y t Az az

N
 Az J cosp Sing
A Cos-
Sing cosf
Az
Ag sicoopng -Sund
fcoap o,Ay
Af
Az
Az
cosy Sinp Ad
unitbeten Relati
on
vectoys?
gni
cal
toan
canteai om
t
2
anetnatien Tra
nt po toPoint
SPHERLCAL coORDI NA TES
(3, e, )

Abount P cam be ubraented as

whu, is fadi al dis teme jom oign
to pod p (or sadis
contrd at origen and' pasing

(co lattude bebeen

z ax is and posjhen Vectos a P
meaun
(azi muth al anye )
 osg<27
A vectoT

Point to point Tramstormahon :

74

&= t a n ' y
2

tan
 8-43 lo =
6.32 aty
Se)
2=3
coT all Cosdinates.
A
is P Evauate
and expess
un A P
(z4) cyindica A=and 6,
3)P(-3,
¤=
Sn -
Az
cof b Ay
9L0sy c sün Sune
np|
A, A
ind -S
Az
Ay cead Ae
-Sun
A Ag
transhrm vecto Vector
to
?en at
3sin y-
 y
cos f+z) Ay
Sung(1+2) yeesgt
+ Ap-
Az
Cep Sing - Ad
Sind
isP)fat
43') 62,
8.lo 64. P{7, =
3)43, 103. 6.3 - 3)
P{P(-2,6,
6462 Jy) = tam' e=
tylt:
3
 VECTDR CALeULUS

The v(Du) oponaor s Vector diyeential ehnahsn

in cateai an coordi nates;

+ +

Thes Del otesabr is not veetor n

but t ehnates en Scalar and veetor held;
and waegu n
) The adi ent q ascalan held (v)
ci) The diwengone avector jeld (A)

a vector jeld )
The a Scalan eld V;
4 V
derie
& Apheni cal coovdinates.
 Se
+

)
)
6

uns
dins

uns +
1 Gradiont acealan
The radiont a Acalen field Vis a vestor
Hhat neasmts both the magnitudi an d

dinectien 4 maxi mum

i ncraae V.

Gradient (n al th
G cordin ates)
a t

aet

v(v+u)
2.
(vo)
3

nyn-!
4. v=
5. man. rate
un V b unt dis tanee
6. (or comfenent)
The brejectiona unit VV un the
vectoy
dinschioncad diechonal dwatie of v
and is
along said to be Scalan bolential
it A= v, via

scalen had;
(a) V= gz cos 2
A
- 2fZ

+ Pze3 2g

Se:
VU = 10 Sin o ces ay
lo Sun 2 ces ao +
2 DWeromce dd a Veetor?
+ Az

y.A= Ax + Az

on
tranormai techniaue;
lao, wung
incglindhical co-ordinate;
? Az
VA = (A)
in sphni cal co-ordinates;

t Ag Sino)

faeponh es:
Tt oducs a sealan d
. ( =V-A + vB
V(VÃ) = VVA +A. VV
4. Ay yecty eld is sid to
The diwence t a¤A tat a wen pount P is the
outward Rr pen unit volme as volwme
shrinks abott P
m

Vyis volwme enelosed by cosed ea S

n which Pis lo cafed.

Posihioe
diyoyonce diwengend.
CURL Oe A VECTOR ?
az
ay
d2.

Ay Ay Az.
 tindical Co-ordinates)

Ap

(un sphni c oodinates)

VXA =

Arg

frounies:
The cud d a veats eld mo then Vey

2.
x( B)
3.
Vx(Ax 8) - A (7- B)- (7. F)
+ (8-V)F - (A) 5
4.
V VXA +
S.
6.
v.(7X =o
vectoy hed is Aaid o b isotah'onal
(or Potential)
+

2(

+
 eld is aaid to he harmani.
harnone n a
Ascalaa
ib lapleei an vamishas.

ie v=D

Tt alao posible to datine, laflai an

vetor A.

&Dedemins the dw ngene and un h gsm

+ gz

casp
So
 if
(Kz- 22) yotanal
Then va o K kz and K3 ane
a.

woo, 3, 2
O.3,D.33, 0.S
t,3, 22
Soen 2-k3) ay
xf= -K, ay t (K2-3 are
(Ks =, K= O, K-3)
 Calculae thu laplaci an o the Bcalan
V=e Sun 2z
coshy
() lor sun
(
NoTE Sun ha = e
2

Cos h z = e + e
2

tam h = Sun hy

Snx j Sun hz Sun hjx= Sinz

cos h Coshj
d Sun bz) = cos bx
(
dz
d Snb

Seuen: (a) -2e Sun 2x Cos hy

(c) (1+ 2C*38)

Difyoanhal engeh ara and vome :

bgenenti al elemane in argth, area

youme ane n Veetor calculus.

n
They catesi an, ylinducal,
an d npheni cal coo di nate aylens.
1. Carteai an coordinate yem 3

A
D

duhguntial angt (o)

dienal diu alecomont :
di=. da az tt dy ay t deai
diuonh al ae a/wtace
A
ds = ds an
elmut
nomal
an: nit v y
to, this tic andvolwme
dihected aey m
1f he Con sud

for PQ RS
dea
du dz
d dz

da dy
ci) aignonhial vome is gwen by
da dy da
ylindhical coondiat ystms

)
d

dy = f df do dz
3. Shnia cooy dinae:
= %dag

de
der

dus glsinodr do d
 ohjsct shnn in hyene, and calcudak:
CD
c (o,5/9)
c) Suitae anea AB CD D

d) Suac area ABD

Suee area tofD
(e)
A
So,)
Se
A(S,0,o) ’A5, 0°, o)
s (o, 5,) ’ e(5, )

D( 5,9, 1°) D(s,0, 10)

BC dl = dz
10
dz 1D
Bc =
L

b)
7/
2-5 7

ds dz

= 257T
 and Z=D

rea ABO =

AofD: da - df dz
area ofD =

S
dw

6 2.5 7
Line and voume inteenal

Line integeal: ai the

he cwe L
Comhonont t alny
d is diynhil
lungtn vecthr
aeng path L.

Path L

areund L
line
is a cldcwe

dosed cenbou nral

A

arbnd L
 s.
tom
S.
taxouthwand not uohich
(dginiga dwnace clerud any for
ormal
s. to seetoy unt n
an ds .
d
l
centaning for
tthe
d
A
Voume vnkyal volume wnterl oh cealan sv

ad
(al ulate circulati en
deed path s hon in

(3)

for patn l: dl dx a

3
X d
 X= 0,2=

Path3:

as dy 0
xd - dz

Aoo z = ’dx=d

(2-) da
1
3
P=
dt- dy as t dz
d=
dz

-y dy

-4+o-t6
Diuegenea o a vector and dwepene Theorern
m

u Vowme enclosed b
whone, av
' ' n woih Point p'is
located.

’ Phyoically, dwengerce dy the vector fild

A at meare

much hu jeld diveges tom tht fout.

V
I t stes that
dwegena Theorem:
To tal outw ard a vechs ld
a clesed e 's' is Aame
volume ntegnad
4
 2
fox a
22
Calcuate the enthe
a a wnit cube DIE Y,ze|
Sse.

-:ds= ( dy dz ay

aa dy a2

- dady
L
dn dy = x2 3 6

Ybotom : z=0; ds'- da dy (-az)

 y=o /
ds = da dz (-ay)

d- dan dz ay
Ads Z dn d
.

dzdz

ds = dydz ax

Yfrert yz2

Y
bock o
 4 6 12

yenilytthe abo ve eultt uain divepene

Theorom_

Ax Ay +
Az

=
yz t 2 yz t
-A dw 2t4y zt x)da dy dz
V

12

Thus,dgenae venited
a= 1oe-2 (sar + az), dtemine
G ow
cyindl s-I, osze |) And conyin
Theoror
disengemce

df dz ay

( , )

+ Ybotom + Ycuwd

Z=| ; ds = 3 ds d az
2

e-2, 27X
10 e-x
2
Yeotom
21
--/[1be-°g aray
= - lox 27T X

= -l07

L
loe-2z dp d

lox 27TX e-27 dz

2071 x e-2 |
-2

*[ee)
totl +
2
+
Vowet

)
+
-(fa)t-
2)
(oe) -()
+
(1oe-)
-P .X l0-e 22 x 2g t 10e X-2
20e-22 20 e- 22

Thus, diwegena Lahsed.

 L
Theoen3 Stokes
max
an
dirton
si whese
direch
on nemmal
wniit
and temds
to
max s
the magibud
og
ter creulaton
ohore vectoYsotahomal arA
STOKES AND VECTOR OF
AHCURL