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TRIBIILIV ;'rlJ I iNiVERSITY
I}iSTITUTE Oi. I\CINEERING
Examination Control Ilivisiori
2$?8 Bhadra
Subject: - Elecro-magnetics (EX 503)
Candidates are required to give^ their answers in their^ orra^ ltords^ as^ frrr^ as^ practicable.
ltte ntpt^ A^ I^ I^ q^ it( itions.
'{he figures
in ihe margin indicate
{S$ Wt'Ls_.
Necessary dfitij a_re a{tached hgls-t+,itk.
,d represent a vector and d*6rrr1rs denotes a unit vector^ aiong the
.lirection (^) given by the
subscript.
Assume suitablc dota dnecess*ry.
i. (^) Givenapoint P (-2, 6, 3) and vector field E= yd+ (xy+z){, express
p
and E in
spherical co-ordinatb system. t5]
- A point^ charge of 6pc located at origiu, iiniforro iinr charge density of 180ne/m lies along
x-axis and uniform sheet charge of 25 clmz lies ol z:0 plane.^ Fhd d at point^ (1,2,4j.^ tI
- Derive the expression for an eiectoic field intensity due to an infinitely long iine charge
with charge density p1 by using Gauss's law. Find the volume charge riensity that is
associated with the field D
Eyzdz + x"yd, (^) + zAz C
/rnz. [4+j]
4" State ccntinuity equation. Given the vector cr.urent de;isit1,
i =
1Ap2frp-,lpsrnz$d6 mA/mz. tretermine the current f<.rllovriirg outrvard the circular
band
P=5, 0<
$<2n,2 il iregion2)
has pr,=^ 10. if there is auniicrm
rrragnetic field H (^) = 56x * 6ey + TezA/m in region 1, find d and }j in region 2. t8l
- Conect the equatioa V x E (^) = 0 for time varying fiel Divergence
cartesian:v D=$p-{.
AxryAz
cylindnca I : v.^ D =
I
a(dD')
1+
p op poq^ sz
.. .- .: 1 f(r2D,)^ I d(D"sins)^ I^
&+
Snhecncal:.lJ= +-_
'--+_--
r' Ar^ rsin^0 eS^ rsin^0 d$
Gradient
carresian, vv^ = s 6,^ *^ *a. *^ *a"
dx6y'az
cvtindrical: vv=9l4, *
1 9Ia. (^) * {a-
0p
o p0{l
Az
av^ Iav. l av^
Sphericai:Vv"* (^) =?6, +:iia. +--:
6 r 6J rsrnu =i-irdO
Curl
c-artesidn:vx fr =fgrr.-
aHi
L -l.g.-fl.]a
.,.[5r.- tr ],
(0Y Az)'taz ax)'\Ax AY)"
cvrindrica r : v. fi
f
9!.
3L'lu *(
Ht
au
"L. *^
lf
'ot'o
4"..|u
'
\ea$
0z)'(az 0P)' P\
0P Ai)'
spher'rcar : V x s
=
I
=[LEtrr
e)
.
1f+=+
g'H,
+.
lf
q('9,
).
,re[--aol-- a+ J"'";L'*s
aS
n, f'
*;[
a'
n (^) )".'
Laplacian:
Cartesian:VzV= (^) ++*
oNL dy. oz"
cyiindricar: vzv
i*(rffi
i#-#
Spherical: vz\r
= i* i" ffi
#*fd
(',ru
iii
.
i0. A 50 f) lossless iransmission line is 0.4 l" long. The line is terminated with a load
Zt:4A +^
j30 A. If the
operating frequency is 30il^ MHz,^ find^ [2+2+4)
a) reflection coeflicient (f)
b) standing u'ave ratio (s) and
c) input i:npedance^ {Z*)
ltr.Explain why TEM-wave doesn't exist in a rectangular waveguide? A rectangular
waveguide has^ dimensions^ a^
:
1 er& b
:
crn. The medium within the waveguides has
€,
: 1, lrr:
1, o
1. Find whether or not tlle signal with^ the frequency^ of^500 MHz^ will
be transmitted in the TE1,6 mode" [2+4]
- $lhat are the parameters^ of^ anterura? List^ out^ the^ different types^ of^ antenna^ you^ have
studied. [1+l
j
Divergence
AD' (^) AD-
Cartesian : V. D (^) =-3 +^
=t
*-
0x AY Oz
I
r6(pDo).1dD0,0D,
CYlindrica I :^ V'^ D =
:
-=--
T --=-
T -;-
' p (^) dp poq oz
.,-i l8(r'?D") I^ a(Dssin0)^
I 6Dr
spnecncal: r"^ u=-T-*6.-^
" r.rB- A
**irin
gE-
Grailient
av^ av^ av^
Cartesian:VV (^) =--O* -a^ *---^d,
AxaY'02'
cylindrical : vu
= $
a, *
!*a,
*a,
o? poq^
sohericat:vv*9Ye.
lav^ I av^
*;H uu *iiln;6:",
Curl
t^'{, oH, )".
(au- (^) oH"f. ,
(au, (^) oH- l" c-artesian'vxfr=l
al' *' !a- "ir
-***o
ur!L)I4rr'"^"-l lu,^ +l^ i-r-gi"^ la,
a),
a,
j"' '[a,
-a*
f'
-l
a" f'
cyrindricar:v,r=[;
r+
E )"r.[.a,.
-?#f (^) r.'
;LTp
(^6) )n,
sphericar : v x^ fr (^) =
-r*
[
aG.:in ol
-. {'-l
gL
a(:ur )lu"
u(t,
'si"e[
ae
a[f'
' i\rsirie a,[ 0r
J-u
'
tl. ar &^ )
Laplaeian:
carresian: v2v^ =
#,
#,
-#
cyiindrieah v21I =;*(-#)
*,#ii
spherieal: vzY
i*(.'11.)
ala('r#)
TRIBHUVAN TNIVERSITY
INSTITUTE
OF ENGiNEERING
Examination Controtr (^) Division
2076 Chaitra
:
gl (^) gg[g-gagjl{[! fsx -i o-ii
/ candiiiates-are required to give their zurswers (^) in their or+:r (^) rvords as far es practicable. t
Atrcmltf Att qttestions.
{. The figures in^ the^ mnrgin (^) inrlicate Full.ltf (^) arks.
,.
"
-lsstintc ,tltit,iblc ,in;o iJiiiliSl-- Unecessary. $I'','i;,*ii1. , ,
-;tffi
_)
?Hm;**l
s
1' (^) Transform the (^) vectar
A
46x
-2ey
*-4ez
into (^) spherical co-orclinates ai a point p(x = -2.
Y
: -3' z': 4')'
lsl 2' (^) An intinite
uniforrn line charge
pL (^) - 2nCrm lies alcng the
x-axis
in fr.ee space, rr4rile
pointcharges
of8'Ceacharelocatedat(0.
0, 1)and(G, (^) 0, -1). (a) Find (^) il at(2,3, -41.
t6l
3" (^) Dbfine uniqueness theorem" Find (^) the energy st.red (^) in free space (^) 1br rhe region 2mm< (^) r< 3iir*r, 0<0<90", 0<0<90", given
thc potential^ field
\r
: , =-*-'
'-'
1MrL
r! (^) r;'v
'vrwrtrroi
ltsru \ - :
l'2*
a) --\r erd (^) b) alcosOy
4' (^) using the continuitv (^) equation elaboi:ie (^) tire coirceiri
cf
Rela:taiion Tirue constant 1RTC)
with nccessary
derivaiions. (^) Ler ;=+- (^) ;,p
Atm2bc rhe cunent
censity (^) in a given
r, Y
region' Ai t: (^) trOms, carcurate
tire arnouni of
current passing through (^) surface p:2m,
00 (^) (region 2) (^) has
Fr
= J
g'
Ifthere is a unifonn
magneric fierd d= (^) 5i* +66., t.75,rNmin region (^) 2. rrnd d
and (^) H in region 2.
12+6) ?- (^) List
out the Maxwelr-eqrrations phasor
forrn for
tim-e varying
case (^) in &ee space. (^) A conducting bar (^) ean slide
fieeiy over two
conducting (^) rails placed at x (^) = 0 and x :^ lgcm.
calculate (^) the induced (^) voitage in the bar if the
bar srides ai
a velocity 3=16agm/s
and
B,=3izmW-h/m
[2+3]
<:
DIVERGENCE
.ARTESTAN v.D^ =
-Y:-
.YLTNDRT*AL o.,^ =|fio
q"l#+*
spHEBrcAL v-o=jflrrr,l*$$(Dpsino).##
GRADIENT
cARrEstAN on^ =#^*#r#'=
ay .Jav^ ,dY
CYL|NDRICAL vV
=?ap+|"0*^" ap'u' pW^ 0z
sPHEBTcAL
on =ff,,l#**#",
CURL
.ARTESTA,N v^ x r{^ = (* -Y)",.
W-*)r.
(*
\
a.rr,.
3y
cyLrNDRrcAL v^ x^ H^ =
G#
-'#)",. (* - #)"
FIBP=I/')
-!!e)"' pL (^) eip da )
spHERrcAL v^ x^ II^ =#
[,(ffl
-W)".itL,*W-ry]"
*ilvy-#7^r
LAPLACIAN
cARrEsrAN v,v^ =ff+#"#
cvLrNDHrcAr-
tr =iAffi
-*#.#
SPHERICAI.
n:i?#).h(,#) .##
TRIBHIJVA}'IUNIVERSITY
INSTITUTE OF^
ENGINEERING
Examination
Control
Division
2075 Ashwin
Exam.
ffit"l
BEL,BEX,
BCT
I
Pass Marl$
I
80
1?
Ixvel
Programme
Year / Part
cs Wsas!
,"**u".esarerequiredtogivetheiranswersintheirownwordsasfaraspractica
{ (^) AnemPt$lquestio'ls'
,
'rir'iffi
in the margin^
indicate (^) fq!'-Ma*r
r (^) iiiriro*
"n
t (^) Rurrre suitable
dataifnecessary'
I. civen
pointsA(p = 5,^
q= 7a',-z=-3)and
B (p^ 1
2'Q=;s^0" z=^
l)'find:
(a)
a unit vecro,^
m .lrt.iiun^
coordinares
at A directeil toward'B;
(b) a unit vector
tsl
in .vtinari.ut^
.ooiainu"t at^
A directed
toward B'
z. Two uniform^
line charges,^
each z^
nc/m, are^ located^
at y = 1,^ z^ =
t1 m' Find
the rotar^
.ru.oi.hio-i;;;ffi&.
surface of^
a sphere
having a^ radius
of 2 m,^ if^
it ir.uitt'oJi'nir'
1' 0)'
e' u^ e'"r'-^ -^
t6l
- ';*;Energy^
Density in
electrostatic^
field'
t7l
- The conducting
planes 2x^
3Y = 12-^ and^
2x
18 a-re:t
notentials of^
100
v and 0, respectivery.^
Let r^
ro rno^
ii,rlill v^ i,^
p (5, 2, E);i)^
E at P(5,2,6)'^ I7l
S.LetafilamentalycurrentofsmAbedirectedfrominfiniwtotheoriginon
thti positive^
r.iil;;;:;b'.k;;;;";;;G
;;ih'
positive x^ axis'^
Find H^
tsl
at P (0, 1,0).
- stare Anrpere,s^
circuirat iaw.^
t et^ the^
perrnittivity
be (^5) uH/m
in regiott A
whetex<'O,.and
pHAn in
"gion
Ei*f"tu i
'
o' If^ there^
is a surface
culTentdensi.ryK=150av-ZOOa'n'rmatx=0'andifHl=300a'-400a'+
500a,tum,
find:. (a)^
lHuh
o) lH"J(";li*r'
to HNel'^
[10]
- srate and^
explain tJre^
Maxwell'S equation
in^
differential
and inteBral
form'
Also define^ the dispracement
currenit*JnJ;dih
;f
penetration.. [10]
s. Establish^
the relation^
for Helmholtz's
equation
for electromagnetic
wave
tsl
propagation.
_ .. r- ^L--_^-
tol
- State and
prove Poynting's^
theorem'
- AloadZr =^
0onalsssless
50-0 line'The
operating frequency.is^
208 MHz,ai^
ii.*uu.t.t
gth on the line^
is? rn"
(a) If
the line is^
1.ti m in^ length,^
ur* ttr.'i*1i1,^
char tJ find^
the input^ impedance'
(b) whar i,^
"'i;t'dhui
ii rr,.^ distance^
from
the }oad to^
the nearest voltage
t?I
maximuri?
11un.o-'o*orectangularwaveguidehasdimensionsa=.2cmandb=1cm.
Determine
the rangE.of
frequenci.;;;;;i;i.tt
tf'* guide^
will operate single
t3l
mode ftEro)
[3x2]
- Write^ short notes
on:'
a) TE^ rnode^
and TM^ rnode
b) Antenna^
ProPerties
TRlBHUV,4N LTNIVERSITY
iNSTITUTE
OF ENGINEE,RING
Exaruination
Control
Division
2{}?5 Chaitra
--;tPt::{:Esryes*sli
cs (EX^ 503)
Candidates are
required to^
give their answers^
in their^ own words
as^
far
AttentPt
All questions'^ "ii*is,r;; ,, t,-..r
in the ruargin^
indicar.e F$tMs'rk*
Tsun* tuitable^
duta if^ necessary^
r
'iirirr*, ^
tkat the^ Bortt^
Faced letter^
represents a^
vector d/?cz^ a5pr561ip
t7l
L1l
vector.
l.Findthevectorthatextendsfrom4(-3'-4'6)to8(-5'2'-8)andexpressitincylindrical (^) li+
cocrdinate 5115{1:6'
rr :- t ''--+^'t (^) '+
+t1a
^ri(
cliarges are^ locate0 is characterized by
zvz:2'LerD1:-30a*+J0a'+70a
'nClnf
and find:-
-^
a) Dg(Tangential^
cornponent of^ D^
in Region 1);
t
b) Polarization^
(P1);
"j
p,, O"""al
component
of E^': Region^
il
grti*gential component of
E in^ Region 2)
- Derive
the Possion's
and Laplace's equations'
Assuming,
that the potential^
v in^ the
cylindricai
"ooraii.ui.
;rr.*;r
rh* #;;;
of 'r' only.^
solve the Laplace's equation by
Integration
Vf"tiroJ
uo derive^ ,ir"^
"rpt.ttion
for ihe^ capacitance^
of the Spherical
capacitor
uri,g d;;^
solution of^
v.'"^"""'"
[2+5]
- Derive^
the equation^
for magnetic
field intensity
in different
regions due^ to^
a co-xial
cabte carryins^
,;;if;if'iirt
iurr.d;;;;;,
I in the inner^
conductor
and -I
in the
t6l
outer conductor'
- Finc the vectcr magnetio
field intensity^
H in catesian^
coordinate
at P(-1'5'
-4' 3) caused
by a cunent frru*."nt^
of 12A in^ the a,
ai'""ti*
on the z-axis and^
extending fi'om z=-3 ta
I6l
z=3.
- Define curl^
and giYe^ the^
physical interpretation^
of the^ curl^
with a suitable
example' [1+3]
B.Auniformplanervaveinfreespaceispropagating'".n:-,1,.u'rectionatafrequencyof
MFIz. If^ E=^
cos
(tot+By) a,/lm','t'rite
the expressions
for electric and^
magnetic
" (^) fields, i.e., E,^
(x,y,z) and^ H.^
(x' y'^
'l '"'pt"ii;"il^
ir:.;il*or forms'^
'
[3+5]
g.DeriveanexpressionforStandingWaveRatio(SWR)indicatingwhereonthez-axis
you,ll get^ the n
ax"i-mu*
a irrioiui'"-oi
"f""toit
held intensity^
E' Assurne that the
boundary
i, at ;;th"^
,*gi"n z<0 is
p-rr* ai.lectricand
the region^
z>a may be^ of^
any
tgl
material.
!
10. Find the^ ampiitude^
of thc displacement
current density in^ an^ air^
space within a^ large
--
;;;;";.f;rrr"rH=trj6cos(3?7t+i.2566x
10-62) *rNm. t6l
1i" A lossless 50-f!^
line is 1.5l,long and^ is^ terminated^
with a pufe^ resistance^
of 100fi' The
load voitage is +olr8"v.^
Find: (a)^ the^ average^
polver delivered to^
the ioad; (b)^ the
magniturie of^
the min'-rmum voltage^ on the
iine' [4+4]
- W,hat^ are^
the advantages and^ disaclvantages^
of *'aveguides-when^
you compare- it-r'vith
transmission lines?^
Eipiain the transverse^
electric (TE)^ and transverse^
magnetic
(TM)
[3+3]
modes used in rectangular waveguides'
1-:. Give the definition^ of^
an antenna and^ explain^
the properties of^ any onff^
type of antenna
rhat you have stu t*
- The velocity of propagation (^) in a lossless transmission line 2.
m/s. Ifthe capacitance
ofthe line is 30 pflm, (^) find: [Z+Z+2+ZJ a) Inductance of the line b) Characteristic impedance c) Phase constant at 100 MHZ d) Reflection coefficient (^) if the line is terminated
with a resistive load of 50Q
I l. What are the advantages (^) of waveguides over tansmission
lines? A rectangular waveguide
' has a moss-section of 2.5 cm x^ 1.2 cm- Find the cut-off frequencies d dominant mode and TE (1,1)
- Write (^) short notes on: Antenna properties
[1+4J
l2l {:
DIVERGENCE
Oartesian: v.il
* +^
93 aLk
dx 0y 0z
Cylindrical: v.D =
]{P
1g*
p (^) op poq (^) oz
sphericai: v.d
#9I9fi,9.
GRADIENT
cartesian: vv^
=xd
#q
cyrindrical: vv
Hq+
l"#eo,
Spherical: Vv^
ffa;
i#a;
##a;
LAPLACIAN
cartesian: v2v^
azv ozv ozv
ffi- ayr- fu
cyrindricar: Yzv^ =;*(t#)
##+fi
Spherical :^ vzv^
=
i,V H #*('i"^
efr) *
CURL
Cartesian:
Cylindrical: V^ x^ H
Spherical: V^
x (^) H
vxr
:(*-T)ai+
(#-*)E+ffi
-T)a
=GW-Hq+W-Har+;eP -#)e
==
;fo
(es#
_
fi')
e? +
i ffiiff
_
#)ad
i(ry-*)q
*d< *
',.*
10. A lossless line having an^ air^ dielectrie^ has a^ characteristics impedance^ of^400 O.^ The^ line
is operating at 200 MHz and zi,
=200 -
j O. Find (a)^ SWR (b)^ Zu^ if^ the^ line is^ I^ m
long; (c)^ the distance from the load to the nearest voltage maximwn. 12+4+
I l. Differentiate between transmission line and waveguide. A rectangular waveguide having
cross-seclion of 2 cmx 1 cm is filled with a lossless medium char4cterized by e = 4eo and
F, =^
I. Calculate the cut-off frequency^ of the dominant mode.^ t4+
- Write short notes on antenna and its properties- t?l
DIVERGENCE
CARTESIAN
CYLINDRICAL
SPHERICAL
GRADIENT
g.5=oD'
*aD' *oD'
AxfuAz
o.D=L
uobr).;#"#
v.o=.!(,,o-\n-
I a('illD').-:
-rt rsin9 Ag rsin90$
CARTESIAN
CYI,INDRICAT-
r;Pt iliRtcAL
CURL
0v" av^
-a
r+
oy (^) -a_oz
tav^ aY-
+-a_
pod' oz
raY^ |^ av^ L--^ .L--n
r O0"o rsin? 0,1"c
At/
vv
=+a,
ox
all
^ vy (^) --d
cp
av"
vy:-a,
or
cARrEsrAN
""
o
=(+-+),,.(+
T)r,"(*-+)^,
cylrNDRrcn L vzH (^) =(1an' -a!t\u
.{9!--gg-.)a..rf
a('?H')
-9!r)u.
\p
ad oz )""
' a" ap )"' pl op
a )
cARTESTAN s'e,^ =g'{ot{ *Ef,
&2 avz 022
cYLTNDRTcAL ,', -Lg( o{)-+.
Pop\
op) p-^ o9- oz-
spHERrcAL v,^ a (^) =--L-lu(u::"') -l?'iu. .:{^
Lary--g('l)la"
rsingl a0 A )
'
rlsind d/ dr )
r{a(,n,1 afl. ).
r[ 0r A0)'
LAPLACIAN
spFrERrcAL o,o (^) =Ig(,'91)-f -1[.r9] *^ .]--!
,' 0r\ dr (^) ) ,'^ sin9^ dd^ A0^ ) r'sin"^0 0Q'
S
Yli
gS| :
Y""tt |rylre*1tig-(W
o:)
{ (^) Candidates are required to give^ their^ answers^
in their ouryt^ words^ as^
far as practicable'
/ (^) Auempt 4ll questions"
{ (^) Thefigures in the margin indicate^ Full^
Matr*s'
t
TreOresent a^ vectorard^
i.uu..rp, anci i,,r,"aprdenotes a unit vector
along the^ direction
gru"i ty the subsript.
Assume suitable data^ if^
necessary.
.*-: pelirr.
a vector field. A field^ vector is given
by an expression
i=#(""-vi,*'i.),transformthisvectorincylindricalcoordinate
system at^ point^
Given the flux^ density^ fr =12ccs0lr';a.+(sin0/13)au
Clm2,evaluate both^ sides^
of the
1r^ (^) , It
divergence theorem^ forthe^ region^
defrned by I^ <^ r <2.^
<0 <-,0
(Region 2) has lrr
= i0. if there is a uniform^ magnetic^ freid^
i=S{+e{+ 7i,Al^in
region l,find Band Hinregion2. [2+3+3]
- Find the amplitude bf the displacement current density^ in^
a metallic conductor at 60 Hz, if
)+
€=€qrp=p0,6=5.8x107S/m, and^
j
=sin(37?t (^) -117.12\a*MA/m2- t5]
ll+sl
Is+3]
?
t dD1-, dDy , dD V.U=--a=:t;
Cartesian:
Cylindrical:
Spherical:
Cylindrical:
Spherical:
0x 0Y 0z vfi-1a(PDpr+1999+ry Y'v-p aP (^) Pa oz -R td(rzD , t altT-9De)- 1 dDO V. D^ = A a, -;trs- ao- - i"o aO w= T,q;# T" w= fri#6##ao Laplacian Gradient
Cartesian:
Curl v,v=#+#+# cyrindricar: vzv =;*(r#) .if"+#
Spherical:
YzY
iV#) . #(sino#) .
Cartesian:
v xE^
-T)a
*)d
- (^) (* -*)a v x n = 1;ffi
#)
;(ry
#)a v x^ n^ == ;k(ry
H)d
iffiH
@)a
i(ry-')'; s ** " 22 TzuBHUVAN UNIVERSITY
INSTITUTE OF^ ENGINEERING
- Express^ the uniform
vector components. Examinatio n Control Division i.t:gry j-.PE]:-!!x:IJf i (^) :' M"'k' j* 2072 chaitr"^ 'if:lgr!--ini{-- (^) ir'g'---._j 3 }
. iflf1-ili{-- (^) ir'1r'---..i
3 hrs'- ---:--- : ------_j:ryr;t r"--*;eps"cle1telp"$ ----':- r' carrdidates are required to give their answers in^ their own rn'ords as^ iar^ as^ practicable' { (^) Anempt Att questions' / (^) Th"
iTur"s
in the margin indicate Full-Mstk' / (^) Neieisary tables are attached^ herewith' { /r"prurent
a rectorarid lo,0, ond
io,o.*p, denotes a^ unit vector along the^ direction given by the subscriPt. / Assume suitable data if necessary' freld i=sa. in^ (a)^ cylindrical^ components
(b) spherical
[2+3]
A .+- Derive the expression^ for the^ electric freld intensity due^ to^ an^ infrnitely long line^ charge with uniform "t -g" density p,^ by^ using Gauss's law.^
A uniform line charge density^
of 20 nClmis located^
at y (^) =3 and z=5^ ' Find^ E at P(5'6'1) [4+4] Derive an expression^
to calculate the potential^
due to a^ dipole^ in^ terms of the dipole
/-r++++ moment (^) f ol o dipole for^ which^ p=3a*-5a'+10a"nC'm
is located at^ the point
/
[4+4] (1,2,-4). Find E at^ P. Assuming that the potential v^ in^ the^ cylindrical coordinate system^
is function of^ p^ only,
solve the Lapplace's equution and derive the^ expression^
for the capacitance^
of coaxial capacitor of iengtn^ f.^ using the
same solution^ of^ V. Assume the inner conductor^ of radiUs ;;;;t"ttt"i iir trt t"ip to the conductor of radius b'^ t6l
- State and^ derive^
expression for stoke's theorem. Evaluate
the^ closed^ line integral of^ H
from
(5,4,i)to Pr(5,6'1)to Pr(0,6,1)to Po(0,4,1)to P,^ using straight^ line segments' if
H = 0.1y3^ a,+0-4xa.Alm^ '
- Define scalar magnetic
potential^ and show^
that it satisfies the Laplace's
[1+3+4] equation. Given the vector magnetic^ potential i=-(Ot t+)i,WAtm,^ calculate the total^ magrretic flux crossingtle surface^ 4:n/Z'
