See 2523 sze 2523 2
Transcript of See 2523 sze 2523 2
I CONFIDENTIAL I
FINAL EXAMINATION SEMESTER ISESSION 2009/2010
COURSE CODE
COURSE
LECTURERS
PROGRAMME
SECTION
TIME
DATE
SEE 2523 / SZE 2523
ELECTROMAGNETIC FIELD THEORY
ASSOC. PROF. DR. NORAZAN BIN MOHDKASSIMDR. YOU KOK YEOWMDM. FATIMAH BT MOHAMAD
SEC / SEE / SEI / SEL / SEM !SEP / SET /SEW
01 - 03
2 HOURS 30 MINUTES
9 NOVEMBER 2009
INSTRUCTION TO CANDIDATE
ANSWER FOUR (4) QUESTIONS ONLY.
THIS EXAMINATION BOOKLET CONSISTS OF 9 PAGES INCLUDING THE FRONTCOVER
SEE 25232
Q I (a) Determine the electric field intensity on the axis of a circular ring of uniform
charge Pi (C1m) with radius a in the .xy plane. Let the axis of the ring be
along the z axis. (7 marks)
(b) At what distance along the positive z axis is the electric field from Ql(a)
maximum, and what is the magnitude of this field? (8 marks)
(c) Find the absolute potential at any point along the z axis referring to Q 1(a).
(5 marks)
(d) Compare the result of electric field using the gradient concept.
(5 marks)
SEE 25233
Q2 (a) With the aid of suitable diagrams and Maxwell 's equations, develop theboundary condition equations for two dielectric materials with differentpermittivities.
(5 marks)
(b) Two concentric spheres are shown in Fig. Q2(b). The spheres are built fromtwo different dielectric materials . Each sphere has been covered with a thinlayer of conductor, in wh ich the th ickness can be ignored .
Fig. Q2(b)
By assuming negative charge on the inner sphere surface:
(i) Determine the electric field intensities in material #1 and material #2.(4 marks)
(ii) Obtain the energy stored in the structure.(5 marks)
(iii) Obtain the resistance between the spheres if the conductivity factor for
the material #2 is assumed to be (Yl !1S/m.(4 marks)
(iv) Given the radius r a"" 10 urn. Determine the range of structural radiu s(range of rb) such that the minimum and maximum charge perunit voltthat can be stored in material #2 are 3 F and 10 F, respectively.
(7 marks)
SEE 25234
Q3 (a) Compare the usefulness of Ampere's Circuital Law and Biot Savart Law in
determining the magnetic field intensity Bof a current carrying circuit.
(5 marks)
(b) A coaxial cable as shown in Fig. Q3(a) consists of a solid cylindrical inner
conductor having a radius a and an outer cylinder in the form of a cylindrical
shell having a radius b. If the inner conductor carries a current of I A in the
form of a uniform current density J == IIrra2 i (Alm2) and the outer
conductor carries a return current of 1 A in the form of a uniform current
density Js = -Il2rrb i (Aim). Determine:
(i) the magnitude of the force per unit length acting 10 split the outer
cylinder apart longitudinally. (7 marks)
(ii) the inductance per unit length of the coaxial cable .
(8 marks)
(e) An infinite cylindrical wire with radius a carries a uniform current density
J == l ltta? i (A/m2) , except inside an infinite cylindrical hole parallel to the
wire's axis. The hole has radius c and is tangent to the exterior of the wire.
A short chunk of the wire is shown in the accompanying Fig. Q3(b) .
Calculate the magnetic field everywhere inside the hole, and sketch the lines
ofBon the figure . (S marks)
· · · ·~· · · · · : Ia
-r----rFig. Q3(a)
y
3D view
Fig. Q3(b)
2D view
x
SEE 25235
Q4 (a) Within a certain region, e = 1O- 11F[m and f.-l = lO-sHlm. If
B; = 2 x 10- 4 cos lOSt sin 10-3 y (Testa):
(i) Use the appropriate Maxwell's equation to find E. (5 marks)
(ii) Find the total magnetic flux passing through the surface
x = 0,0< Y < 40m, 0 < z < 2m, at t = lfiS. (4 marks)
(iii) Find the value of the closed line integral of E around the perimeter of
the given surface at t = lfiS. (5 marks)
(b) A voltage source is connected by means of wire to a parallel-plate capacitor
made up of circular plates of radius a in the z =0 and z = d planes, and
having their centers on the z-axis. The electric field between the plates is
given by
_ n:TE = £0 sin 20 cos wt z
Find the total current flowing through the capacitor, assuming the region
between the plates to be free space, and that no field exists outside the region.
(7 marks)
(c) By analyzing the expression of current in Q4(b) above, give two suggestions
to increase the magnitude ofthis current. (4 marks)
Hint: Judv = uv - Jvdu
SEE 25236
Q5 (a) Derive the vector wave equation for electric field using Maxwell equation.
(4 marks)
(b) The propagation constant y can be written as y = a + jf3, where a is the
attenuation constant of the medium and f3 is its phase constant. Proof that a
and f3 can be expressed as
j1E' ( (E")2 )a=w T 1+"7 -1
j1E' ([ (E")2 )f3=WT~1+7 +1
where OJ is the angular frequency. E;' and E;" are the dielectric constant and
loss factor, respectively. J.1 is the permeability.
(8 marks)
(c) A 20 V 1m electric plane wave with frequency 500 MHz propagates in the z
direction and polarized in the x direction in a medium. The properties of medium
has relative permittivity, e r = 4.5 - jO .02 and relative permeability, J.1 r = 1 .
(i) Write a complete time-domain expression for the electric field, E(6 marks)
(ii) Determine the corresponding expression for the magnetic field.
(7 marks)
SEE 25237
ELECTROSTATIC FIELD I MAGNETOSTATIC FIELD- J dQ " I - f~ XClRCoulomb's Law E::= ---2aR Biot-Savart Law H = 4;d(247!&oR
Gauss's Law db.ds=Qm Ampere Circuital law dH· a1 = len
Force on a point charge F=EQ. Force on a moving charge F = Q(u x B)
Force on a current element F = IdI x 7fElectric field for finite line charge Magnetic field for finite current
E=.--t1..-F(Sina2 + sinal) +~(cosa2 -cosa)}- I .H =-(sina2 +sina1)¢
47l"&o r r 47lr
Electric field for infinite line charge Magnetic field for infinite current
E=J:L,. - I·H=-¢
27l"&or 27lr
Electric flux density D = cE Magnetic flux density B=pH
Electric flux If/E=Q=QD. ds Magnetic flux If/m = JB .ds
Divergence theorem QD .ds= j (v»» Stoke's theorem c}H.dI = j (vx H) dss , I s
Potential difference VAB ::= -1E .dIB
Absolute potential V J dO= 47l"~R
Gradient of potential ~ E = -V' V Magnetic potential, (A)-1 B= V' x AEnergy stored in an electric field Energy stored in a magnetic field
WE =! J(15 .E)1v w; ::= ~ !(B. H)1v2 v
Total current in a conductor
l=fJ·ds where] = 0-£
Polarization vector P =D-8aE Magnetization vector M == XmHwhere Xm =u; -1
Bound surface charge density Magnetized surface current densitypso=P.n J,m=Mxn
Volume surface charge density Magnetized volume current density
Pvo::=-Y'P ]m =YxMElectrical boundary conditions Magnetic boundary conditions
.L1" - Dg, =p, and £'1 =E21 B... = Bz,. and ~I - H2t =J, --R . I Inductance L ::=!:.- where A= 11/,,/'/esistance R=-
as I
Capacitance C =gVao
Poisson's equation V'2V ::= _frE:
Laplace equation v'v = 0
Maxwell equation y. D = Pv. V' x E =0 Maxwell equation V'. B = 0, 'YxH=J
SEE 25238
TIME VARYING FIELD
Maxwell equation V'. D== p, Gauss's Law for electric field
- a8\I x E = -- Faraday's Law
if\1·8=0
- - aD\lxH=J+
if
Gauss's Law for magnetic field
Ampere Circuital Law
Characteristics of wave propagation in lossy medium (a*O,/-l=AA,E=fioE,.)
Electric field, E(z,t) == Eoe-a: cos(a>t - /lz)xMagnetic field, H(z,t) =~e-«" cos(aJt- fJz+ 8n)y
Attenuation constan! a"Q)~~ ~I+(:J -I
where tan 28 =.!!...n OJf:
Phase constant
Intrinsic impedance
Skin depth
Poynting theorem
Poynting vector
Average power
j;i&'7=1 FLBn1+(CT I (j)£)"
I ~ = Eon,o==l/a
.r~ -) - a III " 1 2} I 2I..!\.E xH ·ds=-- -6£ +-pH v- aE dvsa,. 2 2 .'
tJ=ExHP E; -2«" e
avg =-e cos n217
SEE 25239
Kecerunan "Gradient"
Vj =xOf +Yoj+ Zojox By oz
Vj =Poj + i Of + Zojor r or/! 8z
Vj=p aj +~ Of +-L Ofar r ae rsine or/!
Kecapahan "Divergence"
- aA oA, aAV -A =-'+-"+-'
ax By oz
«: =.!.[acrAJ] +.!. OAf +aA,r or r or/! OZ
V -:4=~[O(r'A,)]+_I_ [O(Aa Sine)]+_I_OAfr' or r sine' 8e r sine' or/!
Ikal "Curl"
- _(OA, 8Ay \j _(8A, OA, ) _lOA, oA, JVx A =x 8;- az + Y a;-a; +Z a;--8;
- _[ 18A, oA; j\ "( 8A, OA,) Z[oCrA;) 8A,]VxA =r ---- +r/! --- +- ----r or/! oz oz or " r Or or/!
V x A = _P_[OCAfsine) _8Aa]+~[_1_8A, _8(rA;)] +i(8crAa) - oA, Jrsine ae ar/! r sine or/! or r or oe
Laplacian