Electromagnetic Field Theory: A Problem Solving Approach Part 73 docx

696 Radiationc Repeat a and b if the current is uniformly dis-tributed over a planar slab of thickness 2a: jo eij9-kXi, , -a... while the currents have equal magnitudes but phase diffe

which is plotted versus kL in Fig 9-14 This result can be

checked in the limit as L becomes very small (kL << 1) since the

radiation resistance should approach that of a point dipole given in Section 9-2-5 In this short dipole limit the bracketed terms in (14) are

sin kL - l - (kL) 2

)tL• i coS kL 1

2

kLSi(kL) - (kL) "

so that (14) reduces to

lim R (kL) 2 23L- L(2 = 2 8 0 L 2

(16)

which agrees with the results in Section 9-2-5 Note that for

large dipoles (kL >>1), the sine integral term dominates with

Si(kL) approaching a constant value of 7r/2 so that

lim R -7kL=60 •- r 2 (17)

PROBLEMS

Section 9-1

1 We wish to find the properties of waves propagating within a linear dielectric medium that also has an Ohmic

conductivity or.

(a) What are Maxwell's equations in this medium?

(b) Defining vector and scalar potentials, what gauge condition decouples these potentials?

(c) A point charge at r = 0 varies sinusoidally with time as Q(t) = Re (( e'") What is the scalar potential?

(d) Repeat (a)-(c) for waves in a plasma medium with constitutive law

= w eE

at

2 An infinite current sheet at z = 0 varies as

Re [K 0 e ( ' -k"-)ix].

(a) Find the vector and scalar potentials.

(b) What are the electric and magnetic fields?

696 Radiation

(c) Repeat (a) and (b) if the current is uniformly

dis-tributed over a planar slab of thickness 2a:

jo eij(9-kXi, , -a<z<a

J

3 A sphere of radius R has a uniform surface charge

dis-tribution oy= Re (&o e"•' ) where the time varying surface charge is due to a purely radial conduction current

(a) Find the scalar and vector potentials, inside and outside the sphere (Hint: rep=r 2 +R 2 -2rR cos 0; rQp drQ=

rR sin 0 dO.)

(b) What are the electric and magnetic fields everywhere?

Section 9.2

4 Find the effective lengths, radiation resistances and line charge distributions for each of the following current dis-tributions valid for I zI <dl/2 on a point electric dipole with short length dl:

(a) I(z) = Io cos az

(b) f(z) = Io e- *1 1

(c) I(z)= Io cosh az

5 What is the time-average power density, total time-average

power, and radiation resistance of a point magnetic dipole?

6 A plane wave electric field Re (Eo e i ' ) is incident upon a

perfectly conducting spherical particle of radius R that is

much smaller than the wavelength

(a) What is the induced dipole moment? (Hint: See

Section 4-4-3.)

(b) If the small particle is, instead, a pure lossless dielectric with permittivity e, what is the induced dipole moment?

(c) For both of these cases, what is the time-average scat-tered power?

7 A plane wave magnetic field Re (Ho e••) is incident upon a perfectly conducting particle that is much smaller than the wavelength

(a) What is the induced magnetic dipole moment?

(Hint: See Section 5-7-2ii and 5-5-1.)

(b) What are the re-radiated electric and magnetic fields?

(c) What is the time-average scattered power? How does it vary with frequency?

8 (a) For the magnetic dipole, how are the magnetic field lines related to the vector potential A?

(b) What is the equation of these field lines?

Section 9.3

9 Two aligned dipoles if dl and i2 dl are placed along the z axis a distance 2a apart The dipoles have the same length

while the currents have equal magnitudes but phase difference X

(a) What are the far electric and magnetic fields?

(b) What is the time-average power density?

(c) At what angles is the power density zero or maximum?

(d) For 2a = A/2, what values of X give a broadside or

end-fire array?

(e) Repeat (a)-(c) for 2N+ 1 equally spaced aligned dipoles

along the z axis with incremental phase difference Xo

10 Three dipoles of equal length dl are placed along the z

axis

(a) Find the far electric and magnetic fields

(b) What is the time average power density?

(c) For each of the following cases find the angles where the power density is zero or maximum

(iii) Is = -Is = Io, 12 = 2jIo

2ar

'I

I

1

A di

I dl'

li di

ýp Y

y

(a) Find the far fields from this current sheet.

(b) At what angles is the power density minimum or maximum?

Section 9.4

12 Find the far fields and time-average power density for each of the following current distributions on a long dipole: (a) i(z) Io ( 1 - 2z/L), O<z<L/2

SIo(1+2z/L), -L/2<z<0

Hint:

C e az

Z eaz dz = -(az - 1)

(b) I(z)= Iocos 1z/L, -L/2<z <L/2

Hint:

zi az (a cos pz + p sin pz)

e cos pz dz = e (a2+ p2)

(c) For these cases find the radiation resistance when

kL << 1.

Radiation

11 Many closely spaced point dipoles of length dl placed

along the x axis driven in phase approximate a z-directed

current sheet Re (Ko e'"'i) of length L.

SOLUTIONS TO SELECTED PROBLEMS

Chapter 1

1 Area = •a 2

3 (a) A + B = 6ix - 2i, -6i,

(b) A.B=6

(c) AxB=-14i + 12i,- 18i,

5 (b) Bi1= 2(-i.x+2i,-iý), B,= 5i + i, - 3i

7 (a) A -B = -75

(b) AxB=-100i,

(c) 0 = 126.870

12 (a) Vf = (az + 3bx 2 y)i + bx 3 i, + axi

14 (a) V A=3

17 (b) ' = 2abc

18 (a) VxA=(x-y 2

)ix-yi,-xi.

h au h,av h, a-w

(c) dV = h/h,h, du dv dw

(d) VA-A= - (h,h.Au)+ (huhA)+ (huhAw)

1 8a(hA4) 8(hA,)

25 (a) rQp = i, (b) iQ= rQ i - 5i, + 2i

5i + iy

(c)n -+

Chapter 2

4 wR'pg

3 Eo = 4

4 Q

2=

2reod Mg

4 Q•,=

L47eRE m

700 Solutions to Selected Problems

mlm 2

7 (a) m

(b) v = + 2

2-eom r ro)

r /2/2

(d)t = ro L J

qEoL 2

8 h = 2

mv

6V3

10 (b) q =- Q

12 (a) q = 2Aoa, (b) q = irpoa , (c) q = 2ooabr

15 0 = tan- 1 2 EoMg]

AL

16 (a)

E,= 2reor

18 (a) E,= -Xoa 2

7rEo[Z2 +a2]3/2

Aoa 2

20 (a) E, - 7ro(a22 23/

21 E = o(a2 23/2

27reo(a +z )

23 (c) Po (-d 2 2 ) <d

E,,= 2Eod

25 (c) Por2 r<a

3eoa

E, = 2

poa r>a

3eor>a

pod.

26 E = 1

27 W=- A

4Eo

28 (a) vo- Q ,• (b) r = 4R

2rEsoRm'

29 (a) E=-2Axi., pf= -2Aeo

31 (a) Av =oa

5o

Q

32 (a) dq= dz'

R

33 (c) V oa cos0, (d) r =rosin 0

4reor

qV,

34 (d) q

-V,

36 (a) E,- 2 InrE

-21qeoq

38 (a) xo= q6oEo' (b) vo> Itýq o/4

(c) W=

161reod

43 (e) A= , a =

R 2

2

44 (g) qT= - 47r oR

-Chapter 3

2 (a) p = AoL 2 , (e) p =QR

3Q

4 (a) po =

R-41rvoR Eo

7 (a) d= Q

Q

8 (b) -=2 rEoEo

L

R S

10 (a) Pind=PD

Vo sinhx/ld

12 (a) V(x) = 2 sinhl/ld

2 sinhl/1d

mmRAo"

15 (b) Q = q

702 Solutions to Selected Problems

A

17 (a) Dr

-27rr

A(e 2 -E1) 2E 2 A

19 (a) A'=-A " = (e - E l ) ,

A'= 12E

r<R

EoR E,=

0 r>R

02

s

In-26 (a) R =

ID(o 2 - l)

31 C= 2rl(e 2 a -elb)

e£a

(b -a) In 82

elb

33 oa(r=al)=-a( 1 -e ); T= Ie/

3a,

35 pf=po e-ar/(3sA)

Vo sinh V2RG(z - 1)

38 (a) v(z)=

sinh NRl

41 (b) 2e[E() - E 2 (O)] + ed = J(t)l

1 212

Vo \P I

42 (c) E?= Vo = I

43 (a) W=- p2- E

2P

44 W = 12ER

47 (a) W=

8weoR

48 (a) Wi.i, = 2 C V o , (b) W 4na= 0CVo

49 (b) W=-pE(cos0-1)

50 h = 2(e - Eo) 2

P.Ks

1 roA +Pod 2

52 (b) f =- (s+ d • Vo •

2 (s+d)Lso

TV 0

54 (b) f = ( -eo)

In

-a

1 Eod v

2 s

1 , 2 dC -NVTR'eo

56 (c) T = -v -d

o'Uwt

57 (a) v(t)=

-41reoR

58 (a) p,= Po e-o'lU

59 (a) nCi> + (c) > -, wo

Chapter 4

2 (a) (oo _ >

cos aye x > 0

2Ea

V=

2 cos aye x <O

2ea

sin sinh

nodd n sinh

d

Po

7 (a) V, = 2sin ax

e 0 a

2eo

V(r', )= [_-Eor_+

cos4 r>a

2eor

13 (a) E= Eo 1+ L) cos + t) -Eo -) sin i

A (t)

41TeaEo'

V In r

15 (a) V(r, z) a

Voz

17 (b) Eo - 8p R

27E

22 V(2, 2) = V(3, 2) = V(2, 3) = V(3, 3) = -4.

23 (a) V(2, 2)= -1.0000, V(3, 2)= - 5000, V(2, 3)

= - 5000, V(3, 3) = 0000 (b) V(2, 2) = 1.2500, V(3, 2) = - 2500, V(2, 3)

=.2500, V(3, 3)= -1.2500

Chapter 5

2 (b) B >2mV es (e)

-mg

3 Bo= -mg

qvo eE_

4 (d) e= E, 2

m RBo

8 (c) J = o(E+vx B)

2lAol(a 2 + b)1/2

Mrab

B2> 8b2mVo

e(b - a 2 2 )

(c) B = tan

PoKolr

12 (a) B

-4

clolL

13 (a) B# I

/olod

15 (b) B,= jod

2

,-o-a • a) lyl <a

17 (b) B= 2a 0

Yo

18 (d) y = at x = -o

21 (a) m, = qwaa

704 Solutions to Selected Problems