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wave Guidesand Resonant Cavities`

Electromagnetic fields in the presence of metallic boundaries forma practical aspect of the subject of considerable importance . At high fre-quencies where the wavelengths are of the order of meters or less the onlypractical way of generating and transmitting electromagnetic radiationinvolves metallic structures with dimensions comparable to the wave-lengths involved. In this chapter we consider first the fields in the neigh-borhood of a conductor and discuss their penetration into the surface andthe accompanying resistive losses . Then the problems of waves guided inhollow metal pipes and of resonant cavities are treated from a fairlygeneral viewpoint, with specific illustrations included along the way .Finally, dielectric wave guides are briefly described as an alternativemethod of transmission .

* In this chapter certain formulas, denoted by an asterisk on the equation number, arewritten so that they can be read as formulas in mks units provided the first factor insquare brackets is omitted. For example, (8 .12) i s

dPlosa 1 µcu~

da 4~r 4 ~Hn I2

The corresponding equation in mks form is

dPiosa µr,A

where all symbols are to be interpreted as mks symbols, perhaps with entirely differentmagnitudes and dimensions from those of the corresponding Gaussian symbols .

If an asterisk appears and there is no square bracket, the formula can be interpretedequally in Gaussian or mks symbols .

General rules for conversion of any equation into its corresponding mks form aregiven in Table 3 of the Appendix .

235

236 Classical Electrodynamics

8 .1 Fields at the Surface of and within a Conducto r

As was mentioned at the end of Section 7 .7, the-problem of reflectionand refraction of waves at an interface of two conducting media is some-what complicated. The most important and useful features of thephenomenon can, however, be obtained with an approximate treatmentvalid if one medium is a good conductor . Furthermore, the method, withinits range of validity, is applicable to situations more general than planewaves incident .

First consider a surface with unit normal n directed outward from aperfect conductor on one side into a nonconducting medium on the otherside. Then, just as in the static case, there is no electric field inside theconductors . The charges inside a perfect conductor are assumed to be somobile that they move instantly in response to changes in the fields, nomatter how rapid, and always produce the correct surface-charge density E(capital E is used to avoid confusion with the conductivity a)

in order to give zero electric field inside the perfect conducto r. Similarly,for time-varying magnet ic fields, the surface charges move in response tothe tangential magnetic field to produce always the correct surface currentK :

H47T

K (8 .2)*C

in order to have zero magnetic field inside the perfect conductor . Theother two boundary conditions are on normal B and tangential E ;

nx (E-EI)= Q

where the subscript c refers to the conductor . From these boundaryconditions we see that just outside the surface of a perfect conductor onlynormal E and tangential H fields can exist, and that the fields drop abruptly

to zero inside the perfect conductor . This behavior is indicated schemati-cally in Fig. 8.1 .

For a good, but not perfect, conductor the fields in the neighborhoodof its surface must behave approximately the same as for a perfect con-ductor. In Section 7.7 we have seen that inside a conductor the fields areattenuated exponentially in a characteristic length S, called the skin depth .For good conductors and moderate frequencies, 6 is a small fraction ofa centimeter. Consequent ly, boundary conditions (8 .1) and (8 .2) are

[Sect. 8.1] Wave Guides and Resonant Cavities 237

EH

(a) (b)

Fig. 8.1 Fields near the surface of a perfect conductor .

approximately true for a good conductor, aside from a thin transitional layerat the surface.

If we wish to examine that thin transitional region, however, care mustbe taken. First of all, Ohm's law (7.68) shows that with a finite conduct-ivity there cannot actually be a surface layer of current, as implied in (8 .2).

Instead, the boundary condition on the magnetic field i s

To explore the changes produced by a finite, rather than an infinite,conductivity we employ a successive approximation scheme . First we

assume that just outside the conductor there exists only a normal electricfield El and a tangential magnetic field HIP as for a perfect conductor .

The values of these fields are assumed to have been obtained from thesolution of an appropriate boundary-value problem . Then we use theboundary conditions and Maxwell's equations in the conductor to find thefields within the transition layer and small corrections to the fields outside .In solving Maxwell's equations within the conductor we make use of thefact that the spatial variation of the fields normal to the surface is muchmore rapid than the variations parallel to the surface . This means thatwe can safely neglect all derivatives with respect to coordinates parallelto the surface compared to the normal derivative .

If there exists a tangential HE outside the surface, boundary condition

(8.4) implies the same H,, inside the surface . With the neglect of the dis-placement current in the conductor, the curl equations in (7 .69) become

47r6(8 .5)

Hc__ icO x E

ya)

~ _ 0 -:~b

238 Classical electrodynamics

where a harmonic variation e" has been assumed . If n is the unit normaloutward from the conductor and ~ is the normal coordinate inward intothe conductor, then the gradient operator can be writte n

v

neglecting the other derivatives when operating on the fields within theconductor. With this approximation (8 .5) become :

EC ^~ -C

nx47rQ

is aEHe ^' - n x

E.Iw a~

aH ~

a~(8.6)

These can be combined to yield

2

andn • HC r--J 0

(8.7)

(8.8)

where S is the skin depth defined by (7.85) . The second equation showsthat inside the conductor H is parallel to the surface, consistent with ourboundary conditions . The solution for H, is :

He = H ie Ala e=~la (8 .9)

where H„ is the tangential magnetic field outside the surface . From (8.6)the electric field in the conductor is approximately :

E c ,ua' (1 - i)(n x H , i)e 'l' ez~lb (8 . 10)

87r cr

These solutions for H and E inside the conductor exhibit the propertiesdiscussed in Section 7.7 : (a) rapid exponential decay, (b) phase difference,(c) magnetic field much larger than the electric field. Furthermore, theyshow that, for a good conductor, the fields in the conductor are parallelto the surface * and propagate normal to it, with magnitudes which depend

only on the tangential magnetic field H11 which exists just outside thesurface.

* From the continuity of the tangential component of H and the equation connectingE to V X H on either side of the surface, one can show that there exists in the conductora small normal component of electric field, E , • n - (iivE(47rCr}E'L, but this is of the nextorder in small quantities compared with (8 .10) .

[Sect . 8 .1] Wave Guides and Resonant Cavities 239

From the boundary condition on tangential E (8 .3) we find that justoutside the surface there exists a small tangential electric field given by(8 .1Q), evaluated at ~ = 0 :

EN i}(n x H„) (8.11)8 7T6

In this approximation there is also a small normal component of B justoutside the surface . This can be obtained from Faraday's law of inductionand gives By of the same order of magnitude as E ll . The amplitudes ofthe fields both inside and outside the conductor are indicated schematicallyin Fig. 8.2.

The existence of a small tangential component of E outside the surface,in addition to the normal E and tangential H, means that there is a powerflow into the conductor . The time-average power absorbed per unit areais

dPloss CRe [n • E X H*] = r 1 'uwd 1H"1 2 (8.12)*

da 8,rr L47T 4

This result can be given a simple interpretation as ohmic losses in the bod y

E,H

Fig . 8 .2 Fields near the surface of a goo d, but not perfect, conductor.


of the conductor . According to Ohm's law, there exists a current densityJ near the surface of the conductor :

J = 6E C =~'~ga (1 - i)(n x (8 . 1 3)

The time-average rate of dissipation of energy per unit volume in ohmiclosses is -.1JJ . E* = (1 /2cr) J J12 , so that the total rate of energy dissipationin the conductor for the volume lying beneath an area elemenf. AA is

00J ' P _ AA IH12e2d= DA Hii12AA d1

2d JD 87T fo 16v

This is the same rate of energy dissipation as given by the Poynting'svector result (8.12) .

The current density J is confined to such a small thickness just belowthe surface of the conductor that it is equivalent to an effective surfacecurrent Keff

00Keff = J d~ n x H„ (8.14)*

0 47r

Comparison with (8 .2) shows that a good conductor behaves effectivelylike a perfect conductor, with the idealized surface current replaced by anequivalent surface current which is actually distributed throughout a very

small, but finite, thickness at the surface . The power loss can be written interms of the effective surface current :

dPloss - 1K

C~a 26Sfeff 1 2

This shows that 1 Ja~ plays the role of a surface resistance of the con-ductor. Equation (8 . 1 5), with Kell given by (8 .14), or (8. 1 2) will allow usto calculate approximately the resistive losses for practical cavities, trans-mission lines, and wave guides, provided we have solved for the fields inthe idealized problem of infinite conductivity .

8.2 Cylindrical Cavities and Wave Guides

A practical situation of great importance is the propagation or excitation

of electromagnetic waves in hollow metallic cylinders . If the cylinder has

end surfaces, it is called a cavity ; otherwise, a wave guide. In ourdiscussion of this problem the boundary surfaces will be assumed to be

perfect conductors . The losses occurring in practice can be accounted for

Sect . 8.2] Wave Guides and Resonant Cavities 241

adequately by the methods of Section 8 .1 . A cylindrical surface S of

general cross-sectional contour is shown in Fig . 8 .3 . For simplicity, thecross-sectional size and shape are assumed constant along the cylinder axis .With a sinusoidal time dependence e-11 for the fields inside the cylinder,Maxwell's equations take the form :

px E=i- Bc

pxB =-iµE~E

v•B= o

v •E=o(8 . 16)

where it is assumed that the cylinder is filled with a uniform nondissipativemedium having dielectric constant c and permeability µ. If follows thatboth E and B satisfy

(v2 + ~LGE ~

z E= 0 (8 .17)

G" $

Because of the cylindrical geometry it is useful to single out the spatialvariation of the fields in the z direction and to assume

IF-(X, y ) z, t) = (E(x, y)e ttikz-zW a

B(x , Y, z r t) B(x, y)efikx-amt (8 .18)

Appropriate linear combinations can be formed to give t raveling orstanding waves in the z direction . The wave number k is, at present, anunknown parameter which may be real or complex. With this assumed zdependence of the fields the wave equation reduces to the two-dimens ionalform :

r 2 E~v 2 +

(,€~

C2- k

I2 +

B= 0

where Ott is the transverse part of the Laplacian operator :

v2=v2 a2---toz2

z

(8 .19)

(8.20)

Fig. 8.3 Hollow, cylindrical wave guide of arbitrary cross-sectional shape .


It is also useful to separate the fields into components parallel to andtransverse to the z axis :

E=Ez-f'i't (8.21)where the parallel field is

Ez = (e3 ' E)e3

and the transverse field is(8.22)

Et = (e3 x E) X e3 (8.23)

and e3 is a unit vector in the z direction. Similar definitions hold for themagnetic-flux density B. Manipulation of the curl equa tions in (8.16) anduse of the explicit z dependence (8 .18) lead to the determinat ion of thetransverse fields in terms of the axial components :

Bt _

W

2 [v(!) +iµe c

" e, x p tEz]zp

E 2 _

0)a c

(8 .24)E~ - 2 1 [V t

(o~Ex ~ - i ~ e3 x V tBz

c ) az cy4E

2 -k2

)C

These relations show that it is sufficient to determine Ex and Bz as theappropriate solutions of the two-dimensional wave equation (8 .19) . Theother components can then be calculated from (8 .24) .

The boundary values on the surface of the cylinder will be taken as thosefor a perfect conductor :

a • B =O, n x E =0 (8.25)

where n is a unit normal at the surface . Since Maxwell's equations andthe boundary conditions are internally consistent, it is sufficient to notethat the vanishing of tangential E at the surface require s

Ez I s=O (8 .26)

For the normal components of B, using the expression for Bt (8.24), wefind that n • B = 0 implies

aBz ~ =0 (8.27)an s

where alan is the normal derivative at a point on the surface .The two-dimensional wave equations (8.19) for Ez and B, together with

the boundary conditions on Ez and Bz at the surface of the cylinder,specify eigenvalue problems of the usual sort . For a given frequency cv,only certain values of the axial wave number k will be consistent with

[Sect . 8 .2 1 Wave Guides and Resonant Cavities 243

the differential equation and the boundary conditions (typical wave-guide

situation) ; or, for a given k, only certain frequencies cc) will be allowed(typical resonant-cavity situation) . Because the boundary conditions on

E. and B. are different, they cannot generally be satisfied simultaneously .Consequently the fields divide themselves into two distinct categories :

TRANSVERSE MAGNETIC (TM )

Bz = 0 everywhere

The boundary condition isEzIs= O

TRANSVERSE ELECTRIC (TE)

Ez = O everywhereThe boundary condition is

aBx

an Is= Q

The designations "Electric (or E) Waves" and "Magnetic (or H) Waves"are sometimes used instead of Transverse Magnetic and TransverseElectric, respectively, corresponding to specification of the axial com-ponent of the field. In addition to these two types of fields there is adegenerate mode, called the Transverse Electromagnetic (TEM) mode, inwhich both E. and B. vanish. From (8.24) we see that, in order to havenonvanishing transverse components when both Ez and B. vanish, theaxial wave number must satisfy the condition :

k--~µE c

Thus TEM wavesboundary surfaces .now find

(8 .28)

travel as if they were in an infinite medium withoutFrom the two-dimensional wave equation (8 .19) we

eta ~BETEM

TEM}= 0 (8 .29)

showing that each component of the transverse fields satisfies Laplace'sequation of electrostatics in two dimensions . It is easy to show (a) that

both ETEm and BTEm are derivable from scalar potentials satisfyingLaplace's equation and (b) that BTEM is everywhere perpendicular to E TEM•In fact, from Faraday's law of induction we find

BTEMrCU az

(e3

X ETEM) (8 . 30)

244

With z-dependence e2 v1E '111 , we have

which is just the relation for plane waves in an infinite medium .

An immediate consequence of (8.29) is that the TEM mode cannotexist inside a single hollow, cylindrical conductor of infinite conductivity .

The surface is an equipotential ; hence the electric field vanishes inside.It is necessary to have two or more cylindrical surfaces in order to supportthe TEM mode. The familiar coaxial cable and the parallel-wire trans-mission line are structures for which this is the dominant mode . (See

Problems 8.1 and 8 .2 .)

8.3 Wave Guides

Classical Electrodynamics

BTEM e3 x ETEM (8 .31)*

We now consider the propagation of electromagnetic waves along ahollow wave guide of uniform cross section . With the z-dependence eikx,

the transverse components of the fields for the two types of waves arerelated, according to (8 .24), as follows :

TM wAvES Bt =

L' e3 X Et

ck(8 .32)

TE wAVEs E, ~-' e3 X

Bt

ck

The transverse fields are in turn determined by the longitudinal fields :

TM WAVES :

TE wAwEs

Et = ~k Vt1Y

Bt =i k2 Ot~VY

($-33)

where V is Ez (Be) for TM (TE) waves. The scalar function ip satisfies thetwo-dimensional wave equation (8.19) :

(V t2 + y2)V = 0where

2 w2 k'Y f~F e2 -

subject to the boundary condition :

4, ora

Y =0an 1,s

(8 .34)

(8 .35)

(8 .36)

for TM (TE) waves.


Equation (8 .34) for yr, together with boundary condition (8 .36), specifiesan eigenvalue problem . It is easy to see that the constant y2 must be non-negative. Roughly speaking, it is because V must be oscillatory in orderto satisfy boundary condition (8 .36) on opposite sides of the cylinder .

There will be a spectrum of eigenvalues y 22 and corresponding solutionsVz, A = 1 , 2, 3, . . . , which form an orthogonal set . These different

solutions are called the modes of the guide . For a given frequency co, thewave number k is determined for each value of A :

(,)2

C2 'k

If we define a cutoff frequency a)~,

[c] Ya

then the wave number can be written :

2z11c]-s/wE

(8 . 37)

(8.38)*

(8.39)*

We note that, for w > cv, , the wave number k. is real ; waves of the ~mode can propagate in the guide . For frequencies less than the cutofffrequency, k. is imaginary ; such modes cannot propagate and are calledcutoff modes . The behavior of the axial wave number as a function offrequency is shown qualitatively in Fig . 8.4. We see that at any givenfrequency only a finite number of modes can propagate . It is often con-venient to choose the dimensions of the guide so that at the operatingfrequency only the lowest mode can occur . This is shown by the verticalarrow on the figure .

Since the wave number k ,, is always less than the free-space value

the wavelength in the guide is always greater than the free-space

1

Fig . 8.4 Wave number kj versus

frequency w for various modes A .

wx is the cutoff frequency . W0 col W z Wa W4 W5


wavelength . In turn, the phase velocity v, is larger than the infinite spacevalue :

c~ CvP = - _k~ "/PE

1

w '

c

2

The phase velocity becomes infinite exactly at cutoff.

8.4 Modes in a Rectangular Wave Guide

(8 .40)

As an important illustration of the general features described inSection 8.3 we consider the propagation of TE waves in a rectangular wave

guide with inner dimensions a , b_, as shown in F ig . 8.5 . The wave equationfOT ip = Bzis

a 2 a2

ax2 + ay2 + y2 ) V

(8 .41)

with boundary conditions aVlan = 0 at x = 0, a and y = 0, b. Thesolution for V is consequently

( m 7rx n~ryVmn( x, y) '_ Bp cos ~

)CO S

(bIawhere

2 2m n2 = ,~2yynn a2 + b2

(8.42)

(8 .43)

The single index A specifying the modes previously is now replaced by thetwo positive integers in, n . In order that there be nontrivial solutions, mand n cannot both be zero. The cutoff frequency C.o yln is given by

+ 2.~'TCOmn --° [C] IµE (m2a 2 b

y

Fig. 8.5

(8 .44)*

Q ---~

[Sect. 8 .4] Wave Guides and Resonant Cavities 247

If a > b, the lowest cutoff frequency, that of the dominant TE mode,

occurs for m =_ 1 , n = 0 :Flo = 7rc (8

.45)1/fuEa

This corresponds to one-half of a free-space wavelength across the guide .The explicit fields for this mode, denoted by TEl,o, are :

B z = B0 cos(T )

~eixz-iwtQ

B = - Ika B sin ~

X e i '`4-iwex

71'a

a

E ~. T a ]~ sin X eilcz-iwt~, a

arc a

(8 .46)

The presence of a factor i in Bx (and E) means that there is a spatial (or

temporal) phase difference of 90° between Bx (and Ej and Bz in the

propagation direction . It happens that the TEl,a mode has the lowestcutoff frequency of both TE and TM modes,* and so is the one used in

most practical situations . For a typical choice a = 2b the ratio of cutoff

frequencies wmn for the next few modes to a)10 are as follows :

n --i-0 1 2 3

01

m 234

56

1 .002.003.004.005 .006.00

2.002.242 .843 .614.485 .39

4 . 00 6 .004 . 1 34 .485 .005.66

There is a frequency range from cutoff to twice cutoff where the TEI,o

mode is the only propagating mode . Beyond that frequency other modes

rapidly begin to enter . The field configurations of the TEi o mode and

other modes are shown in many books, e.g., American Institute of Physics

Handbook, McGraw-Hill, New York (1957), p. 5-61 .

* This is evident if we note that for the TM modes Ez is of the for m

m7rx naryEz = Eo sin ~ a sin -

while y2 is still given by (8 .43) . The lowest mode has m = n = 12

greater than that of the TEI, ,, mode by the factor 1 +a

b2~

Its cutoff frequency is


8.5 Energy Flow and Attenuation in Wave Guide s

The general discussion of Section 8 .3 for a cylindrical wave guide ofarbitrary cross-sectional shape can be extended to include the flow ofenergy along the guide and the attenuation of the waves due to losses in thewalls having finite conductivity . The treatment will be restricted to onemode at a time ; degenerate modes will be mentioned only briefly . Theflow of energy is described by the complex Poynting's vecto r

S = 1c

] 1(E X H*) (8.47)*

47r 2

whose real part gives the time-averaged flux of energy . For the two typesof field we find, using (8 .24) :

S=

2E [e,,, IV ~V12 + i VV~V*]

wk k

87ry4 1_~e3

2

o t P l 2 - i ~ V

(8 .48)

where the upper (lower) line is for TM (TE) modes . Since V is generallyreal, * we see that the transverse component of S represents reactive energyflow and does not contribute to the time-average flux of energy. On theother hand, the axial component of S gives the time-averaged flow ofenergy along the guide. To evaluate the total power flow P we integratethe axial component of S over the cross-sectional area A

E

P = S • e3 daa'k4 I1l1(Vv)* • (~',~') da

A 87ry µ A(8.49)

By means of Green's first identity (1 .34) applied to two dimensions, (8 .49)can be written :

EP _ wka

I [*dlJ*v2da]

µ8~ry c an

(8 .50)

where the first integral is around the curve C which defines the boundarysurface of the cylinder . This integral vanishes for both types of field s

* It is possible to excite a guide in such a manner that a given mode or linear combina-

tion of modes has a complex V. Then a time-averaged transverse energy flow can occur .Since it is a circulatory flow, however, it really only represents stored energy and is notof great practical importance .

[Sect. 8.5] Wave Guides and Resonant Cavities 249

because of boundary conditions (8 .36). By means of the wave equation(8 .34) the second integral may be reduced to the normalization integral forip. Consequently the transmitted power i s

2

P-

4~r1 I E \ w~~ ( I2 ~µ

~j 'EcJ 2

)1 y~'~ ~ d a

w fA(8.53) *

where the upper (lower) line is for TM (TE) modes, and we have exhibitedall the frequency dependence explicitly .

It is straightforward to calculate the field energy per unit length of theguide in the same way as the power flow . The result i s

= -(21€)~ da

42 w f,µ

(8 .52)*

Comparison with the power flow P shows that P and U are proportional .The constant of proportionality has the dimensions of velocity (velocityof energy flow) and is just the group velocity :

P _ kc2 =U co JU E

C1 -

(,),1

- vvIfGE CU2

(8.53)

as can be verified by a direct calculation of vg = dw f dk from (8 .39),assuming that the dielectric filling the guide is nondispersive . We notethat vp is always less than the velocity of waves in an infinite medium andfalls to zero at cutoff. The product of phase velocity (8 .40) and groupvelocity is constant :

VA, =C

(8.54)µE

an immediate consequence of the fact that co 0w cc k Ak.Our considerations so far have applied to wave guides with perfectly

conducting walls. The axial wave number k A was either real or purelyimaginary . If the walls have a finite conductivity, there will be ohmiclosses and the power flow along the guide will be attenuated. For wallswith large conductivity the wave number will have a small imaginary part :

where k ( °) is the value for perfectly conducting walls . The attenuationconstant flx can be found either by solving the boundary-value problemover again with boundary conditions appropriate for finite conductivity,or by calculating the ohmic losses by the methods of Section 8 .1 and


using conservation of energy. We will use the latter technique . Thepower flow along the guide will be given b y

P(z) = Poe-~#,, z

Thus the attenuation con stant is given by

1 dP

2P dz

(8.56)*

(8.57) *

where -dPldz is the power dissipated in ohmic losses per unit length of theguide. According to the results of Section $ .l, this power loss is

dP c2 1in x B l2 dl (8 .58) *

~ ~~µ~ Cdz 1677-2 2

where the integral is around the boundary of the guide . With fields (8 .32)and (8.33) it is easy to show that for a given mode :

C 2

aP ~z '~

Wk2 a n- =

2z 27r ~ 1d 3 0~µ 2M. c c2

µECOA

z

2 y dl

- ~~ In x vtV l 2 + ~ l V I 2w 2 r,0 ~

(8.59)where again the upper (lower) line applies to TM (TE) modes .

Since the transverse derivatives of V are determined entirely by the sizeand shape of the wave guide, the frequency dependence of the power lossis explicitly exhibited in (8 .59). In fact, the integrals in (8 .59) may be simplyestimated from the fact that for each mode :

Vt 2+

2) V = 0 (8.60)

(

This means that, in some average sense, and barring exceptional circum-stances, the transverse derivatives of V must be of the order of magnitude

of 1/µE(COj/c)V

~n 2~^-~ Cln x Vt y l2) µEC;~ (~ v 12) (8.61)

Consequently the line integrals in (8 .59) can be related to the normalizationintegral of IV12 over the area . For example ,

G22 LV 2 = ~,µE C J i Vj 2 dLd (8.62)c coy an A A

where C is the circumference and A is the area of cross section, while ~ ,~ isa dimensionless number of the order of unity. Without further knowledge

[Scct. 8 . S] Wave Guides and Resonant Cavities 251

of the shape of the guide we can obtain the order of magnitude of theattenuation constant 0,1 and exhibit completely its frequency dependence .Thus, using (8.59 ) with (8.62) and (8.51), plus the frequency dependence ofthe skin depth (7 .85), we find

2

1 E I ( C~'z + nz

(WA)2]

2)~iµ ~ a x

CO14C77- 6 2A w

w 2

where a is the conductivity (assumed independent of frequency), S, is theskin depth at the cutoff frequency, and qA are dimensionless numbersof the order of unity. For TM modes, = 0 .

For a given cross-sectional geometry it is a straightforward matter tocalculate the dimensionless parameters ~,a and qz in (8 .63) . For the TEmodes with n = 0 in a rectangular guide, the values are $?2,0 = al(a + b)and %, ,o = 2bl(a + b) . For reasonable relative dimensions, theseparameters are of order unity, as expected.

Z3

2 3 4 5w/w X--- )..

Fig . 8.6 Attenuation constant ftxas a function of frequency fortypical TE and TM modes. ForTM modes the minimum atten-

uation occurs at w /wA = V 3 , re -gardless of cross-sectional shape .

The general behavior of ,B A as a function of frequency is shown inFig. 8.6. Minimum attenuation occurs at a frequency well above cutoff.For TE modes the relative magnitudes of ~A and q,; depend on the shapeof the guide and on A. Consequently no general statement can be madeabout the exact frequency for minimum attenuation . But for TM modes

the minimum always occurs at com;, = 1/3 wA . At high frequencies theattenuation increases as co'2. In the microwave region typical attenuationconstants for copper guides are of the order flz , 10-4cU,AJc, giving 1/edistances of 200-400 meters .

The approximations employed in obtaining (8 .63) break down close tocutoff. Evidence for this is the physically impossible, infinite value of(8.63) at cc ) = w.,. A treatment of the problem by perturbation theory

252 Classical Flectrodynamics

with the boundary condition (8 .11) yields the more accurate result ,

k 1 2 = k (0) 2 + 2( t + i )k(O)fla (8.64)

where #,j is still given by (8.63) For k (° ) j #A this reduces to our previousexpression (S.SS) . But at cutoff (kV) = 0) the wave number is now finite

with real and imaginary parts of the order of the geometrical mean ofwAf c and a typical value of P;L, say at w, 2wj.

In the discussion so far we have considered only one mode at a time .This procedure fails whenever a TE and a TM mode have the same cutofffrequency, as occurs in the rectangular guide, for example, with n :5'~ 0,m 0 0. The reason for the failure is that the boundary condition (8 .11)for finite conductivity couples the degenerate modes . The calculation ofthe attenuation then involves so-called degenerate state perturbationtheory, and the expression for P takes the form,

N = WT M + PTE) ± V OTNi - NTE)2 + I K I2 (8 .65)

where PTM and #TE are the values found above, while K is a couplingparameter. The two values of # in (8 .65) give the attenuation for thetwo orthogonal, mixed modes which satisfy the perturbed boundaryconditions . '

8 .6 Resonant Cavities

Although an electromagnetic cavity resonator can be of any shapewhatsoever, an important class of cavities is produced by placing endfaces on a length of cylindrical wave guide . We will assume that the endsurfaces are plane and perpendicular to the axis of the cylinder. As usual,the walls of the cavity are taken to have infinite conductivity, while thecavity is filled with a lossless dielectric with constants µ, E . Because ofreflections at the end surfaces the z dependence of the fields will be thatappropriate to standing waves :

A sin kz + B cos kz

If the plane boundary surfaces are at z ~ 0 and z = d, the boundary,conditions can be satisfied at each surface only if

k=P~ p=0,1,2, . . . (8 . 66)

* For the theory of perturbation of boundary conditions in guides and cavities, see

G . Goubau, Electromagnetic Waveguides and Cavities, Pergamon Press, New York,1961 ; Sect . 25 . Attenuation for degenerate modes in guides is treated by R. Muller,Z . Naturforsch., 4a, 218 (1949), and for the rectangular cavity by the same author in

Sect. 37 of the book by Goubau.

[Sect. 8.6] Wave Guides and Resonant Cavities

For TM fields the vanishing of Et at z = 0 and z = d require s

E. - V~x, y) cos ( ~7TZ~~, P - 0 , 1 , 2, . . .

d

253

(8.67)

Similarly for TE fields, the vanishing of Bx at z = 0 and z = d requires

B z = ip(x, y) sin p = 1 , 2, 3 , . . .~p~zl

Then from (8 .24) we find the transverse fields :

TM FIELDS

pTr zEt = -dye sin d('v p

B t = Iµ 2 cos (P)e3 X O tt'cy d

TE FIELDS

Et ccu2 sin (p ?TZ) ea x 4 th

cy d

B t = cos ()vd

(8 .68 )

(8 .69)

(8.70)

The boundary conditions at the ends of the cavity are now explicitlysatisfied . There remains the eigenvalue problem (8.34)-(8 .36), as before .But now the constant y2 is :

2 ()2

YYE (8.71)

c2 d

For each value of p the eigenvalue y.12 determines an eigenfrequency ofresonance frequency w,p

~2v =I

C2

][y.~'2 +" ()21 (8 . 72)*

,uf d

and the corresponding fields of that resonant mode . The resonancefrequencies form a discrete set which can be determined graphically on thefigure of axial wave number k versus frequency in a wave guide (see p . 245)

by demanding that k = per f d. It is usually expedient to choose the

various dimensions of the cavity so that the resonant frequency of operationlies well separated from other resonant frequencies . Then the cavity will

be relatively stable in operation and insensitive to perturbing effectsassociated with frequency drifts, changes in loading, etc .


z

y

An important practical resonant cavity is the right circular cylinder,perhaps with a piston to allow tuning by varying the height . The cylinderis shown in Fig . 8.7, with inner radius R and length d. For a TM modethe transverse wave equation for V = E, subject to the boundary con-dition Ez = 0 at p = R, has the solution :

V(P ' 0) = Jryri.(YmnP)etamO (8 .73)where

XMn (8.74)R

xm . is the nth root of the equation, T?z(x) = 0. These roots are given on

page 72, below equation (3.92). The integers m and n take on the values

in = 0, 1, 2, . . . , and n = 1, 2, 3, . . . . The resonance frequencies are

given by

x2 27f 2

~mnP = Eel

R

m2 + ~2 (8.75)*

V ~

The lowest TM mode has in = 0, n = 1, p = 0, and so is designatedTMo110o• Its resonance frequency i s

~oio = 2.405 c

(8 .76). .,/fcE R

The explicit expressions for the fields ar e

tZ.4o5n - iWtEx = EaJo R e(8 .77)

BO = - iViWEEoJf 2.405p1e_zwt

R

The resonant frequency for this mode is independent of d. Consequentlysimple tuning is impossible .

x t+' ig. 8.7

[Sect . 8.7] Wave Guides and Resonant Cavities 255

For TE modes, the basic solution (8 .73) still applies, but the boundary

condition on BZ(av

= 0makesP x

yinn = xm R n (8 .78)

where xynn is the nth root of J,,'(x) = 0 . These roots, for a few values ofm and n, are tabulated below :

Roots of Jm'{x) = 0

in = 0 :

m - 1 :

m = 2 :

m = 3 :

xon = 3 .832 , 7 .016, 10 . 1 74 , . . .

xin= 1 .841 , 5 . 331 , 8 . 536 , . . .

X12-n = 3 .054 , 6 .706, 9 .970 . . . .

x3n = 4 .201 , 8 .015 , 11 . 336 , . . .

The resonance frequencies are given by

[C

]xi2 p2~2 f

+ 2 }mnv,/µ

ER

~ 2 d(8 .79)*

where m = 0, 1 , 2, , . . , but n, p = 1, 2, 3, . . . . The lowest TE mode hasm = n = p = 1, and is denoted TE ,rl,z . Its resonance frequency i s

!Iy 1 .841 cWZA~ T R

while the fields are derivable from

R2+2.912-

d2

) e -zU'tB,, = BaJ; ~R P)

cos 0 sin d

(8.80)

(8 .81 )

by means of (8.70) . For d large enough (d > 2.03R), the resonancefrequency 0)111 is smaller than that for the lowest TM mode (8.76) . Thenthe TEI,I,, mode is the fundamental oscillation of the cavity . Because thefrequency depends on the ratio d1R it is possible to provide easy tuning by

making the separation of the end faces adjustable .

8.7 Power Losses in a Cavity ; Q of a Cavity

In the preceding section it was found that resonant cavities had discretefrequencies of oscillation with a definite field configuration for eachresonance frequency . This implies that, if one were attempting to excite aparticular mode of oscillation in a cavity by some means, no fields of the


right sort could be built up unless the exciting frequency were exactly equalto the chosen resonance frequency . In actual fact there will not be a deltafunction singularity, but rather a narrow band of frequencies around theeigenfrequency over which appreciable excitation can occur . An importantsource of this smearing out of the sharp frequency of oscillation is thedissipation of energy in the cavity walls and perhaps in the dielectric fillingthe cavity . A measure of the sharpness of response of the cavity to externalexcitation is the Q of the cavity, defined as 2 7r times the ratio of thetime-averaged energy stored in the cavity to the energy loss per cycle :

Q = wo Stored energy(8 .82)*

Power loss

Here wo is the resonance frequency, assuming no losses . By conservationof energy the power dissipated in ohmic losses is the negative of the timerate of change of stored energy U. Thus from (8.82) we can write anequation for the behavior of U as a function of time :

dU_ cuo U

d t Q (8.83)Wlth solution

U(t) = Uoe wO t/Q

If an initial amount of energy U. is stored in the cavity, it decays awayexponentially with a decay constant inversely proportional to Q. Thetime dependence in (8 .83) implies that the oscillations of the fields in thecavity are damped as follows :

E(t) = doewo t/2 Qe-z cuot (8.84)

A damped oscillation such as this has not a pure frequency, but a super-position of frequencies around w = W . Thus,

E(t) =

where

~ f"QE(co)e-"tdco

E(w) = 1 °°

z~ f E e ~'Oti2Qez(w-wv ) t dt

(8.85)

The integral in (8 .85) is elementary and leads to a frequency distribution

for the energy in the cavity having a Lorentz line shape :

IE((t))1' °c Q )r) - ~o) 2 + (cua 12 2

(8 .86)

[Sect . 8 .7] Wave Guides and Resonant Cavities 25 7

The resonance shape (8 .86), shown in Fig . 8.8, has a full width at halfmaximum (confusingly called the half-width) equal to (f)b I Q . For aconstant input voltage, the energy of oscillation in the cavity as a functionof frequency will follow the resonance curve in the neighborhood of a

particular resonant frequency . Thus, if A(< ) is the frequency separationbetween half-power points, the Q of the cavity i s

Q "' (8 . 87)

Ao)

Q values of several hundreds or thousands are common for microwavecavities .

To determine the Q of a cavity we must calculate the time-averaged

energy stored in it and then determine the power loss in the walls. Thecomputations are very similiar to those done in Section 8.5 for attenuation

in wave guides. We will consider here only the cylindrical cavities of

Section 8 .6, assuming no degeneracies (see the footnote on p . 252). Theenergy stored in the cavity for the mode Z, p is, according to (8 .67)-(8.70) :

E

U

_ a i,r 4

i,u[-4L

(IF7T'IIfjVj2da

yxd A

where the upper (lower) line applies to TM (TE) modes . For the TMmodes with p = 0 the result must be multiplied by 2 .

The power loss can be calculated by a modification of (8 .58) :

loss = [_c22 1Z Cfe

dl fddz I n x B J aides + x ~ daln x167r 2aa,~ o

For TM modes w ith p =,4 0 it is easy to show that

B1ende] (8 .89)*

pions = [_c' + (.E1L)2l 1 + ~z Cd) 1X12 da (8 .90)lb~r2 68,u yxd 4A A

Fig. 8.8 Resonance line shape . The

full width loco at Waif maximum (of

the power) is equal to the centralfrequency wo divided by the Q of the

cavity . co 0 u► ;I-


where the dimensionless number ~ , is the same one that appears in (8 .62),C is the circumference of the cavity, and A is its cross-sectional area . Forp = 0, ~x must be replaced by 2~x . Combining (8 .88) and (8.89) accordingto (8.82), and using definition ('7 .85) for the skin depth b, we find the Q ofthe cavity :

Cd),U'2 1 + ~a 4

A

where ,u , is the permeability of the metal walls of the cavity. For p = 0modes, (8.91) must be multiplied by 2 and ~ A replaced by 2~x. Thisexpression for Q has an intuitive physical interpretation when written inthe form :

V

Q = µ~ sbx (Geometrical factor) (8 .92) *

where V is the volume of the cavity, and S its total surface area. The Q ofa cavity is evidently, apart from a geometrical factor, the ratio of thevolume occupied by the fields to the volume of the conductor into whichthe fields penetrate because of the finite conductivity. For TM modes incylindrical cavities the geometrical factor i s

+2ACd)

(8.93)

(1+A )4A

for p =A 0, and isCd

21 + 2A(8.94)

+ Cd

AlI2

for p = 0 modes . For TE modes in the cylindrical cavity the geometricalfactor is somewhat more complicated, but of the same order of magnitude .

For the TMa,i .o mode in a circular cylindrical cavity with fields (8 .77),$,j = 1 (true for all TM modes), so that the geometrical factor is 2 andQ is :

+dl 6

RI

[Sect. 8 .8] Wave Guides and Resonant Cavities

For the TEl , 1 mode calculation yields a geometrical factor *

2

d (1+o.344)R21 + ---

) 3 1R ( 1 -f- 0 .209

d-F 0 .242R R

3I

and a Q2

d

~

1 + 0 .344 R23 `2

~`,

`~ 1 ~+- 0 .209 d ~{- 0 .242 d3 1R R

259

(8 .96)

(8.97)*

Expression (8 .92) for Q applies not only to cylindrical cavities but alsoto cavities of arbitrary shape, with an appropriate geometrical factor of

the order of unity .

8.8 Dielectric Wave Guides

In Sections 8 .2-8.5 we considered wave guides made of hollow metalcylinders with fields only inside the hollow . Other guiding structures arepossible . The parallel-wire transmission line is an example . The generalrequirement for a guide of electromagnetic waves is that there be a flow ofenergy only along the guiding structure and not perpendicular to it . Thismeans that the fields will be appreciable only in the immediate neighbor-hood of the guiding structure . For hollow wave guides these requirementsare satisfied in a trivial way . But for an open structure like the parallel-wire line the fields extend somewhat away from the conductors, falling offlike P-2 for the TEM mode, and exponentially for higher modes .

A dielectric cylinder, such as shown in Fig . 8 .9, can serve as a wave guide,with some properties very similar to those of a hollow metal guide if itsdielectric constant is large enough. There are, however, characteristicdifferences which arise because of the very different boundary conditionsto be satisfied at the surface of the cylinder . The general considerations ofSection 8 .2 still apply, except that the transverse behavior of the fields is

governed by two equations like ($ . 1 9), one for inside the cylinder and onefor outside :

I NSIDE

c ) 2 E2 (8.98)1 2 8B)

* Note that this factor varies by only 30 per cent as the cylinder geometry is changedfrom dfR > 1 to d/R < 1 .


OUTSIDE! 2

1Vt2 + I ruo ,,o ~ - k2 ~~ j

= 0 (8.99)c

Both dielectric (,ul, f1) and surrounding medium (,uo, Eo) are assumed to beuniform and isotropic in their properties . The axial propagation constantk must be the same inside and outside the cylinder in order to satisfyboundary conditions at all points on the surface at all times .

In the usual way, inside the dielectric cylinder the transverse Laplacianof the fields must be negative so that the constant

2

2

~

= tCGI El 2 - k2 (8 . 100)

is positive. Outside the cylinder, however, the requirement of no transverseflow of energy demands that the fields fall off exponentially . (There is no

TEM mode for a dielectric guide .) Consequently, the quantity in (8 .99)equivalent to y2 must be negative. Therefore we define a quantity X32 :

209

~j2 V "'-JUOEO 2

C

and demand that acceptable wave guide solutions have ,B2 positive (~ real) .The oscillatory solutions (inside) must be matched to the exponentia l

solutions (outside) at the boundary of the dielectric cylinder. Theboundary conditions are continuity of normal B and D and tangential Eand H, rather than the vanishing of normal B and tangential E (8 .25)

appropriate for hollow conductors. Because of the more involved

boundary conditions the types of fields do not separate into TE and TMmodes, except in special circumstances such as azimuthal symmetry in

AG

z

x

Fig . 8 .9 Section of d ielectric wave

guide.

[Sect. 8.81 Wave Guides and Resonant Cavities 261

circular cylinders, to be discussed below. In general, axial components

of both E and B exist . Such waves are sometimes designated as HEmodes.

To illustrate some of the features of the dielectric wave guide we considera circular cylinder of radius a consisting of nonpermeable dielectric withdielectric constant fl in an external nonpermeable medium with dielectricconstant e.. As a simplifying assumption we take the fields to have no

azimuthal variation . Then in cylindrical coordinates the radial equations

for Ew or B. are Bessel's equations :

d2 1 d

dp p dp

d` l d

dp p dp

p C a

p > a

(8 .102)

The solution, satisfying the requirements of finiteness at the origin and atinfinity, is found from Section 3 .6 to be :

jo(YP)ti p < aIL AKo(PP), p > a

(8 . 103)

The other components of E and B can be found from (8 .24) when therelative amounts of E. and B. are known. With no ~ dependence to thefields, (8.24) reduces to

INS I DE

ik aBzBp =

Y2 a p

EO B .

ck

1 ElW aEzB~ = Y

2C ap

ckEP = BOE

I(O

(8 .104)

and similar expressions for p > a . The fact that the fields arrange them-selves in two groups, (Be, E,,) depending on B, and (BO, E) depending on

Ez, suggests that we attempt to obtain solutions of the TE or TM type, asfor the metal wave guides . For the TE modes, the fields are explicitly

Bz = •1a(YP)

ikBp = - ~ fAYA)

Y

Ey j1(YP)cy

P ca (8 .105)

262 Classical Electrodynamic s

and

Bz = AKflOp)

Bp _ ikA KAM p> a (8 .106)

EO =

_ icvA Kj( I~P)

c~

These fields must satisfy the standard boundary conditions at p = a. Thisleads to the two conditions,

AKo(Oa) = Ju(Ya)

Ya) 8_ A KI(fla) J~( . 1 a7)

0 Y

Upon elimination of the constant A we obtain the determining equationsfor y, fl , and therefore k:

Ji(Ya) + Ki«a ) = 0YJo(YQ) #Ko (Oa)

and, from (8.100) and (8 .101), (8.108)

Y

I

+ ~2

= (

E1 -EO)

C

The general behavior of the two parts of the first equation in (8 .108) isshown in Fig. 8 .10a. Figure 8 .10b shows the two curves superposed

~c d

-~ Oa

- •

(a)

rya+~-

_ Jt

yJo

K1

~Ko

(b)

~ 'Yalmax ~E a

V 1- to 1-7

-•

Fig. 8.10 Graphical determination of the axial propagation constant for a

dielectric wave guide .


according to the second equation in (8 .108). The frequency is assumed tobe high enough that two modes, marked by the circles at the intersectionsof the two curves, exist . The vertical asymptotes are given by the roots ofJ0(x) = 0 . If the maximum value of ya is smaller than the first root(xol = 2.405), there can b e no intersection of the two curves for real ~.Hence the lowest "cutoff" frequency for TEo,n waves is given b y

0of =2.405c

(8.109)

At

- Ed a

At this frequency ~2 = 0, but the axial wave number k is stxXl real and

equal to its free-space value VepcvJe . Immediately below this "cutoff"frequency, the system no longer acts as a guide but as an antenna, withenergy being radiated radially . For frequencies well above cutoff, fl andk are of the same order of magnitude and are large compared to y providedFl and Ea are not nearly equal .

For TM modes, the first equation in (8 .108) is replaced b y

,(Y ~ + Eo K(~aa = 0 (8.110)

Y o(Y) i P o(fl

It is evident that all the qualitative features shown in Fig . 8 .10 are retainedfor the TM waves . The lowest "cutoff" frequency for TMfl n waves isclearly the same as for TE ❑ ,n waves . For Ei > EQ, provided the maximumvalue of ya does not fall very near one of the roots of Jo(x) = 0, (8 .110)shows that the propagation constants are determined by J,(ya) -- 0 . Thisis just the determining equation for TE waves in a metallic wave guide .The reason for the equivalence of the TM modes in a dielectric guide andthe TE modes in a hollow metallic guide can be traced to the symmetryof Maxwell's equations under the interchange of E and B (with appro-

priate sign changes and factors of 1I,uE ), plus the correspondence betweenthe vanishing of normal B at the metallic surface and the almost vanishingof normal E at the dielectric surface (due to continuity of normal D with

E1 > co) .If El > Eo, then from 100) and (8 . 101) it is clear that the outside decay

constant fl is much larger than y, except near cutoff . This means that thefields do not extend appreciably outside the dielectric cylinder . Figure8 .11 shows qualitatively the behavior of the fields for the TEo,1 mode . Theother modes behave similarly . As mentioned earlier, modes with azimuthal

dependence to the fields have longitudinal components of both E and B .Although the mathematics is somewhat more involved (see Problem 8 .6),the qualitative features of propagation-short wavelength along thecylinder, rapid decrease of fields outside the cylinder, etc .-are the sameas for the circularly symmetric modes .


B,E

F ig . 8.11 Radial variation offields of TEO,i mode in dielectric

guide. For E, > E,,, the fieldsare confined mostly inside the

dielectric .

Dielectric wave guides have not been used for microwave propagation,except for special applications . One reason is that it is difficult to obtainsuitable dielectrics with sufficiently low losses at microwave frequencies .In a recent application at optical frequencies very fine dielectric filaments,each coated with a thin layer of material of much lower index of refraction,are closely bundled together to form image-transfer devices .* Thefilaments are sufficiently small in diameter (ti 10 microns) that wave-guideconcepts are useful, even though the propagation is usually a mixture ofseveral modes .

REFERENCES AND SUGGESTED READING

Wave guides and resonant cavities are dealt with in numerous electrical and communi-cations engineering books . Among the physics textbooks which discuss guides, trans-mission lines, and cavities ar e

Panofsky and Phillips, Chapter 12,

Slater ,Sommerfeld, Electrodynamics, Sections 22-25,Stratton, Sections 9 .18-9 .22 .

The mathematical tools for the discussion of these boundary-value problems arepresented by

Morse and Feshbach, especially Chapter 13 .Information on special functions may be found in the ever-reliable

Magnus and Oberhettinger .

Numerical values of Bessel functions are given byJahnke and Emde ,

Watson .

PROBLEMS

8.1 A transmission line consisting of two concentric circular cylinders of metalwith conductivity d and skin depth 6 , as shown on p . 265, is filled with a

* B. O'Brien, Physics Today, 13, 52 (1960) .

[Probs. 8] Wave Guides and Resonant Cavities 265

un iform lossless dielectric ( f r. , E ) . A TEM mode is propagated along this line.(a) Show that the time-averaged power flow along the line i s

P = C ~l 7 r Q2(b)

[L/1o

n where H. is the peak value of the azimuthal magnetic fie ld at the surface ofthe inner conductor .

~F--b--~

Sb ]

(b) Show that the transmitted power is attenuated along the line a s

P(z) = Pae -2yzwhere

I E ~a + 6}

~4

c

~1 20 yIn

(b)-a

(c) The characteristic impedance Z . of the line is defined as the ratio ofthe voltage between the cylinders to the axial current flowing in one of themat any position z. Show that for this lin e

z 4 7r 1 In

c 2~r E

(b)

a

(d) Show that the series resistance and inductance per unit length of theline are

R = 11 + 1

27r6O a b

L =[4,g] arc In + ~r~

'r5 1 +

c2 2n(-")a47 r ~ a

where it, is the permeability of the conductor . The correction to theinductance comes from the penetration of the flux into the conductors by adistance of order O .

8 .2 A transmission line consists of two iden tical thin strips of meta l , shown incross section on p. 266 . Assum ing that b j a, discuss the p ropagation

266 Classical Electrodynamic s

of a TEM mode on this line, repeating the derivations of Problem 8 .1,Show that

1, = b y IHa2

47 ]

a

2~ .JE

C 1 E

Y - ~4~~ a6cS it

µ a

Z°-

[4,T]

ce ~b

1R 266b

L ~ 4, µa + µeb

c2] b 1

b

,-~ li Ilo---

where the symbols have the same meanings as in Problem 8 .1 .8.3 Transverse electric and magnetic waves are propagated along a hollow,

right circular cylinder of brass with inner radius R .(a) Find the cutoff frequencies of the various TE and TM modes . Deter-

mine numerically the lowest cutoff frequency (the dominant mode) in termsof the tube radius and the ratio of cutoff frequencies of the next four highermodes to that of the dominant mode .

(b) Calculate the attenuation constant of the wave guide as a function offrequency for the lowest two modes and plot it as a function of frequency .

8.4 A wave guide is constructed so that the cross section of the guide forms aright triangle with sides of length a, a, 'V2a, as shown on p . 267. Themedium inside has ,u = f = 1 .

[Probs. 8 ] Wave Guides and Resonant Cavities 267

(a) Assuming infinite conductivity for the walls, determine the possiblemodes of propagation and their cutoff frequencies .

Ta

(b) For the lowest modes of each type calculate the attenuation constant ,assuming that the walls have large, but finite, conductivity . Compare theresult with that for a square guide of side a made from the same material .

8 .5 A resonant cavity of copper consists of a hollow, right circular cylinder o finner radius R and length L, with flat end faces .

(a) Determine the resonant frequencies of the cavity for all types of

waves . With (cl N/1ceR) as a unit of frequency, plot the lowest four resonan tfrequencies of each type as a function of RIL for 0 < RfL c 2. Does thesame mode have the lowest frequency for all RJL ?

(b) If R = 2 cm, L = 3 cm, and the cavity is made of pure copper, whatis the numerical value of Q for the lowest resonant mode ?

$.6 A right circular cylinder of nonpermeable d ielectric with , dielectr ic constant Eand rad ius a serves as a dielectric wave guide in vacuum ,

(a) Discuss the propagation of waves along such a guide, assum ing thatthe azimuthal variat ion of the fields is e

(b) For m = ± 1, determine the mode with the lowest cutoff frequenc yand discuss the propert ies of its fields (cutoff frequency, spat ial variation ,etc.) , assuming that e > 1 .

t om- a -~1

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