Joule Loss in a "Perfect" Conductor in a Magnetic Field

by

J . M . Goodman and C . R . Legendy

1 May, 1964

Cornell University

Ithaca, New York

Report #201

Issued by

The Materials Science Center

Joule Loss in a "Perfect" Conductor in a Magnetic Field*

J . M . Goodman+ and C . R . Legendy

Laboratory of Atomic and Solid State PhysicsPhysics Department, Cornell University

Ithaca, New York

ABSTRACT

We present calculations of the skin effect in the

presence of a magnetic field exhibiting some novel features

in the dependence of the power absorption on frequency

and conductivity . In particular, in the limit of classi-

of finite electron mean free path and scattering time are

neglected) the loss tends to a nonzero limit independent

of the conductivity and is confined to the surface . For

available fields and sample conductivities, this limiting

case is a good approximation at low frequencies and is

relevant to many studies of helicons in metals . We also

present and analyze data from a preliminary experiment on

sodium at 4 .2o in a 55 kG magnetic field for frequencies

below a few kilocycles .

2 -

I . INTRODUCTION

When an alternating electromagnetic field is incident upon a

metallic surface, there will be induced currents and a consequent

Joule loss . If there is no static magnetic field present, as is the

case in the usual skin effect, this loss is proportional to the

square root of the resistivity . In this paper we discuss how this

result is modified by the presence of a static magnetic field . We

calculate the Joule loss directly, from the integral of j 2 p over t e

volume, and also nd rectly, by calculat n t e power absorbed rom

t e sources o t e nc dent eld . We nd t at w en t e sur ace o

t e metal s parallel to t e stat c ma net c eld t e loss, nstead

o decreas n w t t e res st v ty, tends to a nonzero l m t n value

n a certa n conduct v ty low requency l m t w c we call t e

class cal "per ect" conductor . T s l m t n case s not a super-

conductor (obey n t e London equat ons), nor s t a normal conductor

(or "coll s onless plasma") w t an n n te scatter n t me and mean

ree pat or ts carr ers . Instead, by class cal "per ect" con-

duct v ty we mean t e ypot et cal l m t n w c t e macroscop c

conduct v ty s allowed to approac n n ty, but not to become so

lar e t at t e electron mean ree pat , cyclotron rad us, or

scatter n t me becomes comparable w t or lar er t an t e mportant

d mens ons or t mes n t e problem . T at s to say, we are concerned

only w t t e case o extremely conduct v ty n t e class cal

re me, and not n t e anomalous l m t . T e mportance o suc a

mat emat cal l m t s t at t e actual loss approac es t e class cal

asymptot c value very closely or atta nable elds and conduct v t es

w c are st ll well s ort o t e anomalous l m t . In part cular,

- 3 -

t s l m t s appropr ate to our exper mental case o pur ty

sod um at 4 .2 °K n a 55 kG ma net c eld or requenc es below a

ew k locycles .

II . NATURE OF THE EFFECT

It can be s own t at t e pecul ar constant loss occurs only

w en t e ma net c eld s parallel to t e metal's sur ace . To

ex b t t e nature o t e e ect n t e s mplest case, we s all

cons der a sem - n n te metal ll n t e re on x 1 0 c aracter-

zed by an sotrop c res st v ty p and Hall coe c ent R n a

un orm stat c ma net c eld B.0 parallel to t e z ax s . T e

const tut ve equat on

E = pj -RjxB114

V 1W

olds n t e metal except n t e anomalous l m t . (T e ne lect o

nert al terms s val d as lon as =<Q .) Wr t n B = B +b(r,t)~ 0 A4, A,

and comb n n (1) w t Maxwell's equat ons, ne lect n d splacement

current, we nd t at b ns de t e metal sat s esre.

A RB

AAN

V

AV

L V X (Vxb)6t

110 ~w

6 z

~Lo ~

Aw /VV

w ere we ave ne lected (Rjxb) relat ve to (pj-RjxBO ) n (1) . T s,v NV

W ~ NV

l near zes t e const tut ve equat on and s a ood approx mat on

n two d erent mportant cases :

t s paper we are pr mar ly concerned w t t e case jbjQjB,0 1, nAV

&

w c l m t Eq . (2) s val d ndependent o p . We s all assume t at

t s cond t on olds w enever we let p tend to zero to obta n results

or t e l m t o class cal "per ect" conduct v ty . It s wort not n ,

(1)

(2)

JbIQB0 1 and jbjQp/Rj . InAM

my

W

owever, t at all our ormulae w t t e except on o (10) and t ose

n w c t e asymptot c orms (10) are used are also val d as lon

as jbjQp/Rj no matter w at t e ma n tude o B0 By replac n (10)/vv

A-V

w t t e correspond n orms or t e l m t AMB,--p- 0, t ere ore, ourA14

mulae y eld t e usual zero eld sk n e ect results .

Now cons der t e e ect o an mposed quas -stat c electroma net c

eld . We s all treat only one Four er component o t e eld

b '-,t) = b 0 exp[ (wt-k-r)] w ere w s t e pos t ve real requencyVL"I

and k s t e complex wave vector w t components V ' P,y) . Assume

t at

0 and t at y s real and pos t ve . T us t e mposed eld

component s a wave travel n n t e z d rect on w t a veloc ty w/y .

ne lect o any y dependence o t e elds s only or s mpl c ty .

treatment nclud n y dependence leads to s m lar results . In t e

analys s o t e t n slab and n t e prel m nary exper ment d scussed

below, a stand n wave s used w c s obta ned by summ n over + y .

We may now wr te eac eld component n t e orm

b(r,t) = b(x) exp[ (wt-yz)] .

(3)

T

S nce t e vacuum eld s quas -stat c ( .e ., d splacement

current s ne lected w c s val d as lon as w/y<<c), t must

t e sum o two part al elds, one w t

be

b(x) = b o Q, 0,1) exp (yx)

(4a).1"

and one w t

b(x) = b 3 (-- ,0,1) exp(-yx) .

(4b)

T e part al eld (4a) can only be due to currents outs de t e metal,

w le (4b) can only be due to nternal currents . T ere ore, t e

ampl tude b 0 s xed by t e exper mental cond t ons . Our pro ram s

now to wr te down t e orm o t e nternal elds and t en solve or

t e r ampl tudes (and t at o t e re lected eld (4b) des red) n

o t e nc dent ampl tude b 0 us n t e boundary cond t on t atI

= 0 all components o .1 are cont nuous .

We s all t en nd t e

currents correspond n to t e nternal elds and evaluate t e Joule

eat n pj 2dT d rectly .

Insert n t e eneral Four er component nto (2), one obta ns

t e d spers on relat on

cu (p/q0)(uyk+ k 2 ) ; u QRB0/0 . (5)

T e parameter u s t e tan ent o t e Hall an le and n t e ree

electron approx mat on equals wc T . In a metal suc as sod um w ere

t e carr ers are electrons, u s a pos t ve number . In nd um, or

example, w ere t e carr ers are oles, u s ne at ve . In t s paper

u s always assumed to be pos t ve . T e results obta ned, owever,

are also val d or ne at ve u .

S nce t e nc dent eld determ nes w and y, t ere are two

solut ons o (5), k and k2 . For eac o t ese t ere are two

correspond n values o a sat s y n ant = kn 2_ y 2 (recall n t at

= 0), but t e p ys cal requ rement t at t e eld n t e metal

s all not d ver e at x = - oD perm ts only t e a's w t a pos t ve

ma nary part . T e nternal eld s t ere ore a l near comb nat on

o t e two elds av n

b(x) = bn ( y,kn - an )exp(- anX)

(6)OW

w ere n = 1 and 2 . 2

- 5 -

w ere A

T e ex

We may wr te t e total current dens ty n t e orm

( U-071

l m t I ; a

I

= kj A j +k

w ere use as been made o t e relat on

+1k, I Nja

PO)n

~= Vxb

k b,n

nAvn

w c can be ver ed by d rect calculat on . Us n (7), we may wr te

t e t me avera e Joule eat n per un t volume as t e sum o t e

ollow n t ree terms :

12+N 12)exp [21m(a,

*+a02)exp[[Im(a

RB 0b 0 2A,2 )12k Ik2 />(k

a1 =

val d u(l-n`)>>1

- 6 -

exp [ 21m

µ

7IM-1 2)1 / 2 _q 3u -l (1-7~

I

Q+ Rj

2

ss on or a 2 requ res more care, and d e

o t ree urt er l m ts--

a ]x

11

In order

to evaluate t ese express ons, we obta n approx mate express ons or

and a W

It s conven ent to de ne a reduced requency q = (w/cu w ere

(0 1 =(-RB 0 7 /q,Y L ke t e parameter u, t e requency w l s pos t ve

or sod um, and n w at ollows we s all assume t at t s pos t vee

n t e l m t u>>1 and u>>41 ( .e ., su c ently conduct v ty and

su c ently low requenc es), we obta n t e asymptot c orms

-V )y, k9 = ( n- nVu)7^

ant tot mus t

- 7 -

l m t II : a2 = ('-1)u-1/27

val d lu(o-1) l<<l ;

l m t III : a 2 =

2 )1/2+ j 2u-1 (1-0 -2 ) -1/2 ]

val d uQ2_1)>>l .

(10b)

Compar n (10) w t (6), we see t at ^b 1 s con ned to a sur ace.,

layer w ose t ckness s o order (107), w le or 1>1, b, extends

nto t e metal or a d stance proport onal to u . T e "propa at n "

solut on, b21 s t e el con wave o A ra n, 3 and muc work as been~

done on t e propert es o t s mode, n part cular n t n plates o

metal perpend cular to B0 *4-7 In onosp er c p ys cs, t s wave s

known as t e "w stler" and as also been extens vely stud ed? In

t s paper we ocus our attent on pr nc pally upon t e ot er mode .

T e p ys cal s n cance o t e l m ts n (10b) s as ollows : w en

Q1, t e el con wave (b penetrates t e sample, w le or q<l t e

el con wave veloc ty RBO, s less t an t e veloc ty o t e nc dent

eld w/7, and so t e nc dent eld s totally re lected . We may

de ne or eac mode an e ect ve sk n dept bn as [Im(an)]-

For

all requenc es or w c our approx mat ons (10) old, 5, = (107) .

are ven nT e express ons or 5 2 as well as a plot o 5 2 versus

F . 1 .

Insert n (10) n (9), and nte rat n over x rom -co to 0,

we obta n t e power loss per un t sur ace area or t e sur ace mode,

I

or t e el con wave, and a cross term . We s all denote t ese

P2 , and P 3 respect vely . T ese a a n must be ven by separate

express ons or eac o t e t ree l m ts n (10b) . For l m t I we

obta n

as Pl,

(-RB07)(b o /40 )

/u(1-,2)1/2

[l+(1-71 2 ) 1/2 ]

or l m t II we obta n

A = QRBOy)(bO/VO) 2

-1/2

2Pl (l+u") " ,

and or l m t III we obta n

term P3

P 1

a power absorpt on w c ,

Uny)(bn/4n) 2 [l+(l-j- 2 )1/2

1-1

1-1 -2 ) 1/2

P n2[IM-0

n t e l m t u

r1`` [ 1+(l-j'

u +n

1

w ere n eac express on only t e lowest term n (1/u) as been

reta ned .

T e above express ons or P1 ex b t t e advert sed result,

oo, s ndependent o

ne l ble or all moderately values o u . In

F 2, t e requency dependences o p1 and p2 n t e l m t o

(12)

(13)

/2 ] Yu1+n 2 [l _ (l _ V -2 ) l/2 ] 1 --1 '

conduct v ty and s con ned to a sur ace layer w ose t ckness

s o order (1/u7) . T e loss p2 s l kew se ndependent o u ( or

1>1), but represents a power loss spread over a dept 5 2 w c s

proport onal to u and, t ere ore, t e loss n any sur ace layer

w ose t ckness s less t an 5 2 w ll decrease as (1/u) . T e cross

- 9 -

n n te u are presented . It s nterest n to note t at or ->l,

t e r sum s constant .

At t s po nt t s per aps wort restat n t e restr ct ons on

t e val d ty o our calculat on . We ave assumed t at t e metal's

sur ace s parallel to t e stat c ma net c eld B . A rou cal-

culat on culaton s ows t at t e sur ace s t lted rom t e eld d rect on

by an an le less t an (1/u) t e p enomena are not apprec ably c an ed,

but or lar er an les ne t er mode s con ned to t e sur ace and so

t e loss, at least n a n te sample, w ll tend to zero as p---O .

T e rou ness o t e sample sur ace must be small compared to t e

s ortest d stance n w c t e elds c an e apprec ably, w c s 8 1 .

G ven t ese eometr cal cond t ons, t e p enomena ollow d rectly

rom (1) and Maxwell's equat ons and so w ll occur w enever (1) s

val d . T e const tut ve relat on (1) w c we ave wr tten assumes,

n add t on to wT<<I, a local relat on between E and j and s t ere ore^Q

AV

val d only t e spat al var at on o t e elds alon BO s small n

one electron mean ree pat , and across B 0 s small n a cyclotron

rad us . S nce (al/71 = u 1 coat, t ese cond t ons appen to be dent -

cal and requ re t at 71<<I .

III . THEORY FOR THIN SLAB

We commented above t at t e power loss n a sur ace layer o t e

sem - n n te metal w c s muc t cker t an 5 1 but muc t nner

t an 5 2 s essent ally equal to P1 . However, n a sample o n te

t ckness w t a eld o t e orm (3) nc dent symmetr cally rom

bot s des, t e s tuat on s not so s mple . In suc a sample t ere are

our elds o t e orm (6) s nce bot values o an are perm tted or

eac value o kn . I t e sample t ckness s 2a and uya>>l, two

o t ese elds (correspond n to bl above) are con ned to t e sur-

acesaces and do not nteract w t eac ot er s n cantly . However,

loss s not s mply 2P1 because o t e nter erence o t e el con

2waves penetrat n rom bot sur aces . I also Q 7a/u)<<1 and

(ya/u)<Q, t e el con wave solut ons (correspond n to b 2 above) are~essent ally undamped and contr bute very l ttle to t e power loss .

But s nce at eac sur ace t e total nternal eld must matc t e

total external eld, t e construct ve and destruct ve nter erences

o t e el con waves ve r se to a modulat on o t e sur ace mode

ampl tudes and ence o t e power loss . T s appears as an osc llat on

o t e power loss w t requency and w ll occur n any eometr cal

con urat on n w c t e propa at n mode can orm stand n waves .

In t e ollow n calculat on we develop t ese deas n su c ent

deta l to perm t compar son w t exper ment . In add t on to t e

calculat on or a t n slab presented ere, we ave done t e equ valent

calculat on or a cyl nder parallel to Q0 and ave obta ned s m lar."

results .

Let t e re on jx1ja be lled w t a metal c aracter zed by an

sotrop c res st v ty p and Hall coe c ent R n a stat c ma net c

eld B0 parallel to t e z ax s and surrounded by a co l w t aAyw nd n dens ty n(x,z) = On cos(yz)[5(x-d)-5(x+d)1

exc ted w t a

y s1(t) = 1 0 e Wt w ere y s a un t vector n t e y d rect on and

5(x) s t e D rac delta unct on . T e current dens ty s t en j(x,z,t)1W(x,z)I(t) . T s w ll produce t e des red nc dent eld, and t e

power del vered to t e co l, apart rom t e res st ve losses n t e

w re o t e co l, w ll be equal to t e power absorbed n t e metal .

- 10 -

A su table measure o t s loss s t e ma nary part o t e mutual

nductance between t s co l and an dent cal secondary wound closely

over t . Our calculat on s, t ere ore, d rected toward nd n t e

complex nductance o t e co l per al wavelen t and un t w dt .

T s w ll not be a completely correct representat on o t e corre-

spond n p ys cal system w t a n te w dt and len t , but t e

errors are expected to be small as lon as t e w dt s lar e

to a and (1/y) and t e len t s several wavelen t s .

From t e nvar ance o t e system under 180° rotat on, we know

t at or t e eld ns de t e metal t e z component must be

x, w le t e x and y components must be odd . T s perm ts us to sum

pa rs o nternal elds correspond n to t e same kn ,, and t en wr te

t e problem only or pos t ve x . T e elds we obta n ave

w ere

(6) above .

= B

b ll (x )/W

s n (a n

n, and at

To nd t e

over t e area o eac turn, or

d

x

0,1) e -7x , or x>d,

0

Q, 0, 1) e' - , or d>x>a,

, a101(x))

(x)), or a>x>0,

even

(14)

COWA, and kn and a are as nTT

T e boundary cond t on at x = d s bQ) -b(x) 1W

) 111 ) 11s b(x-b(xM

/W

compared

ce we nte rate t e z component o t e eld

or s mpl c ty we may nte rate over

t e re on outs de eac turn . (T s s perm ss ble because all o

t e eld l nes close upon t emselves .) T s allows us s mply to cal-

culate B 1 , rat er t an all o t e ot er our eld coe c ents . T e

nductance s t en t e lux per un t current . Solv n t e boundary value

problem and per orm n t e nte rat on we et

M = (1/2)woy- 2n2Y[1-e-2y(d-a) (1-F)l

(15)

w ere

2y[(l/k2)-(l/kl ) l

7[(l/k2) -(I/kl ) "(a2A 2 )ctn (a 2a)-(a l /kl )ctn(a l a)

or t e complex mutual nductance per al wavelen t and w dt Y .

W en a complex nductance s measured n an a-c br d e, one obta ns

d rectly t e real part o t e nductance and a res stance . T s

res stance, r, s (-u ) t mes t e ma nary part o t e nductance .

T ere ore, t e ormula w c we s all use to compare w t exper ment

s

r = (v/2)(-RBon2Y)e- 27(d-a) ImQF) .

(17)

As quoted, t s s an exact ormula w t no adjustable constants . It

s, owever, str ctly val d only or a str p o w dt Y n an n n te

metal slab, and s not necessar ly val d or an solated sample w ose

total w dt s Y . But we expect t at t e qual tat ve eatures pred cted

by t e n n te slab model w ll be observed n t e correspond n n te

system .

IV . EXPERIMENTAL TEST OF THEORY

For our n t al test o t e above t eory, we used t e con urat on

s own n F . 3 . T e co l was wound as a ser es o spaced co ls w t

t e w nd n d rect on alternat n rom sect on to sect on . T s does

- 1 2 -

- 13 -

not prov de a eld w c s str ctly o t e orm (3), but t w ll

conta n er spat al Four er components as well . For t e co l

sect on spac n and value o y(d-a) w c we c ose, owever, t e

max mum res stance (17) due to all ot er components t an t e unda-

mental s less t an 27 o t e max mum due to t e undamental . T e

exc t n co l cons sts o ourteen sect ons, eac a 21 turn s n le

layer o 40 au e w re . T e secondary co l cons sts o s x sect ons,

eac 56 turns o 40 au e w re n t ree layers closely wound over

t e central sect ons o t e exc t n co l . T s arran ement m n m zes

any end correct ons . T e ent re co l s ree-stand n n epoxy

res n . T e d mens ons are ven n t e capt on o F . 3 .

T e sample was a p ece o very pure polycrystall ne sod um

metal . T e exper ment was per ormed at 4 .2 ° K w ere t e conduct v ty

was about 8000 t mes t e room temperature value . T e t ckness was

measured w t a m crometer be ore t e exper ment, and a terward a

port on o t e sample was used n a el con wave cav ty resonance

exper ment, as n re erence 5, to obta n a measure o t e t ckness

under t e ox de and also to obta n values or t e parameter u at t e

elds used n our exper mental test o t e t eory .

T e mutual res stance, r, was measured as a unct on o requency

on a Harts orn br d e or t ree elds : 55 .0 kG, 36 .7 W, 27 .5 kG, and

also or B 0 = 0 . T e zero eld result measures t e convent onal sk n

e ect loss . T ese eld rat os are accurate to better t an Wand t e

absolute eld accuracy s only sl tly worse . T e values o t e

parameter u n t e sod um at t ese t ree elds were approx mately 107,

82, and 64 respect vely .

T e or nal mutual res stance data are presented n F . 4 .

Eac po nt s uncerta n by about 0 .2 M . In F . 5, t ese data ave

been replotted a ter scal n n accord w t t eory . T e sol d curve

n t at ure s ven by (17) w ere (16) as been evaluated by use

o t e asymptot c orms (10) . T e parameters are t ose ven n t e

capt on o F . 3 w t Y1 used or Y . T e uncerta nty n t e exper -

mental parameters mpl es about a 77 uncerta nty n t e e t w c

s ould be ass ned to t e t eoret cal curve, w le t e or zontal

scale actor s n doubt by about 4% .

T e exper mental results a ree w t t e t eoret cal pred ct ons

n several mportant respects : T e osc llat on o t e loss w t

requency s clearly present . T e ampl tude and requency o t e

loss curve depend l nearly on t e ma net c eld . T e requency o

t e rst max mum a rees w t t e t eoret cal pred ct on w t n

exper mental error . On t e ot er and t ere s one de n te qual ta-

t ve d sa reement w t t eory ev dent n F . 5 . T e t eory pred cts

t at or 17a>>l, t e loss s ould be proport onal to s n 2 Qya) . T e

exper mental results, owever, seem to nd cate t at t e loss

osc llat on ampl tude d m n s es at er requenc es alt ou t e

avera e value appears to be rou ly constant and very close to t at

pred cted by t eory . T s e ect s probably not due to a breakdown

o t e approx mat ons nvolved n (10) s nce t e curves or 27 .5 kG

and 36 .7 kG, or w c t e u values were nearly 307 d erent, seem

to s ow substant ally t e same be av or . Per aps t e most l kely

source o t e d screpancy s t e ne lect o any y dependence o t e

elds n t e t eory . We bel eve t at t e s arp d ver ence o t e

55 .0 kG curve above rya = 3 s due to a spur ous add t onal power

- 15 -

absorpt on mec an sm . At around 1300 cycles, n t e system we used,

t ere was a stron mec an cal resonance (w ose requency was

essent ally ndependent o eld and w ose e ects, t ere ore, do not

s ow up on t e lower eld curves n t e normal zed plot) w c

absorbed a reat deal o power rom t e co l and t ereby masked t e

ot er e ects . At er requenc es (up to 3000 cps) st ll ot er

mec an cal resonances were observed .

We also ave prel m nary data rom a s m lar exper ment us n a

cyl nder parallel to B0 . T e loss curve or t at exper ment d splays~all t e eatures noted n t s case, and t e a reement w t t e

correspond n t eory appears to be comparable . We plan a uture ser es

o exper ments to clar y t e requency be av or .

V . SUMMARY

We ave presented calculat ons o t e sk n e ect n t e presence

o a ma net c eld, ex b t n some novel eatures n t e dependence

o t e power absorpt on on requency and conduct v ty . In part cular,n t e l m t o class cal "per ect" conduct v ty ( n w c but

t e e ects o n te electron mean ree pat and scatter n t me are

ne lected) and or su c ently low requenc es t e loss tends to a

nonzero l m t ndependent o t e conduct v ty and s con ned to t e

sur ace . T s results rom t e act t at apply n t e ma net c eld

produces two mod cat ons o t e electroma net c modes n t e metal .

T e sk n dept o one electroma net c mode n t e metal s d rectly

proport onal to t e res st v ty, nstead o be n proport onal to t e

square root o t e res st v ty as s t e case or t e normal (zero

ma net c eld) sk n e ect . T e ot er mode (t e el con wave) pene-

- 1 6 -

trates t e metal to a d stance nversely proport onal to t e res st v ty .

T e bas c p ys cal reason or t ese mod cat ons o t e modes by t e

stat c ma net c eld s t at n ts presence t e current and electr c

eld are not parallel . In our pur ty sod um metal at 4 .2 ° K and

n a ma net c eld o 55 kG, we can obta n values o t e parameter u

n excess o 100, or w c t e t eory nd cates t at t e loss s ould

be v rtually equal to t at n a ypot et cal "per ect" conductor . In

our prel m nary exper mental test we ave obta ned reasonable a reement

w t t e t eoret cal pred ct ons .

ACKNOWLEDGMENTS

We w s to t ank J . A . Krum ansl and R . Bowers or st mulat n

d scuss ons .

-17-

FOOTNOTES

* This work was supported by the U. S. Atomic Energy Commission and the

Advanced Research Projects Agency .

+ National Science Foundation Predoctoral Fellow

1 . The justification of this boundary condition and a discussion of

the general boundary value problem are contained in a paper by

C . R . Legendy (to be published) .

2 . The general case (P O) of the form was given by Bowers et al .,

reference 4 .

3 . P . Aigrain, Proceedings of the International Conference on Semi-

conductor Physics, Prague, 1960 (Czechoslovak Academy of Sciences,

Prague, 1961), p . 224 .

4 . R . Bowers, C . Legendy, and F . E . Rose, Phys . Rev . Letters 7, 339

(1961) .

5 . F . E . Rose, M . T . Taylor, and R . Bowers, Phys . Rev . 127, 1122

(1962) .

6 . R . G . Chambers and B . K . Jones, Proc . Roy . Soc . (London) A270, 417

(1962) .

7 . J . R . Merrill, M . T . Taylor, and J . M . Goodman, Phys . Rev . 131,

2499 (1963) .

8 . L . R . 0 . Storey, Phil . Trans . Roy . Soc . London 246A, 113 (1953) .

-18-

CAPTIONS

F ure 1 . Penetrat on dept o el con wave versus requency or a

metal sur ace parallel to B 0 . T e normal z n constant'Iw

s t e z component o t e wave vector .

F ure 2 . Frequency dependence o t e power loss per un t sur ace

area o a sem - n n te metal n t e two modes or t e

l m t o n n te u .

F ure 3 . Sc emat c d a ram o co l and sample . T e arrows s ow

t e d rect on o w nd n n eac co l sect on . T e values

o t e parameters are : 2a = 1 .05±0 .05 mm ; 2d = 2 .00±0 .04

mm ; Y 1 (t e sample w dt ) = 17 .5±0 .5 mm ; Y2 (t e co l w dt )

= 20 .0±0 .5 mm ; L 50 mm ; A = 6 .35±0 .06 mm ; and m

0 .340±0 .003 .

F ure 4 . Or nal mutual res stance data . Eac po nt s uncerta n

by about 0 .2 W .

F ure 5 . Data o F . 4 replotted a ter scal n n accord w t t eory .

T e sol d curve s Eq . (17) w ere (16) as been evaluated

us n (10) . Bot scales are as ven by t eory us n t e

exper mental parameters w t no adjustment .

1 .0

2.0

3.0

4.0

5.0

6.0

REDUCED FREQUENCY t

-=~--~----

--

T ~~ --~=~_----~--- --~ ~

-- -J---

SCALED FREQUENCY ]~[5~~ .~~~~

~~~~~ .~ ._~_~^ .~ .

~~~~ .~~~~~ (cps)