1
On the construction of Tesla transformers Period of oscillation and self-inductance of the coil.
(Zur construction von Teslatransformatoren. Teslatransformatoren.
Schwingungsdauer und Selbstinduction von Drahtspulen)
By P Drude1 Annalen der Physik, 1902, vol. 1! (! th series, vol. 9), Part ". issue 10, #29-9 $ Plate 1, Part "". " ". issue 11, #%90-&10. 'ranslated y David *ni+h *ni+htt2, and oert eaver . 1th ay. 201&. Added y the translators/ Table of Contents
'ranslators coentary......................................... coentary................................................................ .............................................. .............................................. ................................2 .........2 otes on the translation....................................... translation.............................................................. ............................................... .......................................................... .................................. "ntroduction ........................................................................................................................................! ". 3scillation #eriod of 4ire coils .......................................................... ................................................................................................% ......................................% 1. 56#eriental ethod ............................................. ..................................................................... ............................................... ...............................................% ........................% 2. 'ransfer 'ransfer to lar+e coils of the results otained on sall coils ............................................ .......................................................10 ...........10 . 5ffect of of the nature of the coil coil core and and its area on the self-resonance #eriod .....................10 !. 5ffect of of 4ire insulation on the self-resonance #eriod of the coil ...............................................1! ...............................................1! %. 7oils 4ith uneven #itch ............................................ ................................................................... .............................................. ............................................ .....................1% 1% &. 'he 'he nuer of characteristic #araeters a coil of constant constant #itch ............................................. ...............................................1% ..1% Plate 1 ............................................ ................................................................... .............................................. .............................................. ............................................... ..........................18 ..18 . 7oils on 4ooden cores ............................................ .................................................................... ............................................... .............................................19 ......................19 8. 7oils on hollo4 cores (tues) .............................................. ...................................................................... ........................................................20 ................................20 9. 7oils 4ithout cores ........................................... .................................................................. .............................................. .................................................... .............................22 22 10. 'ales for calculatin+ the self-4avelen+th of a coil .............................................. ................................................................ ....................2 ..2 11. A##ro6iate theory of self-oscillation of a lon+ thin thin coil ............................................ .........................................................2& .............2& 12. 7oils of fe4 turns and si#le loo#s loo#s ............................................. .................................................................... ..................................... ....................... .........2 2 1. A s#ot-check of the tales usin+ a coil 4ith 2 self-resonance half-4avelen+th ...................! ................. ..! 1!. 3vertones of coils ............................................. ..................................................................... ............................................... ..................................................% ...........................% 1%. "ncrease of the #eriod of coils due to a##lied ca#acitance ......................................................... .........................................................& & "". elf-inductance of the coil ........................................................ ............................................................................... ......................................... ......................!0 ....!0 1&. ethod of easureent .............................................. ..................................................................... .............................................. ..................................... .................!0 ...!0 1. i#le loo#s ............................................ ................................................................... .............................................. ............................................................ .....................................! ! 18. ectan+les ........................................... .................................................................. .............................................. .............................................. ..........................................!& ...................!& 19. 7oils ............................................ ................................................................... .............................................. .............................................. ..................................................!8 ...........................!8 20. i+orous testin+ and a##lication of the the forulae in t4o 'esla transforers .............................% uary of results .............................................. ..................................................................... .............................................. ......................................................%& ...............................%& A##endi6 1. ecalculation of the tale on #28................. #28........................................ ......................................................... .......................................%8 .....%8 A##endi6 2. 3scure or osolete osolete :eran 4ords and areviations........................................... areviations................................................%9 .....%9 1 Paul Paul *arl *arl ;ud4i+ ;ud4i+ Drude, Drude, 18& 18& - 190& 190&.. 2 3ttery 3ttery t ary ary, Devon Devon,, 5n+la 5n+land. nd. htt#/<<+ynh.info< orcid.or+<0000-000-0!99-98 aska askatoo toon, n, ask askatc atche4 he4an, an, 7anada 7anada.. htt#/<
2 Translator's commentary
'he follo4in+ #a#er descries P * ; Drudes 1902 investi+ation into the factors affectin+ the selfresonant ehaviour of sin+le-layer coils. 'his 4ork, lar+ely i+nored i+nored y the 5n+lish-s#eakin+ 4orld th in the second-half of the 20 7entury, 7entury, is #articularly #articular ly relevant to the construction of 'esla transforers, ut it is also of +eneral #ractical and theoretical interest. 'he e6#eriental ethod 4as that of e6citin+ coils y eans of an induction loo# 4ith a variale resonatin+ ca#acitor, this circuit c ircuit ein+ ener+ised y an induction coil and a 'esla transforer 4ith oth #riary and secondary s#ark +a#s. esonance of the coil under investi+ation 4as detected y holdin+ an electrodeless sodiu-va#our dischar+e tue near to the coil. avelen+th caliration involved reovin+ the coil and usin+ usin+ the loo# to e6cite a #arallel-4ire transission line 4ith a oveale shortin+ stra#, resonance ein+ detected y #lacin+ the dischar+e tue at the volta+e anti-node. 'he self-resonance #eriod of a coil 4as 4as found to increase 4ith the dielectric constant of the the core aterial= ut this 4as less than #ro#ortional to the s>uare root of the dielectric constant (as 4ould e the case for iersion in a hoo+eneous ediu). 'he dielectric effect of the core 4as also found to e +reater as the hei+ht to diaeter ratio 4as reduced, ecause of the increasin+ density of electric field lines on the inside of the coil. ?ollo4 cylinders had less effect than solid cylinders. ire insulation 4as also found to increase the self-resonance #eriod, and the effect a+ain increased as the hei+ht to diaeter ratio 4as reduced. 7oils 4ere characterised y eans of a function f , , 4hich is defined as the ratio of the selfresonant half-4avelen+th to the 4ire len+th. 56cludin+ dielectric effects, effects, f is #riarily de#endent on the coil hei+ht to diaeter ratio ( h<2r ),), its value ein+ lar+e 4hen h<2r is is sall. 'he effect of the #itch to 4ire-diaeter ratio is relatively sall, and the nuer of turns has little effect #rovided that there are ore than 1. A +ra#h of e6#eriental values of f vs. h<2r , , for coils on eonite cores (@ 2.9) and for coreless coils, is +iven in Plate 1. "n accountin+ for the relationshi# et4een coil #araeters and self-resonance, it 4as noted that 4hen a current is induced in a disconnected lon+-thin coil, the current 4ill e at its a6iu in the iddle re+ion and ero at the ends. 'his causes electric char+e to i+rate i+rate to4ards the coil ends, inducin+ a #otential difference. difference. "f the resultin+ char+e dis#laceent dis#laceent is considered to e localised localised on t4o s>uat cylinders located at the ends of the coil, the ca#acitance can e calculated in ters of s#herical haronics. 'he resultin+ calculated value of f 4as 4ithin %C of the oserved over an h<2r ran+e fro aout aout 2.2 to 1.0. 3vertone resonances 4ere 4ere also investi+ated, the node #ositions ein+ located usin+ the sodiuva#our dischar+e tue. tue. 3vertones are not haronically related to the fundaental resonance. Drude ar+ues that 4hen a coil is oscillatin+ at its first overtone, the fact that it does not ehave as t4o se#arate coils is due to the a+netic cou#lin+ et4een the t4o halves. hen ca#acitance is added to a coil, such as y the addition of a conductin+ s#here at one end, end, the #eriod of oscillation is increased, ut never ore than douled. 'his effect can e >uantified y considerin+ the resultin+ shift in the volta+e node. "n #art "", the difficulty of calculatin+ hi+h-fre>uency inductance (i.e., the reduction due to skin effect and non-unifor current distriution) 4as overcoe y #lacin+ fi6ed ca#acitances in #arallel 4ith coils and loo#s and akin+ resonance easureents. (D* une 201%)
Notes on the translation
1) 'ranslators 'ranslators coents and additions are +iven +iven in the te6t in Es>uare racketsF. "t is recoended that this docuent is read in conGunction 4ith the ori+inal :eran #a#ers, 4hich are otainale in a sin+le #df file fro/ htt#s/<
uare rackets, i.e., E#29F to E#9F and E#%90F to E#&10F. ote ho4ever that sentences that 4ere s#lit over t4o #a+es #rior to translation are no4 #laced entirely either efore or after this nuer. nuer. ote also that source #a+e nuers are not al4ays in nuerical order and ay soeties a##ear t4ice H this is ecause soe tales have een oved to i#rove te6t flo4 and #lace #lace the close to the te6t that refers to the. Iootnotes are nuered se>uentially in this docuent. Ior cross-referencin+ #ur#oses, the ori+inal #a+e nuer and footnote nuer is +iven at the e+innin+ of the footnote, e.+, E29-F indicates that the follo4in+ te6t is a translation of footnote ) on #29. #29. "n the asence of such a cross-reference, the footnote has een added in translation. ) Drude used the the c+s syste of units. "n soe cases, such as ca#acitances and inductances in c, the result in rational units has also een +iven +iven in s>uare s>uare rackets. 'o translate an inductance in c to rational ks ("), convert the len+th into etres and ulti#ly y J0 < !K 10 - ?<. 'his eans that 1 c L 1 n?. 'o translate a ca#acitance in c to ", convert to etres and ulti#ly y !K @0 111.2&%00%& 111.2&%00%& #I<. #I<. 'hus 1 c L 1.112&%00%& 1.112&%00%& #I. #I. Ire>uencies in ? are also +iven in soe #laces usin+ f 0 c < M , 4here c 299 92 !%8
! - E#29F !. On the construction of Tesla transformers Period of oscillation and self-inductance of the coil By P. Drude. "ntroduction
'he construction of 'esla transforers involves rin+in+ a #riary circuit fored y a coil of 4ire of fe4 turns 4ith a##lied end ca#acitance into electrical resonance 4ith a coil of any turns 4ithout an a##lied end ca#acitance. "t 4ill, es#ecially in the construction of lar+e and vi+orously actin+ transforers, involve uch tie-consuin+ e6#erientation if the #eriod of oscillation of the secondary coil and the inductance of the #riary #riary coil cannot e calculated in advance. 'his atter 4ill e addressed in the follo4in+ article. A rational a##roach to the diensionin+ of 'esla transforers, and to their theory, 4ill e covered in a later #a#er !. H 'he kno4led+e of the natural #eriod of coils can also e a##lied to the construction of the i#ortant i#ortant ne4er devices for 4ireless tele+ra#hy, althou+h one should e careful that the electrical conditions ay e essentially different if the coil does not have free ends ut has has ca#acitance or strai+ht 4ires connected. 'he resultin+ chan+es can e estiated theoretically, ut the #eriod of oscillation of the free-ended coil ust first e kno4n. ote that here 4e discuss only coils of 4ire in a +iven 4indin+ sense, i.e., 4ith lar+e lar+e selfinductance, as they are #riarily the 'esla 'esla coils of i#ortance. H Ior the #ur#ose of 4ireless tele+ra#hy,, and also for tele+ra#hy f or laoratory e6#erients 4ith 'esla coils, 4ire coils 4ith different 4indin+ sense, i.e., saller self-inductance, are soeties useful. 'o kee# this docuent docuent reasonaly sall, such coils are e6cluded fro this discussion.
EA #hoto+ra#h of a##aratus siilar to that used in this #a#er (the induction coil on the left is alost certainly the sae one). 'his #icture is fro Zur #essung der Diele$tricit%tsconstante vermittelst ele$trischer Drahtwellen (easurin+ the dielectric const. y eans of electric 4ire 4aves (i.e. standin+ 4aves) ), P. P. Drude, Ann. Phys. 1(&) th (! series vol. 8). #&-!. 1902. 'his is Ii+. 2 fro #a+e !0.F
!
&ationelle $onstruction von Teslatransformatoren, P Drude.
Ann. Phys. Nol. Nol. 21(1), 190%, #11&-1
% - E#29!F ". Oscillation period of wire coils . ()perimental method
'he e6#eriental ethod 4as that the coil to e e6ained, S , 4as e6cited inductively y the electrical oscillation of an e6citer E (Ii+. (Ii+. 1), 4hich consisted of t4o sei-circularly curved thick co##er 4ires that s#anned a circular area of 21 c diaeter. diaeter. 'he e6citer 4ires 4ere held y t4o thick slotted eonite Ehard ruerF su##orts, H . At one end they 4ere ent do4n so that they 4ere iersed in a +lass o4l filled 4ith #etroleu. 'his end had sall rass alls of 0.% c diaeter, the se#aration (aout 0.2% 0.2% ) easily varied y a shift of the eonite su##ort H . 'he e6citation s#ark et4een the rass alls took #lace under Petroleu. 'hey 4ere connected to the ends of the secondary coil, T , of a 'esla transforer %, 4hich 4as fed y an "nduction coil, J , 4ith strikin+ distance of 0 c, havin+ a rotary ercury reaker. Z is the inc s#ark +a# for 4ave e6citation in the #riary #ri ary coil of the 'esla transforer, L is the ;eyden Gar of its #riary circuit. 'he other ends ends of the field 4ires lead to t4o 9 c lon+, lon+, 0.% thick co##er 4ires, a, a, 4hich connect to a #etroleu-iersed circular #late ca#acitor, C ( ( Ii+. 2). - E#29%F A #etroleu ath P for C is e6treely convenient, ecause it allo4s the distance of the #lates to e reduced to 1 4ithout the occurrence of s#ark or corona dischar+e. 'he ca#acitance can therefore e varied over a uch 4ider ran+e than 4hen C has an air environent. environent. 'he #lates of C 4ere 10 c in diaeter, their se#aration could e u# to % c. 'hey 4ere attached, 4ith vertical eonite su##orts e, e, to t4o t4o horiontal horiontal ars, h , h, one of 4hich 4as oveale and had a scale for easurin+ #arallel dis#laceent. 'he distance of the ends of the e6citation 4ires, fro 4hich the thin co##er 4ires 4ere an+led, 4as % c. 'he coil to e tested, S , 4as #laced vertically in the centre of the e6citation loo# on 4ooden locks. De#endin+ on the circustances, the distance 4as % c - 0 c fro the e6citation #lane, and a vacuu tue 4as #laced at the end. 'he tue 4as #laced in the o#en, 1 c - 2 c fro the end of the coil. 5lectrodeless tues y the +lass lo4er *raer in Ireiur+ in #articular are hi+hly % E29!-1F E29!-1F By insertin+ insertin+ a 'esla 'esla transfor transforer er et4een et4een inductor inductor and e6citin+ e6citin+ s#ark s#ark +a# , the intensity intensity of the the electric electric 4aves is +reatly increased. 'he diensions of the 'esla transforer , 4hich are >uite irrelevant if i f it is sufficiently vi+orous, 4ere as follo4s/ econdary coil 100 turns of 1 thick (4ith insulation 2 thick) co##er 4ire on a 4ooden cylinder of diaeter of 9 c and 20 c hei+ht. hei+ht. 'he ;eyden Gar L 4as 11.% c in diaeter, 4ith 19 c overla# hei+ht, and % thick +lass.
& recoended. 'hese have a thin layer of electrolytically de#osited sodiu. hen the induction coil is o#erated in a darkened roo, and there is a s#ark dischar+e et4een the alls of the e6citer, the vacuu tue 4ill not +enerally e lit. 3nly at a certain se#aration of the #lates of ca#acitor C 4ill it li+ht u#. - E#29&F & 'his se#aration corres#onds to the resonance et4een the 4ire coil and e6citation circuit. 'his resonance #osition of the ca#acitor C is deterined y adGustin+ a horiontal ar on 4hich a #late of C is fi6ed. 'he resonance ecoes shar#er as the inductive e6citation (a+netic cou#lin+) of the coil y the e6citer is reduced, i.e., the hi+her it is aove its level, #rovided that the resultin+ illuination of the vacuu tue is not too 4eak. eak a+netic cou#lin+ et4een the t4o systes is ho4ever necessary, ecause other4ise (e6ce#t for the attenuation of the e6citation virations, see footnote E29&-1F ) a6iu e6citation of the coil 4ill not e6actly occur at resonance ecause of the retro-action of the coil on the e6citer circuit. 'his #ullin+ effect ho4ever, 4as not +enerally a #role, ecause usually the diaeter of the test coil (2 c - c) 4as uch saller than the diaeter of the e6citation loo# (21 c), so that the nuer of flu6 linka+es 4as sall. "n any case ho4ever, the distance et4een the coil and the e6citer 4as al4ays lar+e, and the s#ark +a# 4as ke#t sall to achieve #referaly 4eak rather than stron+ illuination of the vacuu tues. 'his is ecause, if the tue +lo4s stron+ly, the conductivity of the +as is si+nificantly increased, and if this is a##lied at one end of the coil, its #eriod of oscillation is sli+htly reduced in co#arison to a coil 4ith t4o free ends 'here 4as thus otained, in a coil of 0 c len+th and 1. c diaeter, 4hich consisted of 100 turns of 1 thick are co##er 4ire, a resonance distance d et4een the #lates of the ca#acitor C , d 18. , that is, M<2 28& c, 4hen the tues +lo4ed stron+ly. 7ontrast that to d 21.0 , that is, M<2 2 c, if the tues +lo4ed 4eakly. - E#29F 'he influence of ca#acitance increase caused y the +lo4in+ tues is reduced as the self-resonance 4avelen+th M of the coil +ets lon+er. 'he resonance #ositions of the ca#acitor C 4ere adGusted several ties (usually & ties) and the distance of the ca#acitor-#lates 4as easured usin+ a vernier scale 4ith 0.1 accuracy. An e6#erient 4ith a #articular coil 4as terinated only after settin+s 4ere found such that a sall chan+e in the intensity of the inductive e6citation, or in the distance 8 of the vacuu tue fro the coil end, and thus the li+ht intensity of the vacuu tue, had no noticeale influence in the natural oscillation #eriod of the coil 9. 'he coil ends 4ere often held y sall 4ire #ins. 7onfiratory tests sho4ed that the sae settin+s 4ere otained 4hen the coil ends 4ere ceented 4ith sealin+ 4a6, or held y notches in the coil core or y t4ine. 'he oscillation #eriods associated 4ith each s#acin+ of the ca#acitor #lates C 4ere otained y & E29&-1F 'his does not strictly a##ly if the attenuation of the e6citation virations is very si+nificant (see/ Periode f*r welche die +mplitude einer er,wungenen Schwingung ein ma)imum wird EPeriod for 4hich the a#litude of forced oscillation is a a6iuF, . ien , Ann. Phys 29!(8) 189& (ied. Ann. %8) #2%-28). "t is so sall here ho4ever that it can e ne+lected. As calculated fro the attenuation of the e6citation 4ithout a##lied end ca#acitance O 0.1% (see Theorie stehender electrischer Drahtwellen E'heory of electric 4ire standin+ 4avesF, P Drude , Ann. Phys. 29&(1) 189 (ied. Ann. &0), #1-!&, see #1) k 0.0% n accordin+ to ein. o4, if the attenuation 4ith a##lied end ca#acitor is lar+e, it 4ill still only +ain influence (0.% C), 4hen three ties as lar+e, i.e., 4hen O 0.!%, and this is certainly not the case. E29-1F "t 4as even #ossile to estiate to 0.02 y usin+ a a+nifyin+ +lass. 8 E29-2F "n the case of the +reat intensity of electrical oscillations this distance could e c= e.+., 4ith a 1&c lon+ coil, this 4as #ossile even if the lo4er coil end 4as 1% c aove the e6citer #lane and the u##er end 4as 1 c aove it. 9 E29-F Althou+h instead of li+htin+ a vacuu tue, a sall s#ark +a# at the end of the coil could e used as a 4ave indicator and +ave the sae resonant distances d of the ca#acitor #lates. #ark +a#s ho4ever are not such sensitive indicators as vacuu tues.
caliratin+ the a##aratus in the follo4in+ anner/ After easurin+ a coil, S , a lon+ transission-line, D, ade of t4o are 1 diaeter co##er 4ires stretched taut 4as arran+ed 1% c aove the level of the e6citation 4ires (Ii+. .). - E#298F 'he 4ire s#acin+ 4as 2. c. 'hey 4ere shorted close to the s#ark +a# (the e+innin+ of the line D ). At the other end they 4ere shorted y a slidin+ etal stra# B. 'he shortin+ stra# 4as oved y hand (in a darkened environent) so that a vacuu tue V #laced a##ro6iately half 4ay et4een B and the e+innin+ of the line +lo4ed at a6iu ri+htness. 'his occurs 4hen the resonance10 of the line D is consistent 4ith the oscillations of the e6citer E . 5ach s#acin+ of the #lates of the ca#acitor C therefore corres#onds to a #articular resonance #osition of B. Because of the 4eak a+netic cou#lin+ et4een E and D, these resonance settin+s are very shar# (0.2%C to 0.%C of the distance fro the e+innin+ of the line D to the stra# B ). 'he half-4avelen+th of the electrical oscillation is e>ual to the distance of the shortin+ stra# B fro the e+innin+ of the line, increased 11 y the len+th of the shortin+ stra#s= #lus a sall addition due to ca#acitance of the +lo4in+ vacuu tue. 'he latter 4as noticeale here ecause, 4hen oservin+ lon+ 4aves, the vacuu tue had to e reoved so far ( ) that the faint +lo4 4ould no lon+er e #erceived. Both corrections can e deterined e6actly (at shorter 4avelen+ths), y leavin+ V 4here it is and ovin+ B further ack to the ne6t resonant #osition. - E#299F 'he distance et4een the first and second resonance #ositions of B is e6actly one half-4avelen+th. 'he correction so otained 4as an addition of 9 c12, althou+h it de#ends to soe e6tent on the actual 4avelen+th. 'his latter variation ho4ever is so sall that it 4as 4ithin the oservational error (0.2% C) and could e ne+lected. 'his caliration of the e6citer 4as carried out al4ays iediately efore and after an oservation of a coil S . 'he caliration results chan+ed si+nificantly only 4hen the #late-ca#acitor C 4as taken a#art and reasseled. 'he follo4in+ tale contains the results. d is the s#acin+ of the ca#acitor #lates e6#ressed in illietres, M<2 the corres#ondin+ half-4avelen+th of the e6citer oscillation in centietres .
10 E298-1F Due to the lar+e distance et4een E and D and ecause of the sall relative distance et4een the t4o 4ires D, the a+netic cou#lin+ et4een e6citer E and line D is so 4eak that a reaction fro D to E is not noticeale. 'hus the #osition of B for 4hich V +lo4s ost ri+htly really corres#onds to the resonance. 'his 4as #roved y the fact that the #osition of B is not de#endent on the distance et4een D and E . 11 E298-2F ee Theorie stehender electrischer Drahtwellen E'heory of electric 4ire standin+ 4avesF, P. Drude, Ann. der. Phys, Nol. 29&(1) (ied. Ann. &0), 189, #1-!&, see #1!. 12 E299-1F c 4as oitted fro this correction due to the #ro6iity of the 4ooden easurin+ rod (2 c) aove 4hich the #arallel 4ires 4ere strun+. Because this distance of 2 c 4as increased to &.% c, so there 4as only & c additional correction , instead of 9 c. 'herefore, the shortin+ stra#s contriute c, the ca#acitance of the +lo4in+ vacuu tue another c, and the #ro6iity of the 4ooden easurin+ rod c. 'he additional correction (9 c), 4hich is al4ays a##lied in the follo4in+, +ives the correct 4avelen+th in free air, ecause the #ro6iity of 4ood for the rear #arts of the secondary line 4as avoided.
8 De#endence of the 4avelen+th M of the e6citer on the distance d of the ca#acitor #lates. d < % 9 11 1 1% 1 19 22 2& 1 9 %0 M<2 < c %8% !& !08 &9 ! 2! 09 29& 28 2! 2&2 2%1 2 22& f 0 < ? 2%.& 2.1 &. !0.& !. !&. !8.% %0.& %2.2 %!. %.2 %9. &.2 &&. "t 4as difficult to easure the asolute value of d accurately1, so there is an uncertainty of u# to 0.1 in d . - E#00F 'his is not ovious ecause the #late s#acin+ d , 4hich 4as read on the scale on the horiontal ar of one of the #lates of the ca#acitor C , 4as the sae et4een caliration and oservation of a coil, #rovided that the ca#acitor C had not een taken a#art, and #rovided that only a short tie1! had ela#sed et4een the t4o oservations. As said aove, the ca#acitor #lates 4ere scre4ed to t4o vertical eonite holders e, e, and these in turn 4ere scre4ed to t4o horiontal etal ars h, h, 4hich rested in slidin+ +uides on insulatin+ +lass #illars g , g (Ii+. 2). - E#01F Durin+ the resonance settin+ for a coil under oservation S , the slidin+ horiontal ar (adGusted y icroeter scre4) 4as touched y a hand, i.e., shunted to earth. 'his caused the ca#acitance of the ca#acitor C to e soe4hat increased, as if the t4o horiontal ars 4ere insulated. ?o4ever, this 4as only noticeale 4hen the distance of the ca#acitor #lates C 4as lar+e (a##roachin+ % c), i.e., 4hen the ca#acitance 4as sall. hether this 4as si+nificant or not, could e easily detected durin+ the caliration #rocedure= y notin+ 4hether the resonance #ositions for the shortin+ stra# B 4ere different if the horiontal ars su##ortin+ the #lates of the ca#acitor C 4ere insulated or 4ere 1 E299-2F 'o the nearest 0.1 , the asolute values of d are aout ri+ht. Ior sufficiently lar+e ca#acitance of C , i.e., sufficiently sall d , M<2 ecoes #ro#ortional to C , i.e., inversely #ro#ortional to d . Ior d , %, 4e +et (M<2).d as 1012, 10!, 108 = this inconsistency is ecause the a##ro6iation forula C r 2 < !d , 4here r is the radius of the ca#acitor #lates, is used instead of the ore accurate forula (see/ e.+., B I *ohlrausch, ;eitfad. d. #rakt. Phys. 8th ed. #!09)/ C (r 2
-1 $ E δ
4here δ is the #late thickness. 'he diensions 4ere δ 1 , r % c. 'he 4avelen+th is yet to e ulti#lied y @ , 4here @ is the dielectric constant of #etroleu. "t 4as found that @ 1.!1, this ein+ the ratio of the e6citer 4avelen+ths 4hen co#ared 4ith usin+ the ca#acitor in air. "f 4e calculate C usin+ the aove forula and ulti#ly y 1.!1 4e otain/ % 9 11 1 1% 1 19 22 2& 1 9 (M<2)<C 8%.& 8%.9 8&.% 8.2 88. 89.! 90. 90.% 91. 92.! 9.9 9&.1 98.% d
i.e., it is actually the case that M<2 S C , and the deviation for lar+er values of d is considerale ecause the forula for C is then still too i#recise. 'he re+ular increase of the >uantity (M<2)/C ho4ever su##orts the reliaility of the oservations. "f 4e take the value 8%.& for d as reasonale, 4e can use the forula M 2K LC , 4here L is the self-inductance of the e6citer loo#. 'his +ives the value L !! c E !! n?F. Accordin+ to ein, (Ann. Phys. 289 (ied. Ann. %). #91. 189!), for a 4ire of len+th ℓ and thickness 2 ρ, s#annin+ a circular area of radius r = 4e have L 2ℓ ( lo+eQ8r < ρR - 2). ?ere 4e have ℓ 2.2 c, as each half of the e6citer loo# 4as 2 c lon+, 2r 21 c , ρ 1.% . 'o this value of L, the self-inductance of the t4o 0.% thick, 9 c lon+ 4ires a is still to e added. Ior t4o #arallel 4ires of len+th ℓ' , thickness 2 ρ' , 4hose relative distance is d , 4e +et (see P. Drude, Physik des Aethers, #&!) L !ℓ' lo+e(d < ρ) . ?ere 4e have ℓ ' 9 , ρ' 0.02% , d % . 'hus the su L %%! $ 188 !2 c. 'his is in #recise co#liance 4ith the value of L (!! c) otained fro M<2 and C , ut 4ith soe error as the 4ires could not e run e6actly #arallel ecause of their connection to the ca#acitor #lates. 1! E00-1F "f the ca#acitor C reains for several days in #etroleu, then the thick, 1% 4ide, 12 c lon+ eonite ars carryin+ the #lates of the ca#acitor end noticealy. ithin the oservation tie et4een t4o easureents ho4ever (2 hours), such deflection is not noticeale.
9 earthed. At a #late distance d !.8 c the follo4in+ 4ere otained/ M<2 22!.% c if oth horiontal ars 4ere insulated= M<2 22%.0 if one horiontal ar 4as +rounded= M<2 22.0 if oth horiontal ars 4ere +rounded. - E#02F Ior saller #late distances d , the chan+es of M<2 y +roundin+ the horiontal ars 4ere sall or i#erce#tile. ince, durin+ the oservations of the coils, only one horiontal ar 4as +rounded, and the distance d 4as alost al4ays saller than c, the resultin+ ca#acitance chan+e 4as ne+li+ile assuin+ that easureent accuracy of 0.2%C is acce#tale. H "n contrast, such a ca#acitance effect 4as very noticeale 4hen the eonite holders e, e (Ii+. 2.) 4ere re#laced y etal stri#s, 4hile e', e' (Ii+. 2 ) 4ere ade of eonite. "n that case= at d !.8c/ M<2 2% c 4ith the horiontal ars isolated= M<2 2!%.% c 4ith one horiontal ar +rounded= M<2 2%.% c 4ith oth etal ars +rounded. Tsually the vertical rass holders 4ere re#laced y eonite holders. 'he results of caliration 4ere #lotted as a +ra#h, and the corres#ondin+ M<2 at any d taken fro it. Ii+. ! is a scaled re#roduction of that curve.
Ii+. !. E#01F A second caliration ethod of 4ave e6citers in the ran+e M<2 & to M<2 12 is discussed later (in Part "") Esee #%98 - %99F.
10 . Transfer to large coils of the results obtained on small coils
ince the natural resonances of the test coils could not e6ceed the corres#ondin+ half-4avelen+th of M<2 & or 12 , de#endin+ on 4hether the first or the second e6citer caliration ethod 4as used, only relatively sall coils 4ere e6ained. "t is #ossile ho4ever to transfer the results otained for the to lar+er, +eoetrically siilar coils, ecause a conse>uence of a64ells electroa+netic-field e>uations is that the natural oscillation periods of geometrically similar systems scale exactly in proportion to the physical dimensions 1%.
!. (ffect of the nature of the coil core and and its area on the self-resonance period
By 4indin+ a #articular ty#e of 4ire in e6actly the sae +eoetrical arran+eent on cylinders of different aterials, the self-resonance period of the coil increases with the dielectric constant of the coil core = ut the rate is slo4er than #ro#ortional to the s>uare-root of its dielectric constant. - E#0F 'his is easy to understand, since the #eriod of the coil ust increase #ro#ortionally 4ith the s>uareroot of dielectric constant of the environent 4hen the coil is in an infinite hoo+eneous ediu. 'he fundaental electric oscillation no4 takes #lace in such a 4ay that, in the iddle of the 4ire len+th the oscillatin+ current has a6iu a#litude of viration, in contrast to the #otential at the ends. 'he ends therefore have #eriodically-varyin+ #ositive and ne+ative free electric char+e. Bet4een the coil ends therefore, there are induced electric field-lines, ostly outside of the coil, ut to a sall e6tent also inside the coil= and in the latter case the ore so the shorter the coil is relative to its diaeter. "f no4 the dielectric constant increases in the interior of the coil, it ust increase et4een the coil ends, thus increasin+ the self-resonance #eriod of the coil 4hen the dielectric constant of the core is lar+e= s#ecifically, ecause of the increasin+ density of electric-field lines in the interior of the coil, the ca#acitance increases as the coil ecoes shorter relative to its diaeter. Coils on hollow cylindrical insulating material therefore have shorter self-resonance periods than coils on solid cylinders= the ore so, of course, the thinner the E4all of theF hollo4 cylinder is. If the coil is immersed in a bath of liquid insulator (#etroleu) instead of air, the selfresonance period will increase in consequence (ecause of the electric field lines outside the coil). Some e)amples to illustrate this proposition/
A coil of 100 turns of 1 thick are co##er 4ire, of 1% internal diaeter and 2& c hei+ht, 4as #roduced. Denotin+ the 4avelen+th corres#ondin+ to the natural electrical oscillation in air M (4here M ×1010 T , and T is the self-resonance #eriod), it 4as found that, if the coil 4as in air, M<2 2& c Ef 0 %!. ?F. But 4hen the coil 4as lo4ered into an 11 c 4ide +lass container filled 4ith #etroleu, it 4as found that M<2 &0 c Ef 0 !1.& ?F. 'he ratio of the 4ave len+ths &0<2& 1.1 is soe4hat saller than the s>uare root of the dielectric constant of #etroleu (@ 1.!1) ecause #art of the coil (2 c lon+) 4as still stickin+ out of the #etroleu. - E#0!F "f the coil 4as reoved fro the #etroleu and #ushed onto a +lass tue of 1% outer diaeter and 1.2 4all thickness (4ithout chan+in+ the #itch or len+th of the coil), it 4as found that M<2 290 c Ef 0 %1. ?F. 'he sall increase of M<2 fro 2& c to 290 c is caused y the 1% E02-1F (lictrischen Schwingungen um einem stabfrmigen /eiter0 behandelt nach der #a)well'schen Theorie E5lectrical oscillations around a rod-sha#ed conductor, treated y a64ells theoryF, . Araha, Ann. Phys. 02(11) 1898 (ied. Ann. &&). P!%-!2, see #!!2.
11 sall nuer of the electric field lines fro the coil that run in the +lass 4all #arallel to the coil a6is. hen #etroleu 4as #oured into the interior of the +lass tue, then the self-resonance 4avelen+th M of the coil 4as not si+nificantly increased (ecause the coil is very lon+ and the +lass 4all is rather thick in co#arison to coil diaeter)= ut 4hen distilled 4ater 4as #oured into the +lass tue, M<2 increased to %! c Ef 0 !2. ?F. H o4 the 4ater 4as #oured out a+ain, and the e#ty +lass tue 4ith the 4ound coil 4as #laced in the iddle of a thin 2 c hi+h, ! 4ide cylinder of thick +lass. 'he half-4avelen+th 4as then M<2 20 c Ef 0 !&.8 ?F. 'he increase fro 290 c to 20 c is caused y the electric field lines on the outside of the coil, 4hich #artly run in the 4all of the outer +lass cylinder. Petroleu 4as then #oured into this, and the half-4avelen+th increased a+ain to M<2 !0 c Ef 0 !!.1 ?F. ?o4ever, if a lar+er outer +lass cylinder 4as used (11 c diaeter) , fillin+ it 4ith #etroleu +ave M<2 &! c Ef 0 !1.2 ?F. As another e6a#le, a coil 4hich 4as short co#ared to its diaeter 4as chosen. 10 turns of 1 thick co##er 4ire, 4hich 4as 2 thick includin+ its insulation, 4ere 4ound onto an eonite cylinder of %.9 c diaeter and 2. c hei+ht 1&. 'he individual turns 4ere #laced close to+ether, so that the overall hei+ht of the coil 4as 2 c. 'he len+th of the co##er 4ire 4as 192 c. A coil of e6actly the sae len+th 4as 4ound on a +ood dry oak cylinder of the sae diensions. Both cores 4ere then ored out, so that the coils 4ere co#ared on hollo4 cylinders, of l.% 4all thickness for eonite, and .% thickness for 4ood. - E#0%F 'he follo4in+ self-resonance 4avelen+ths 4ere otained/ 5onite hollo4 cylinder ood hollo4 cylinder 5onite solid cylinder ood solid cylinder
M<2 < c &% 8& !0& !!0
f 0 < ? !1.1 8.8 &.9 !.1
"t follo4s that 4ood has a lar+er dielectric constant than eonite Ehard ruerF. o4 this is indeed the case, as 4as estalished directly y cuttin+ thin 0.% #lates, ade fro the sae #iece of 4ood, and co#arin+ the 4ith eonite #lates et4een the diaeter 1 holdin+ #lates of ca#acitor 18 , 4hich accordin+ to a #reviously descried ethod 19 is o#erated at the resonance line of a sall Blondlots e6citer 20, 4hich +enerated electrical oscillations of c 4avelen+th easured in air E!11 ?F. 'he ca#acitor sho4ed the +reatest ca#acitance (and si+nificant electrical asor#tion) for the 4ood fires cut #er#endicular to its #lates, lo4er ca#acitance (and no electrical asor#tion) for 1& E0!-1F 'he ends of the coil 4ere held in #lace y sall indentations in the eonite cylinder. 1 'his is an e6treely sall ca#acitor. 'he diaeter 4as #ossily c rather than . 18 E0%-1F 'he #lates had the follo4in+ thicknesses/ ood, #er#endicular to the fires 0.!28 ood, #arallel to the fires 0.!!2 ood, #arallel to the fires 0.!% 5onite disk 0.!&% 'hey fitted ti+htly et4een the ca#acitor #lates 19 E0%-2F (ine methode ,ur messung der Dielectric. const. . . , P. Drude, Annalen der Physik, 29() 189 (ied. Ann. &1). #!&&-%10. 20 ee also Blondlots ori+inal #a#er Sur un nouveau proc1d1 pour transmettre des ondulations 1lectri2ues le long de fils m1talli2ues0 et sur une nouvelle disposition du r1cepteur E3n a ne4 ethod for transittin+ electrical 4aves alon+ etal 4ires, and a ne4 receiver arran+eentF= Blondlot, 7o#tes endus de lAcadUie des ciences, vol. 11!, 1892, #. 28 - 28&.
12 fires of 4ood cut #arallel to its #lates, and the sallest ca#acitance for the eonite. 'he dielectric constant of eonite is only sli+htly saller than that of the 4ood #lates cut 4ith fires #arallel, ut on the other hand it is sustantially saller than the dielectric constant of the 4ooden #late cut 4ith fires #er#endicular. 'he Dielectric constant of the 4ood is thus +reatest 4ith fires #arallel to the ca#acitor #lates, ut also still +reater than the Dielectric constant of 5onite 4hen the fires are #er#endicular. 'his is in a+reeent 4ith the easureents on 4ood ade y i+hi21 and ack 22 usin+ electric irefrin+ence Edoule refractionF. - E#0&F 'he latter has oserved the t4o electrical refractive indices, in fir 2 #articularly/ n 1=1.%
, n 2=2.1% ,
2
n1 =.0& 2 n1 =
#er#endicular to the fires !.&2 #arallel to the fires
#ecifically2! " have not easured the dielectric constant of 4ood= ecause it 4ould e necessary to ierse it in a li>uid of the sae dielectric constant, and it 4ould not e #ossile to assess safely the chan+e to the dielectric constant that the 4ood under+oes throu+h the ca#illary action of the li>uid. 3n the other hand " have deterined the dielectric constant of eonite, usin+ this ty#e of null ethod2% (y iersion in enene - acetone i6tures), @ 2.9, and that 4as e6actly the sae value for t4o eonite #ieces of different ori+in, 4hich 4ere used in the coil cores, oth in the direction #arallel to the a6is of the eonite cylinder rather than in the direction #er#endicular to the a6is. 'he electrical asor#tion of the 4ood in directions #arallel to the fire 4as noticeale in the coil2&/ 4ith the coil on the solid 4ooden cylinder the e6citer loo# had to e closer (1 c), than 4ith the coil on the solid eonite cylinder (the distance 4as 21 c fro the e6citer loo#) to otain e>ually distinct resonance indication in the vacuu tue. 5ven the inductive e6citation of coil on the thin hollo4 4ooden cylinder 4as noticealy 4eaker than 4ith the coil on the eonite cylinder. For the construction of Tesla transformers it is therefore best to avoid wood cores, and preferably to use no cores 2 or cores made from ebonite, or possibly also glass rods (or tubes 28.
- E#0F hen a +ood conductor is #laced in the interior of the coil, the intensity of the e6citation is consideraly reduced and also the self-resonance 4avelen+th of the coil is shorter. o, in the coil on the hollo4 4ooden cylinder, M<2 of 8& c Ef 0 8.8 ?F decreased to M<2 !! c Ef 0 !.& ?F, as a c hi+h 0.% thick hollo4 rass cylinder of %2 outer diaeter 4as 21 E0%-F Doppelbrechung der electrischen Strahlen 3The birefringence of electrical rays4 , A i+hi, Ann. Phys., Nol. 291(&) 189% (ied. Ann. %%), #89-90. Also #ulished as/ A. i+hi, e. . Acc. della c. Bolo+na !, 189!. #!8. 22 E0%-!F Doppelbrechung der electrischer Strahlen0 *. ack, Ann. Phys., 290(-) 189% (ied. Ann. %!), #!2- . Eee also revie4 Double refraction in wood, ?allock, cience, Au+. 2, 189%, #29-2!0.F 2 E0&-1F Doppelbrechung der electrischer Strahlen0 *. ack, Ann. Phys., Nol 292(12) 189% (ied. Ann. %&), #1-2. ee #a+e 29. 2! E0&-2F 'he t4o dielectric constants of ash are crudely Gud+ed to have the values s#ecified y ack for fir. 2% E0&-F ee/ #ethode ,ur bestimmung der Dielectricit%tsconstanten fester 5rper Eethod for deterinin+ the dielectric const. of a solid odyF, ?. tarke, Ann. Phys. 29&(!) 189 (ied. Ann. &0), #&29-&!1 = also/ ()perimentel-untersuchung *ber electrische dispersion einiger Organiscuer S%uren0 (ster0 und von ,ehn 6lassorten E56#eriental study of electrical dis#ersion of soe or+anic acids, esters, and 10 ty#es of +lass.F, *. I. ;V4e, Ann. Phys. 02 (11) 1898 (ied. Ann. &&), #90-!10, %82-%9&. 1898. ee #!02. 2& E0&-!F Ior very lon+ thin coils, less= ut in shorter ones, ore. 2 E0&-%F 'he coil can e su##orted usin+ soe thin eonite rods, or even thin etal rods. 28 E0&-&F 7ardoard tues also asor to soe e6tent.
1 inserted into the coil interior, and at the sae tie the distance et4een the coil and the e6citer loo# had to e reduced fro 1 c to 1 c to restore clear indication fro the vacuu tue. 'his rass cylinder 4as also introduced into the hollo4 eonite cylinder 4ound 4ith thinner insulated 4ire (22&.% c 4ire len+th), resultin+ in the decrease of M<2 fro %& c Ef 0 2&.! ?F (4ithout rass cylinder ) to M<2 !1%c Ef 0 &.1 ?F (4ith rass cylinder). Both results, oth the 4eakenin+ of e6citation and the reduction of the natural #eriod, can e e6#lained y the induced current in the rass cylinder (tertiary) flo4in+ in the o##osite direction to the coil current and hence reducin+ the self-inductance of the coil (as does the secondary current of a transforer). hen chan+in+ the nature of the coil core, the sae chan+es in M occur in these short coils as in the lon+ coil discussed earlier on #0= e6ce#t that the effects are even clearer ecause the coil is short and 4ide, so there are ore electric field lines inside the coil (see aove #0). "f, for e6a#le, the coil on the 4ooden hollo4 cylinder (M<2 ori+inally 8& c Ef 0 8.8 ?F ) 4as #ushed onto a & c tall +lass eaker of 1 4all thickness (4ith the 4ooden cylinder still #resent), so M<2 increased to 9 c Ef 0 .8 ?F. hen #etroleu 4as #oured into the eaker, so M<2 increased further to !12 c Ef 0 &.! ?F (as on #0! for the thin coil, the introduction of #etroleu into the +lass tue had ne+li+ile effect). hen 4ater 4as #oured into the eaker instead of #etroleu, M<2 increased still further to %11 c Ef 0 29. ?F. "n the coil on the hollo4 eonite cylinder, a 4ooden core ( hornea) that fitted 4ith 1 #lay 4as inserted= M<2 then increased fro &% c Ef 0 !1.1 ?F to !11 c E&.% ?F, i.e., the coil took an interediate #osition et4een the solid eonite cylinder and the solid 4ooden cylinder. - E#08F educin+ the len+th of the coil on the hollo4 eonite cylinder, either y 4indin+ fe4er turns, or the sae nuer of turns of thinner 4ire, reduces the effect of the inserted 4ooden core= ecause for a short coil, the electric field lines run ore at the coil surface , i.e., in the eonite cylinder, as the coil ends +et closer to+ether. 'he follo4in+ tale +ives inforation/ is the hei+ht of the coil (i.e., the distance et4een the centres of the end turns, see Ii+. %.), 2r is the avera+e diaeter (4hich is found usin+ 2K r · n ℓ , 4here n is the nuer of turns and ℓ is the len+th of the coil 4ire). M1 is the self-resonance 4avelen+th of the coil on the hollo4 eonite cylinder, M2 is the self-resonance 4avelen+th after insertion of the 4ooden core. h
(ffect of a wood core in a hollow ebonite cylinder. h < c h<2 r M1 M2 M2 < M1
2.0 1.2 1.0 0.&& 0.%%
0.2 0.20 0.1& 0.11 0.09
&% %& %08 !02 &0
!11 &2 %!9 !1 9
1.1 1.10 1.08 1.0! 1.0%
1! 7. (ffect of wire insulation on the self-resonance period of the coil Thin sil! insulation exerts no influence on the natural period of the coil, whereas thic!er insulation increases the natural period by " - # $, specifically, the more so the shorter the coil relative to its diameter . Ior e6a#le, a coil of hei+ht h 1!.9 c, 4ith n !8 turns of 1 thick are 4ire of len+th ℓ !&1 c, 4ith an e6actly constant #itch 29 of .1, 4as 4ound on a
4ooden core of 2.9& c diaeter= the self-resonance half-4avelen+th 4as then M<2 ! c.
- E#09F o4 this 4ire 4as un4ound and re#laced y a 1 thick 4ire 4ith 4a6ed cotton douleinsulation, of 2.1 thickness overall= the len+th 4as a+ain ℓ !8 c, and M<2 &8 c. ?ence/ M<2 / ℓ 0.% 0.0
Plain 4ire "nsulated 4ire
h/2r
!.8 !.1
'he ratio h/2r , the coil hei+ht to the coil diaeter, is no4 not the sae in the t4o cases= and since M<2/ℓ de#ends on this relationshi#, this ust e considered in order to assess the effect of the 4ire insulation alone. 7orrectin+ for this (as e6#lained elo40) +ives/ h<2r
Plain 4ire "nsulated 4ire
!.8 !.8
M<2ℓ 0.% 0.&
p < C
1.8
i.e., the sole effect of the 4ire insulation causes a 1.8 C. increase in the ratio M / 2 ℓ . 'his effect could e verified in another 4ay/ An eonite cylinder of 2.2 c diaeter 4as 4ound, on the lathe, 4ith a 0.! thick co##er 4ire 4ith thin silk insulation and a 0.& thick cotton thread 4hich lay Gust et4een the turns of 4ire. After the self-resonance #eriod 4as deterined, the thread 4as un4ound 4hile the 4ire turns e6actly retained their ori+inal #ositions. 'his al4ays resulted in a si+nificant reduction in M<2. 'he results 4ere as follo4s/ - E#10F n
%% 29 1
h<2r
M<2ℓ 4ith thread 2.1 Q 1.0 1.0&0 4ithout thread 4ith thread 1.0 Q 1.!1% 1.9% 4ithout thread 4ith thread 0.2 1 Q 2.!2 2. 4ithout thread
p < C
1.& 1.! .8
Thus cotton insulation, which is about as thic! as the wire, increases the self-resonance period in coils which are at least as high as wide by about "%& $, and with shorter coils more (i.e.,
aout !C). "t is assued that the insulated 4ire turns touch each other, or at least that their distance is not +reat. - 'his result can e easily understood fro section , since the insulation has a +reater dielectric constant than air. 29 E08-1F Ior this #ur#ose, a shallo4 thread 4as cut in the 4ood core on the lathe. 0 56#lanation of the correction #rocedure is issin+ fro the ori+inal #a#er. 1 E10-1F 'his oservation refers to a thick eonite cylinder of %.0 c diaeter.
1% 8. 9oils with uneven pitch
i6 turns of 1 thick are co##er 4ire, 4ith a constant #itch of % , 4ere 4ound on an oak core of 12. c diaeter. "t 4as found that M<2 !&2 c Ef 0 2.! ?F. 'he 4ire ends 4ere held 4hile the iddle turns 4ere co#ressed to #itch and the #itch at the end coils increased= this increased M<2 to %%! c Ef 0 2.1 ?F. ?o4ever, 4hen the end coils 4ere co#ressed to #itch, 4hile the #itch of the iddle turns increased, M<2 chan+ed to !!! c Ef 0 .8 ?F. 't a fixed coil height h and wire-length ℓ , a coil with narrowed central turns has a slower oscillation period, and a coil with narrowed end turns a faster oscillation period, compared to a coil of constant pitch . 'his result can e easily understood on the asis that the ca#acitance of the coil de#ends essentially only on the coil hei+ht h, it ein+ created y the electric field lines s#annin+
fro the one end of the coil to the other, 4hereas the self-inductance of the coil arises fro the current-carryin+ turns in the iddle. - E#11F 'hus, if the #itch g is decreased 4hile h reains constant , the self-inductance of the coil increases, 4hereas the ca#acitance reains constant2= hence M ust increase. "n order to otain definite conditions, the coil should therefore e ade 4ith constant #itch, as is, in #ractice, al4ays the case in coils of insulated 4ire 4ith the turns #ushed close to+ether. 'he follo4in+ considerations relate only to coils 4ound 4ith constant #itch. 'he constancy of #itch 4as achieved either y carefully cuttin+ a coarse thread into the 4ood core (not dee#) on the lathe, or (for insulated 4ire) y #ushin+ the turns close to+ether.
:. The number of characteristic parameters a coil of constant pitch
'he follo4in+ #araeters are re>uired for a coil of constant #itch surrounded y air/ n nuer of turns, g #itch E+an+hVheF, h coil hei+ht, 2r coil diaeter, ℓ 4ire len+th, W 4ire thickness, also thickness and ty#e of 4ire insulation, @ dielectric constant of the core, also its thickness if it is a hollo4 cylinder. By se#arate oservations, it 4as found that the location of the coil on a lon+er core (4hether in the centre of the core or at the end) had no effect= and this also a##lied to the aterial of the su##ort on 4hich the coil core rested , at least if this su##ort 4as also ade of insulatin+ aterial (4ood). o4 to address the >uestion of ho4 the self-resonance #eriod T of the coil, or the self-resonance 4avelen+th M, is deterined y the #araeters of the coil. 'he #araeters are not all inde#endent, ecause there are the follo4in+ relations/ h (n- l) g , ℓ 2r K n
Accordin+ to the siilarity rule +iven earlier on #02, M ust no4 +ro4 in #ro#ortion to the len+th of 4ire ℓ if n reains constant, 4hereas h, r , ℓ , g and W +ro4 in the sae #ro#ortion. 2 E11-1F oeties the ca#acitance also increases, ecause the end turns ecoe closer to soe e6tent even if h reains constant. E11-2F M also reained the sae 4hen the coil 4as not #laced on a su##ort, ut ke#t a4ay fro other oGects y ein+ hun+ u#. 'his Elack of effect of neary oGectsF does not a##ly to coils 4ithout cores, see elo4.
1& - E#12F -
"t ust therefore e that / !
(A)
M<2 ℓ X f (n , h<2r , g
here f is a function of n, h<2r , g
h < c
0.% 1.11 1.22 1.19
W < g < 0.! 0.%2 0.! 0.%1 0.9 2.0 0.! 0.99
n
ℓ < c
12 2 1
10% !1% 10 2%
M<2 < c 28 1100 0 %&9
h<2r
f
2.&% 2.&% 2. 2.!2
g
0.20& 1.29 R thin insulation 0.19 1.2 0.20 2.2& R thick insulation 0.20 2.!
As 4ill e e6#lained later, f M<2/ ℓ is y far ainly de#endent on h<2r . "n the second oservation h<2r is sli+htly saller than the other oservations. educin+% this second oservation to the coon ratio h<2r 0.20&, the value f 2.&! is otained. Iurther, as is already evident fro this tale, f is soe4hat de#endent on g
h<2r
g
f
12 2 1
0.20& 0.20& 0.20 0.20
1.29 1.29 2.2& 2.2&
2.&% 2.& 2. 2.!
- E#1F "ncreasin+ the nuer of turns n fro 12 to 2 has therefore reduced f y less than 1C= 4hereas the increase of n fro to 1 has increased f y 2.%C. 'he last oservation is ho4ever not e6actly co#arale to the #enultiate, ecause the insulation aterial 4as sli+htly different in oth cases&. Ior lar+er values of n, the de#endence of f on n is still insi+nificant, and al4ays reains elo4 1C= as is deonstrated y the follo4in+ tale, 4hich refers to coils 4ith 4ooden cores. 'he sae thin insulated 4ire 4as used in all three cases. 2r < c 1.91 2.0 2.9
h < c
8.0% 9.%1 11.%
W < 0.! 0.! 0.!
g <
n
ℓ < c
1.!0 2.0 .1
%8.% !8.% .%
%0 %0 %0
M<2 < c h<2r M<2 reduc. 02 !.21 02 0! !.1! 0 01 .90 29%
g
.% %.8 8.0
! E12-1F 'here is a factor Y attached to M, ecause then, in a strai+ht thin 4ire in air, f 1. % Drude uses reducirt not in the sense of akin+ saller ut in the atheatical sense of data reduction, i.e., co#actin+ a data set y reducin+ the nuer of #araeters associated 4ith it. & E1-1F "n the latter case , the 4ires 4ere insulated fro each other y a cotton thread. "f 4ound the sae, then f for the n 1 case 4as 1.% C saller than for n .
1 hen ℓ is constant, M<2 is therefore alost constant, i.e., inde#endent of n. 'he increase of +
M<2 ℓ X f (h<2r , g
'he follo4in+ series no4 refers to t4o different values of constant g
2r < c
2.0 to .0
%.8 to &.1
n
h<2r
f
& &0 % !! 0 %% 2 29
%.!0 !.11 .& .01 2.% 2.10 2.10 1.&1 1.0%
0.!1 0.88 0.82& 0.888 0.9&& 1.0&1 1.0& 1.190 1.!0%
10 1
0.2 0.20 0.20
2.11 2.8 2.8
(bonite core g
2r < c
n
h<2r
f
p < C
10
!.11
0.808
2.%
9
.01
0.92!
!.0
%!
2.10
1.110
!.0
!2 29 22 1&
1.&1 1.0% 0.9 0.%&
1.2 1.%21 1.% 2.0!
.& .9
12
0.20
2.80
1&
11 10 &
0.18 0.1& 0.11 0.092
2.88 2.99 .28 .!
2.0 to .0
%.8 to &.1
Iro this tale it is clearly seen ho4, on the one hand, at constant g
18 ;late
19 - E#1%F Ior h<2r [ 0.& , the values of f should e accurate to at least 1 C, as is also clear fro the sooth course of the curve, and fro the fact that 4hen re#eatin+ an oservation (re4indin+ the coil) 8 the differences is less than 1C. "n the #late there is also a third curve #lotted for g
1.0 .00
f
1.09 2.80
1.2 2.&!
h<2r 2.10
2.! 2.8
1.08 1.12
1.2! 1.10
2.! to 2.8 1.0&
. 9oils on wooden cores
As 4as discussed earlier on #0, the self-resonance 4avelen+th of a coil on a 4ooden core is +reater than that of an other4ise identical coil 4ound on an eonite core, and the ore so the saller the value of h<2r. "n addition, there is an effect due to the the ty#e of 4ood= +ood dry (seasoned) cores of ash, eech, hornea, and oak 4ere used. 'he fires ran #arallel to the coil a6is. "f 4e denote f in forula (B ) for a 4ood EholF core as f h , and for an eonite core as f e , and define/ p =
f h− f e f e
. 100
as the #ercenta+e increase of f in chan+in+ fro eonite core to 4ood core, 4here p is inde#endent of g
20 dielectric constant is sustantially lar+er in the direction of the coil a6is than in the #er#endicular direction, then due to the relatively sall nuer of electric field lines that run #arallel to the coil a6is on the inside, there ust e, at lar+e h<2r , a fairly stron+ increase in the ca#acitance of the coil, i.e., an increase in M<2 4ill occur. Ior saller h<2r , the internal electric field lines of the coil 4ill run #artly out of #arallel 4ith the coil a6is, i.e., in directions havin+ a saller dielectric constant. 'herefore, the ore the dielectric constant of the core is +reater in the direction of the a6is than in the #er#endicular direction, the less 4ill e the increase in p 4ith decrease of h<2r . By +ra#hical adGustent, the follo4in+ values of # have een taken fro the tale #rovided, and these are the asis of later calculations. - E#1F p = h<2r
& % ! 2 1.% 1 0.& 0.! 0.2 0.1 0.0%
f h− f e f e
. 100
for 4ood s#ecies.
Ash \ Beech ?ornea % .% %.% ! &.% % & 8.% 9.% 8 10.% 8.% 11 9 11 9 11 9 11 9 10.%
3ak & &.% .% 8 9.% 11 12 12.% 12.% 12.% 12.% 12
<. 9oils on hollow cores =tubes>
ith coils on hollo4 cores, a#art fro the conditions h<2r and g
f h− f e f e
. 100
the result # is inde#endent of g
p <
C
-&.1
-&.1
-&.1
-!.0
21 - E#18F 9oils on glass tubes (eakers). w!r 1<% w!r 1<9 w!r 1<%0
h<2r p <
%.!% -.!
C
2.0 -0.9
0.1 $%.&
0.0!% $.1
9oil on cardboard tube, 4
p < C
-!
9oil on ash-wood tube, 4
p < C
-!.
"t follo4s fro this, as has already een said in section , that the natural #eriod is reduced 4hen w!r is sall. Iro the first series of oservations listed here, coils on eonite tues9, at a certain value of w!r , a##roach in their natural #eriod that of +eoetrically siilar coils on solid eonite cores as h<2r ecoes saller. 'his is also consistent 4ith the oservation ade on #08, 4hich is that the effect of #ushin+ a 4ooden core into the eonite tue is saller the saller the value of h<2r . Ior increasin+ h<2r at constant values of w!r , coils on tues ecoe ore like coils 4ithout solid core, and indeed this is ore the case as w!r ecoes saller, and also as the dielectric constant of the tue aterial ecoes saller. "n fact, 4e see this confired y the coils on eonite tues (and +lass), of w!r 1<20. As 4e 4ill see in the ne6t section, for a coil of h<2r 0. 4ithout core, the value p -1 C. Ior the coil on an eonite tue 4ith this value of h<2r , the value of p -10C , and for the coil on a +lass tue p -!C. - E#19F Ior h! 2r %.&, for the coil on a +lass tue, p -&.1 C. Ior a coreless coil is p -.% C. Ior eonite tue 4ith w!r 1<20, at h<2r %.&, p ust therefore lie et4een -.% and -&.1C, at aout p -C. Assuin+ that, et4een h<2r %.! and h<2r 0.2 4ith the eonite tue, p chan+es alost!0 linearly fro p -C to p -10 C, 4e otain the follo4in+ tales for p / 9oils on ebonite tubes, w!r 1<20
h<2r p < C h<2r p < C
0.0! -
0.0% -!
0.0& -%
0.0% -&
0.09 -
0.10% -8
0.1 -9
0.1& -10
0.2 -10.%
0.2 -10
1.0 -9.%
1.% -9
2.2 -8.%
.2 -8
!.2 -.%
& -
9oils on glass tubes, w!r 1<20
h<2r p < C
0.0! $10
0.0& $9
0.08 $8
0.1 $
0.1%
%$0.2 $
0.2% 0
0. -2
0.% -!
0.! -%
0. - &.0 -&
9 E18-1F 'he oservations on the coils on +lass tues are not as +ood in co#arison 4ith each other, due to variation of dielectric constant 4ith +lass ty#e. !0 E19-1F 'he tale 4as otained y +ra#hical adGustent Ei.e., soothin+F. 'he error 4ill not e6ceed 0.%C.
22 ?. 9oils without cores
A ethod for #roducin+ such coils that 4orked >uite 4ell 4as first to 4ind the on a solid core, then reove the carefully and co#ress the turns to+ether 4ith li+ht #ressure y indin+ the 4ith three #ieces of t4ine, so that a +ood cylindrical sha#e 4as restored. In coils without solid (or li>uid) cores the shortest self-resonance periods are to be expected . 'hey also 4ork 4ell in fact. Due to the asence of any asor#tion, the res#onse of the coil at resonance is of course fla4less, and also a coil of this construction has the sallest #ossile ca#acitance= so secondary coils without a core are best for Tesla transformers . - E#20F ('here is ho4ever the >uestion of ho4 to #roduce the est coil technically, 4ithout it ein+ too easily deforale.) 'he #ercenta+e chan+e p of the coefficients f in e>uation (B) #1 at the transition fro coil 4ith eonite core ( f e ) to +eoetrically siilar coil 4ithout core ( f 0 ), 4ill e denoted a+ain y p =
f 0− f e f e
100
'he follo4in+ results 4ere otained ( g
h<2r p <
C
!.1 -8.!
2.0 -9.1
1.&8 -12.
1.08 -1!.%
0.19 -1.1
'hat p ecoes steadily saller Eore ne+ative]F 4ith decreasin+ h<2r is to e e6#ected, ecause a coil core increases the natural #eriod ore as h<2r ecoes saller. Plottin+ the values of p +ra#hically, 4e otain the follo4in+ re#resentation/
Ii+. &. 'he oserved values are arked y crosses ^. Iro this curve , the follo4in+ tale sho4s the calculation of the coefficients f 0 for coreless coils. f 0= f e 1− h<2r p <
C
0.2 1
0.! 1&.%
0.& 1&
0.9 1%
1.2 1!
1.% 1
p 100
1.8 12
2.1 11
2.% 10
.0 9
!. 8
&.0 .%
2 - E#21F 'hese coils 4ere freely sus#ended y a cotton thread. hen they were placed on ebonite, wood or glass, the period was increased as a consequence , s#ecifically/ 4ith h<2r 1 4ith h<2r 0.2
around %C restin+ on eonite around 8C restin+ on 4ood or +lass around !C restin+ on 4ood
"f h<2r is very sall, the value of p +iven in the tale should only e a##lied 4hen the 4ire insulation is not too thick (not lar+er than the 4ire thickness), ecause other4ise the coil 4ill ehave as thou+h it is 4ound on a hollo4 core, i.e., p 4ill e saller.
@. Tables for calculating the self-wavelength of a coil
'he tales set out here for convenient use 4ere otained y +ra#hical inter#olation fro the oservations of coils on solid eonite cylinders (see #1!), ecause these can e 4ound e6actly and the aterial of the coil core is 4ell defined. Accordin+ to sections , 8 and 9, after the values of f 4ere calculated for 4ood cores and hollo4 cores and vice versa, the oservations of 4ood and hollo4 cores 4ere used to su##leent the oservations of eonite cores 4ith very sall h<2r . Ior 4ooden cores, oservations 4ere also ade 4ith lar+e values of g
!1 E21-1F "f the 4indin+s lie in 4ood, then f is aout 2C +reater than if the turns are #ushed to+ether 4ith cotton insulation. "f the corres#ondin+ value of f fro the tales is not stated directly, it is easy to see that the values of f in the 7oluns a) need to e enlar+ed y aout 2C.
2! - E#22F -
2% - E#2F -
2& - E#2!F 'he data in the tale on #22 are ore reliale than those in the tale on #2, in 4hich there ay e errors due to varyin+ 4ood te6ture. 'he ost reliale data for f are for solid eonite cores= 4here for h<2r [ 0., the accuracy is at least 1C, and for h<2r Z 0. it is at least 2C. Ior 4ood cores of sall h<2r (Z 0.1) deviation of the data in the tale is #ossily as uch as %C due to variale 4ood te6ture, ut in +eneral, includin+ 4ood cores, the deviations of the data in tale are 4ithin 2C. Ior tues of saller 4all thickness than w!r 1<20, f naturally lies et4een the values that the tale +ives for tues 4ith w!r 1<20 and for coreless coils.
. +ppro)imate theory of self-oscillation of a long thin coil
hen the current in the coil is constant, then (for lar+e values of h<2r strictly, and at least a##ro6iately 4ith saller h<2r ) the self-inductance of the coil is/ (1)
2
n L =! π " h
Ec+sF
,
4here > K r 2 is the coil cross section. 'herefore Esince the 4ire len+th ℓ _ K 2r n F/ L ℓ 2 < h
'here is still a factor (2
2 2 ℓ L =
πh
'he ca#acitance of the coil can e evaluated in the follo4in+ 4ay/ 5lectric char+e i+rates to the coil ends. ;et us ia+ine the char+e `e on t4o s>uat cylinders (ut 4hose hei+ht ay include several turns) lyin+ at the ends of the coil= 4ith the se#aration of the cylinders e>ual to the coil hei+ht h, and their radii e>ual to the coil radius r . "f 4e vie4 these short cylinders as infinitely thin circular rin+s (circles) of radius r , the #otential is easily co#utale. "n the centre of the circle 4e construct a line of len+th a #er#endicular to the #lane of the loo# ( Ii+. ) and #er#endicular to this a line of len+th r' . - E#2%F At the end #oint P of this line is then the #otential +enerated y the circular line, 4hich can e calculated usin+ s#herical haronics fro the forula/ V = 2 π
∑=0 ( ρ ) P (0 ) P (μ ) ∞
r '
n
n
n
n
4here is the char+e #er unit len+th of the circular line, and a
μ= cos β= ρ ,
2
ρ
=a 2+ r 2 .
2 o4 all s#herical haronics of ar+uent ero of odd order n are e>ual to ero / ( n)
P μ
=0 , if n is odd.
Iurtherore/ ( 0)
=1
( 2)
= μ2 −
P μ P μ
2
% 2 !
(!)
% 2 2
1 2 !
= . μ !− . μ 2 + .
P μ
9 11 & % 9 ! % 2 1 % μ − . . μ + . . μ − . . 2 ! & 2 2 ! 2 ! 2 2 ! &
( &)
P μ
1 2
= . .
etc.
"f the #oint P is very close!2 to the circle, then 4e have/ a0
,
r' r ρ
, J0,
hence/
{
V = 2 π 1+
1 9 + + . . . ! &!
}= π
2 .2
,
or , if 4e introduce the char+e e of the 4hole circle/ e 2 K r X V =
,
2e r
- E#2&F 'his co#onent of the #otential occurs in the coil in addition to the co#onent that results fro the -e char+ed circle at a distance a h . Because r' r and a h , this co#onent is +iven y/ e ∞ 1 ( 2n ) ( 2n ) V ' =− P 0 P μ 2 2 n r n=0 ( 1+ h / r )
∑
,
in 4hich !2 E2%-1F "nside the circle itself, the series for V 4ould e diver+ent, as there is a hy#er+eoetric series (b Y , O 1, 6 1) and this diver+es, see :ausss 4ork on the hy#er+eoetric series, section 1%. "n reality, of course, 4e do not have V , ecause the char+e is not on an infinitely thin circular line. 'he finite sie of the char+e sustituted is therefore reasonale, ecause P only ecoes lar+e near the circle. 'hen the series for V has a##ro6iately the value V 2 K . 2, strictly V 2 K . 1.9.
28 μ 2=
1 2
1+ r / h
.
2
'herefore, the total electric #otential at one end of the coil is ( 2n)
( 2n)
∞ P P e μ 0 V 1=V +V ' = 2− 2 2 n r n=0 ( 1+ h / r )
∑
,
and at the other end of the coil the #otential is V 2 - V 1
'he #otential difference et4een the coil ends is therefore V 1−V 2= 2V 1=
e C
,
here C is the ca#acitance of the coil. 'herefore C =
{
r (2n )
∞
2 2 −∑ n= 0
( 2n )
P 0 P μ
( 1 +h 2 / r 2 )n
}
'he nuer of factors is no4 in need of correction ecause of the assu#tion that the entire char+e of the coil should e concentrated on t4o circles at the ends. As the char+es of the coil are distriuted, not on t4o circular lines, ut on several turns of 4ire, one can think of it as havin+ een re#laced y t4o circular cylinders of finite 4idth, and so the ca#acitance 4ill e soe4hat lar+er than in the aove forula. e can therefore #ut ()
C =
{
α r (2n )
∞
2 2 −∑ n= 0
( 2n )
P 0 P μ
( 1 +h 2 / r 2 )n
}
,
in 4hich b [ 1. 'he nuerical factor b 4ill e all the +reater than 1 as h!r increases, ecause the char+es of the coil 4ill then e s#read over ore turns of 4ire. - E#2F o4, althou+h the nuerical value of b is not deterined 4ith certainty, it is still reasonale to assue that b is soe4hat de#endent on h!r , so 4e can 4rite/ (!)
C r (r!h)
,
4here is a function of the ratio r!h.
29 o4 the electrical resonance #eriod T of a syste of self-inductance L and ca#acitance C m is deterined (in electroa+netic units) y the 'hoson-*irchhoff forula/ T = 2 π √ L C #
,
'herefore the resonant 4avelen+th M is +iven y the forula (%)
λ=2 π √ L C ,
4here C is the ca#acitance in electrostatic units. Tsin+ the calculated values of L and C here, 4e +et (&)
√
2 2 ℓ λ= 2 π π
r h
φ( r / h ) = ℓ χ( r / h )
i.e., the result is the formula (B) of #1 when the dependency of M on +
()
2
2
2
2 +h / r + r / h C = 2 α r , 2 2 2 2 10 + !h / r +r / h
i.e., accordin+ to (2) and (%)/ (8)
√
2 2 2 2 λ = 2 ℓ α π r . 2+ h / r +r / h 2 h 10+ !h 2 / r 2 +r 2 / h 2
Iro this it follo4s/ (9)
f =
ℓ 2λ
=
√
2
2
2
2
r 2+h / r + r / h 2 α π . h 10+ !h 2 / r 2 +r 2 / h 2
'his forula is co#ared elo4 4ith the e#irical results for coreless coils 4ith g
0 - E#28F 'he follo4in+ values for 2bK are otained! / h<2r
& %.% % !.% ! .% 2.8 2.& 2.2 2.2 2 1.8 2bK !.& !.&! !.%2 !.!2 !.2 !.22 !.1 !.1& !.1 !.10 !.0 !.0! !.01 h<2r
2bK
1.& .98
1.! .9
1.2 .89
1.0 .88
0.9 .8
0.8 .82
0. .82
0.& .9
0.% .&9
0.! 0.% .%& .8
0. .1
'hus 4e see that, as h<2r increases, b also increases soe4hat, as e6#ected. ithin the interval 2.2 h<2r 1.0 forula (9) is fulfilled to 4ithin % C, and the avera+e value of b 4ould e thus/ 2bK .9 , b 1.2& . 'he theoretical considerations therefore a##ly a##ro6iately. H 'he a+reeent is even etter for the coils on solid and hollo4 cores, as is deonstrated in the follo4in+ tale Eon #29F, in 4hich the values of f .h!r for +
f .h!r bK
"n fact it is #articularly evident for the coils on 4ooden cores that the Product f .h!r is constant over lar+e intervals of h!r , so that the table can well be used for calculating the value of f at any h!r that is not listed in the tales on #22 and 2 !!.
! 'he values in the tale have soe istakes and roundin+ errors. ecalculated values are +iven in A##endi6 1. !! E28-1F 3n the other hand, it ecoes very #ractical to adGust the oservation error y adGustentg of the value of f h!r . 'his 4as #artially done in the #re#aration of the tales on #22 and 2 . g E +ra#hical adGustent, i.e., soothin+ the data. F
1 - E#29F -
2
- E#28F . 9oils of few turns and simple loops
As the tales #22 and 2 sho4, for soe sall values of h<2r (i.e., h<2r 0.08 to 0.0%, de#endin+ on +
n
h<2r h < c
W < 0.! 0.! 1.0 0.! 0.! 0.! 2.0 2.%
ℓ <
M<2 < c 20 &22 10 % 10 !10 10 !09 18 2!% 18 2% 18 2% 2! 2%9
f
2.0 2.20 2.!1 2.!0 1.1 1.!0 1.!0 1.0&%
core 3ak ,, ,, ,, ed eech ed eech, 4ire in +roove ed eech, 4ire in +roove Air (no core)
'he last four ro4s a) ) c) d) of this tale are ased on n 1, i.e., on the natural wavelength of a simple loop !%. 'he 4ire circuit 4as nearly closed, the distance $ of the 4ire ends 4as chan+ed fro 2 c to 0.% c, 4ithout affectin+ f . )either (as the tale also sho4s) does the oscillation period of a simple circuit depend on the wire thic!ness !&. "n case a), the 4ire lay on a 2.% c thick %.% c 4ide 4ooden rin+= in cases ) and c) in a 0.% c dee# seicircular +roove in this 4ooden rin+. "n cases ) and c), f a##ears to e sli+htly lar+er than in case a)= ecause the 4ire, lyin+ in the +roove, is ore surrounded y 4ood, 4hich has a dielectric constant lar+er than that of air. !% E0-1F ince the 4avelen+ths of these si#le loo#s 4ere uch saller than those of the coils, the easurin+ ca#acitor C 4as used 4ithout Petroleu fillin+. !& E0-2F 'hat is, Gust as a##lies for a strai+ht 4ire, as lon+ as the 4ire thickness is ne+li+ile co#ared to the 4ire len+th. ee/ Die electrischen Schwingungen um einen stabfrmigen /eiter0 behandelt nach der #a)well'schen Theorie' E5lectrical oscillations around a rod-sha#ed conductor treated accordin+ to a64ells 'heoryF . Araha, Ann. Phys. 02(11) (ied. Ann. &&). # !%-!2, 1898. ee #!1.
- E#1F "n case d) the 4ire 4as su##orted only y four thin 4ooden s#okes= this case therefore corres#onds to ein+ surrounded Gust y air, 4ith the loo# alost closed !. et for this f 4as &.% C lar+er than 1, 4hile for a strai+ht thin 4ire f 1. The self-resonance half-wavelength of the nearly-closed thin!8 wire loop is *%& $ larger than its length . 'he increase of the #eriod of a strai+ht 4ire y endin+ it into to a circle is >uite understandale, since the self-induction 4ill therey decrease alost i#erce#tily, 4hile the ca#acitance 4ill increase!9.
! E1-1F 'he distance et4een the loo# and the e6citer level 4as &% c, and even then the intensity of the oscillations in the loo# 4as so lar+e that the vacuu tue 4as oserved to +lo4 at a distance of 1 c fro one end of the 4ire. 5ven if, instead of a vacuu tue, s#arkin+ et4een the #ointed 4ire-ends (4hich 4ere se#arated y a##ro6iately 0.% ) 4as used as the 4ave indicator= this yielded M<2 2%9 c, the sae value as 4ith the vacuu tue as indicator. 'herefore, this does not si+nificantly increase the ca#acitance of the loo# (see earlier, #29&). !8 E1-2F 'he e6#eriental conditions 4ere such that the 4ire thickness 4as sall enou+h to +ive the values otained for f as those a##licale to any thin 4ires= this follo4s #ractically fro the tests ) and c), 4here f is inde#endent of W. After Araha (see #revious citation, sae #a+e), M<2 for a strai+ht 4ire of 2.% thickness and c len+th is calculated as 0.8%C +reater than its len+th ℓ . hen the 4ire is circularly ent, the correction is the sae as for the strai+ht 4ire= so therefore for a very thin 4ire loo# it should e #ut that f 1.0% and not 1.0&%. But notice that the accuracy of the M co#arison in the tests ) and c) 4as 0.2%C = therefore, it sees that the value f 1.0&% for an infinitely thin 4ire loo# is correct. !9 E1-F "f, ho4ever, the 4ire is ent to for t4o #arallel conductors runnin+ ne6t to each other, f l 4ill e otained a+ain, #rovided that the 4ires are lon+ enou+h in relation to their distance= ecause the self-inductance is then reduced in the sae ratio as the ca#acitance has +ro4n. 'his 4as verified usin+ a !2 c lon+ #arallel line (4ire distance 2. c), 4hich +ave M<2 !2& c. 'he difference of c is caused y the #ro6iity (2.% c) of a 4ooden easurin+ rod. ('he data in the tales are not affected y such errors as the #ro6iity of the the 4ooden easurin+ rod). - hen endin+ the 4ire into a circle ho4ever, only the ends of the 4ires, 4hich are char+ed ut carry no current, a##roach each other, 4hile the current-carryin+ iddle #arts are only sli+htly chan+ed in sha#e. 'herefore the self-inductance reains unchan+ed, 4hereas the ca#acitance increases.
hen finally you end a 4ire in the for sho4n in the illustration aove, so f ust e Z 1 , i.e., its half4avelen+th is shorter than its len+th, since only the current-carryin+ iddle #arts coe close, that is, the selfinductance ecoes saller, 4hile the char+e-earin+ ends do not coe near, so that the ca#acitance does not increase. "n fact, a ent 4ire in this for, in 4hich the ends dra4n horiontally in the illustration 4ere each 2 lon+, and the vertically-dra4n lines 4ere 12 c lon+, and the A6ial distance 4as 8 (4ith 4ire thickness 1 ) +ave M<2 %%& c. ince here 4e have ℓ &&% c , then f %%&/&&% 0.8!. 'he easureent ethod 4as that first a coil of M<2 %%& c 4as #laced over the e6citer circuit. 'hen, at one end of the coil, an end of the easurin+ 4ire syste 4as connected, and no4 the 1c lon+ shortin+ stra# over the #arallel section of the 4ire syste 4as adGusted so that a vacuu tue #laced on the free other end of the 4ire syste +lo4ed at a6iu ri+htness.
! - E#2F !. + spot-chec$ of the tables using a coil with ! m self-resonance halfwavelength
As a s#ot-check of the usefulness of the tale= a coil of 0.! thick silk-insulated co##er 4ire, havin+ the follo4in+ #araeters, 4as 4ound on a +lass cylinder of 1.% thickness and % c diaeter/ h<2r
h < c
0.89
!.%%
2r < c %.1
g <
n
ℓ < c
g
w!r
0.!9
9!
1%00
1.2
1<1
Accordin+ to the tale on #22, 4hen w!r 1.20, f 1.!, i.e., M<2 1/! ^ 1%00 c 22.0 . uch a lon+ 4avelen+th could not e easured easily. 'he solution ho4ever 4as to re#lace the #etroleu fillin+ of the easurin+ ca#acitor C 4ith distilled 4ater. Due to the considerale ca#acitance increase therey induced, the 'esla transforer e6citer s#ark could only e #roduced usin+ a very sall s#ark +a# (0.1 ]) %0, ut 4ith a sufficiently stron+ e6citation of the inductioncoil (0 volts) and an increased #riary s#ark +a# of the 'esla transforer (-! c) it 4as #ossile to #roduce >uite a +ood e6citer s#ark. - E#F 'he resonance #osition of the ca#acitor sho4ed very 4ell at d 9.& . ith Petroleu fillin+, this d 4ould have a corres#ondin+ 4avelen+th M<2 &0 c (see the curve of Ii+. ! on #01 Eoved to #02F )= 4ith 4ater fillin+ therefore, the 4avelen+ths ust e ulti#lied y the ratio 9/1.!1, Ethese ein+F the s>uare roots of the dielectric constants of 4ater and #etroleu %1. 'herefore it follo4s that λ / 2= &0 .
9 c = 2 1.!1
'he a+reeent of these nuers 4ith the calculated (22 ) has to e called +ood, es#ecially considerin+ the fact that ecause of the restriction of the 4ater fillin+ EvolueF of the ca#acitor (see Ii+. 2, #29%), its ca#acitance 4ill have increased soe4hat less than 4ould e in accordance 4ith the dielectric constant of 4ater (es#ecially as the #late s#acin+ d 9.& 4as relatively lar+e), so that M<2 ust have een sli+htly saller than 2 , and that the calculated value of M<2 22.0 , is therefore sli+htly too sall ecause w!r 1<1, and not 1<20 as the tale on #22 i#lies. 'he ethod used here, that of chan+in+ the ath li>uid of e6citer ca#acitor C , is +enerally convenient for varyin+ the #eriod over lar+e intervals%2. ith air fillin+ and % c #late distance of the ca#acitor C , M 00 c = 4ith 4ater fillin+ and 1 #late distance, M 100 c 1 . %0 E2-1F ith direct connection of the e6citer 4ires to the secondary #oles of the induction coil only feele s#arks could e #roduced, even thou+h the strikin+ distance of the induction coil (!0 c) 4as uch +reater than the strikin+ distance of the 'esla transforer. 'he reason a##arently is that the volta+e et4een the e6citer alls never ecoes very hi+h due to the conductivity of the 4ater, ecause 4ith direct connection of the induction coil they receive only a co#aratively slo4 feed Erise tieF. ith the inter#osition of a 'esla transforer, this volta+e e+ins suddenly and is then not so stron+ly reduced y the conductivity of the 4ater that an e6citer s#ark cannot e #roduced. %1 E-1F 'he Dielectric constant of #etroleu, accordin+ to footnote E299-2F has een found to e 1.98. %2 E-2F 3ne such ethod, y 5. ar6 (achs. Ber. ath.-#hys. *l., ession v. 21 3ctoer 1901) has hitherto een #ortrayed as issin+.
% - E#!F "n the latter case, it is necessary to use a sufficiently #o4erful induction coil and 'esla transforer ecause of the lar+e ca#acitance of C , so that s#arkin+ for the oscillatory dischar+e of C takes #lace. H "f the #riary loo# is ade lar+er than has een done here (21 c diaeter), the 4avelen+th can of course e further increased.
7. Overtones of coils
ith sufficiently lon+ coils, overtone resonances can e easily verified usin+ the ethod +iven here. ith a coil% on a solid eonite cylinder, of 2r 2.8 c and h 11.%2 c, that is, h<2r !.0, there 4ere three resonance settin+s of the ca#acitor C = the first (the stron+est resonance) +ave the 4avelen+th/ M<2 1 c (fundaental) the second/ M1<2 !&& c (1st overtone) the third/ M2<2 %1 c (2nd overtone) 'hat these 4ere overtones could easily e verified y ovin+ the vacuu tue alon+ the coil. 'here 4ere, on settin+ the ca#acitor C to M<2 !&& c, t4o null locations on the coil, 4here the vacuu tue 4as not lit%!. Ior the second overtone, there 4ere three null #oints, one in the iddle of the coil , and t4o at 1 c fro the coil ends. 'his distance is less than half the distance et4een t4o e>uivalent nulls (!.&<2 2.! c), so the #otential nodes do not share the coil in e>ual intervals of Muencies of the fundaental oscillation to the overtones is soe4hat de#endent on h<2r , since it is M / M1
h<2r !.1
h<2r 2.0
1.&%
1.&9
% E!-1F "t had n 10, W 1 , g
& 8. "ncrease of the period of coils due to applied capacitance
'he #araeters for a coil 4ound on a eech core 4ere/ h<2r
h < c
%
1%
2r < c
g <
.1&
W < ℓ < c M<2 < c !8 1 !&1 ! n
at one end of a 4ire a rass hollo4 s#here of .8 c in outer diaeter 4as connected, this caused the 4avelen+th to increase fro M<2 ! c to M<2 !2 c. At the sae tie the #otential node oved fro the iddle of the coil to aout c fro the end of the coil to 4hich the s#here 4as attached, i.e., the #otential node 4as aout !.% c fro this coil end and aout 10.% c a4ay fro the free coil end. 'he location of the #otential node 4as a+ain reco+nied y ovin+ the vacuu tue alon+ the coil 4hile resonance virations 4ere +enerated y it. 'he vacuu tue is then e6tin+uished at the #otential node #oint. ;ike4ise, the half-4avelen+th of 9 c lon+ coil of 1.8 c diaeter ( h<2r % ) increased fro M<2 21 c to M<2 1! c, 4hen an 18 c rass #late 4as connected to the end of the coil. 'he chan+e in the #eriod of a coil 4ith ca#acitance attached at one end can e derived theoretically in the follo4in+ anner/ 'he a6ial direction of the coil is taken as the coordinate, and the current % at any #oint in the coil is +iven y (11)
{ } { }
% = &. sin 2 π
π ( cos T 2a
. - E#&F -
'he current anti-node Ea6iuF occurs at ( 0 , i.e., the node at ( a is the current node, i.e., the #otential anti-node. 'his is the free end of the coil, 4hile at ( -a a ca#acitance C' ay e a##lied. Desi+natin+ the #otential of the coil at an aritrary #oint ( as V = at the coil end ( -a, at 4hich the #otential is identical to the #otential of the a##lied ca#acitance, the follo4in+ condition ust e fulfilled/ (12)
% =−C '
∂ V for ( -a , ∂
4hen the #ositive current direction is reckoned accordin+ to the #ositive -a6is direction, i.e., directed a4ay fro 7. Desi+natin+ no4, at an aritrary #oint ( , the electric char+e #resent on the len+th d ( of the coil as . d ( , it ust therey result that at this #oint less current e6its than enters. 'he relation therefore arises/ (1)
−
∂% ∂ = ∂ ( ∂
.
3n the other hand, for each #oint ( of the coil
(1!)
ℭ .V ,
4here ℭ is the ca#acitance #er unit len+th of the coil (d ( 1) at the #oint ( . Iro (11), (1) and (1!) 4e otain (1%)
ℭ.
{ }
{ }
∂ V π = &sin 2 π . π sin ( ∂ T 2 a 2a
,
and therefore it follo4s that (12) satisfies the condition/ (1&)
cos
{ } π a '
ℭ
2a
(2 )
ℭ
2a
{ }
π = C ' . π sin a ' 2a
,
or (1)
π . tan π a ' =
2a
a
.
C '
'his e>uation can first e to used, for an oserved a and a, to find the ca#acitance #er unit len+th ℭ of the coil. "n the case aove, for e6a#le, there 4as a 10.%, a !.%, C .9 c (e>ual to the radius of the rass all). "t follo4s fro (1) therefore ℭ=.9 .
π tan 8.%k =0.!&%
21
, - E#F -
for the ca#acitance #er unit len+th of the coil at ` !.% c. o4, accordin+ to e>uation (), #2, the 4hole ca#acitance C of a coil of len+th h 2a is +iven. 'his ca#acitance C is not s#read over the len+th a h<2 e>ually ho4ever, ut rather the eleents d ( of the coil have ore 4ei+ht the further they are a4ay fro the #otential node. 'he ca#acitance #er unit len+th ℭ of the coil at the #oint ( a is therefore otained y dividin+ the total ca#acitance C y a len+th that is saller than a, naely the len+th E+iven y/F a '
∫ sin 0
( ) π (
2 a '
2 d( = π a '
.
'herefore, fro e>uation ()/ (18)
C π r . =α π ℭ( = a )= a ' 2 a '
{
2+( 2 a ' / r )2+( r / 2 a ' )2 10 + ! (2 a ' / r )2+ (r / 2 a ' )2
}
.
"n this case b 1.8 %% , r l.%, a !.%, hence ℭ
0.!&% ,
i.e., the calculated value a+rees 4ell 4ith that derived fro the oservations. o4, althou+h such e6act a+reeent ay e soe4hat accidental, it sho4s nevertheless that e>uation (18) is useful for %% E-1F 7alculated fro the tale on #29.
8 evaluatin+ the coil ca#acitance #er unit len+th. 'herefore, the theoretical calculation of the chan+e of M<2 of a coil y connectin+ a ca#acitance on one side is no4 achieved y first calculatin+ a, i.e., the location of the #otential node, fro (1) and (18) and the overall len+th a $ a of the coil. o4 M<2 is very easy to find y notin+ that the coil ust have the sae #eriod as a free coil of total len+th h 2a. - E#8F 'herefore, in +eneral, the self-resonance 4avelen+th M 4ith attached ca#acitance C' is +iven y/ (19)
λ ' / 2= ℓ .
2a . f ( 2 a / 2 r , g / δ , ε) h
,
and (20)
λ ' 2 a f ( 2 a / 2 r , g /δ , ε) = . λ h f ( h / 2 r , g /δ , ε)
4here f can e taken fro the tales on #22 and 2, and h is the coil hei+ht. ince 2a is al4ays +reater than h, so the period of a coil with a capacitance C' attached at one end is always increased, but always less than doubled (i.e., even 4ith 7 , a h), ecause f (2a<2r ) Z f (h<2r ). ith the coil under discussion 4e had 2a 21 c, 2 r c and h 1% c, therefore λ ' 21 0.0 = . =1.2& λ 1% 0.8
,
4hereas the oserved 4as M
C ' =ℭ.
h ζ π π tan 2 ζ = ℭ. a ζ=ℭ. 2
2a
( )
4here h is the coil hei+ht. Iurtherore M<2 f X ℓ ,
M<2 f X ℓ X 2a!h ,
i.e., M / M 2a / h or, ecause a $ a h , it follo4s that 2a - a
h
2a( 1 - <2 ) ,
,
9 λ ' ζ = 1+ λ 2
,
ζ
λ ' − 1 . λ
=
2
'herefore (21) +ives/ (22)
C ' =ℭ. h
λ ' −1 λ
,
or, 4hen the value ℭ fro (18) is used and h ( 2a ) is lar+e co#ared to 2 r (the coil diaeter)/ (2)
C ' = α π . 2 r
!
λ ' − λ 1
, - E#9F -
Through these equations very small capacitances can easily be determined+ e%g%, the capacitance increase due to an intensely glowing vacuum tube%
Ior e6a#le, in a coreless coil of 100 turns of 1 thick are co##er 4ire, of hei+ht h 0 c and diaeter 2r 1. c, the value M<2 2 c 4as otained 4ith 4eak illuination of a vacuu tue at the coil end, a+ainst the value M<2 28& c 4ith stron+ illuination (see #29). 'herefore the ca#acitance increase fro usin+ the stron+ly +lo4in+ tue, since b is a##ro6iately e>ual to 2, is evaluated as/ C ' =r π .
9 =0.09 c 2
(7ontinued in the ne6t issue.) (eceived, 2&th une 1902)
E0.1 #IF
!0 - E#%90F 7. On the construction of Tesla transformers.
Period of oscillation and self-inductance of the coil. By P. Drude. (7ontinued fro #29)
"". Self-inductance of the coil
Ior constant currents, a64ell, ;ord aylei+h , ?. eer and . tefan have calculated forulas for the self-inductance of the coil. Ior fast current oscillations, the inductance ust e saller. "nstead of calculatin+ this, " a +uided y e6#erient.
:. #ethod of measurement
'he ethod of easureent 4as that the coils 4ere connected at their ends to the #lates of a ca#acitor ) of constant ca#acitance, and then this syste 4as e6cited inductively usin+ the e6citer descried on #29!. 'he distance d of the #lates of the e6citer ca#acitor 4as adGusted y icroeter so that a vacuu tue a##lied to a #late of the ca#acitor ) +lo4ed at a6iu ri+htness. 'his settin+ d corres#onds to the resonance et4een the e6citer and the receiver. A+ain, since (as efore) the a+netic cou#lin+ et4een the e6citer and receiver 4as chosen to e very 4eak, so the res#onse 4as e6treely shar# (d could e read to 0.02 , the easureent accuracy 4as 0.C), and the tue only lit u# at all if d 4as very close to the resonance #osition. H Iro d , usin+ the caliration ethod descried earlier, the 4avelen+th M of the oscillation 4as then otained. o4, since/ (2!)
λ= 2 π √ LC
if self-inductance is L, and C is the ca#acitance (in electrostatic units) of the receiver, the selfinductance L follo4s fro M and C . - E#%91F 'he 7a#acitance C is calculated%& usin+ the forula >uoted in footnote E299-2F. 'he ca#acitor ) consisted of t4o circular rass #lates% of 99.2 diaeter, et4een 4hich 4ere three sall eonite #lates (s>uares of side len+th) 4hose thicknesses 4ere/ 1.01 1.020 1.02 ean/ 1.019 . %& E%91-1F ith the fast oscillations, the ca#acitance of a #late ca#acitor is sli+htly less than for static conditions, due to the concentration of the electric field lines to4ards the ed+e. ?o4ever, this correction is ne+li+ile here= see Aeber die ;eriode sehr schneller electrischer Schwingungen E'he #eriod of very fast electrical oscillationsF 5. 7ohn and I. ?eer4a+en, Ann. Phys. 29(&) (ied. Ann. !). #!-0, see #&2. 1891. % E%91-2F "n the centre, t4o holes of % diaeter 4ere drilled. 'he resultin+ reduction in ca#acitance is included in the calculations. 3ne of the ca#acitor #lates 4as attached y a scre4 #assin+ throu+h its central hole to an eonite cylinder of 2 c in diaeter, so that the ca#acitor could e used in oth horiontal and vertical #ositions. "n the latter case the second ca#acitor #late 4as held y three sall eonite rackets.
!1 "n a second case, the thicknesses 4ere/ 0.%20 0.%2 0.%8 ean/ 0.%8 (the thickness of the ca#acitor #lates 4as 1.% ). 'he ca#acitance, calculated%8 accordin+ to the forula >uoted aove is/ C
&.! c E0.%! #IF
and in the second case is/ C 119.% c E12.9& #IF
'he e6#eriental deterination of C can #roceed y connectin+ si#le closed 4ire loo#s or 4ire rectan+les to C , and then deterinin+ M. ince it is strai+htfor4ard to calculate the selfinductance L of circles and rectan+les, the ca#acitance C then follo4s%9 fro M. - E#%92F 'his value of C 4as found&0 to e, in the first case/ C &.1
c E0.21 #IF
and in the second case&1 / C 119.1 c
E12.%2 #IF
i.e., saller than the calculated value 7, in the first case y 0.&C, in the second case y 0.C. o4 Ere+ardin+F C ein+ soe4hat saller Ethan calculatedF, 4e can e account for this shortfall ecause the #lates 4ere not #olished, ut had sall scratches, 4hich could have 1C i#act even 4hen they 4ere Gust 0.01 dee#. "n fact, after #olishin+ the rass #lates, this ethod #roduced C &.% at 1.019 #late s#acin+, i.e. atchin+ 4ithin 0.1&C of the theoretical value. hen usin+ forula (2!), it is taken into account that it only strictly a##lies 4hen the current in the entire closed loo# is constant, 4hich is only likely to take #lace if the 4avelen+th M is very lar+e co#ared to the 4ire len+th ℓ of the closed loo#. trictly &2, M is to e calculated fro the forula/ (2%)
)
2 π ℓ ℓ π λ tan λ = !;7
ℓ
"f the value of the ri+ht-hand side of this e>uation is sall (e>uate to a2 ), then a is a##ro6iately e>ual to K ℓ < M . o 4e can #ut %8 %9 &0 &1
E%91-F 'he increase in ca#acitance due to the dielectric constant (2.9) of the eonite #late is taken into account. E%91-!F Iorula (2!) is not used directly, ut 4ith consideration of the correction discussed further elo4. E"n the ori+inal, an entry in rackets says see elo4 in section 10 , ut that section does not e6ist.F E%92-1F 'he ca#acitor 4as Ein #arallel 4ithF a 2 thick circular loo# of len+th ℓ %!. c. "t 4as found that M<2 && c. 'he ca#acitor has not een used for other e6#erients 4ith such a sall #late distance. &2 E%92-2F 7o#are/ : *irchhoff, :es. Ahandl. #11, 1%!, 182= P Drude, Phys. d. Aethers #8, Iorula (&).
!2 ℓ
π λ =a ( 1−δ) tan ε=ε+ 1 ε
and, since/
if @ ZZ 1 ,
it follo4s fro (2%)/ - E#%9F a
i.e.
2
2
( 1−δ)
2
( 1+ 1 a 2)= a 2= ℓ !;7
δ= 1& a 2 ℓ
2
π = ℓ ( 1 − 1& a 2) λ 2 √ LC
"t is therefore finally (2&)
λ =π √ LC 1 +
2
ℓ
2
2!;7
λ =π √ LC +
π ℓ 2
or (2&' )
2
2! √ LC
'his e>uation is used to calculate M fro L and C . "n order to calculate, vice versa, LC fro M, it follo4s (2)
π2 ℓ 2 λ 1− √ LC = 2π & λ2
or (2' )
2
2 − ℓ L = λ2 ! π C 12 C
! . Simple loops
i#le circular loo#s 4ere ade of co##er 4ire of thicknesses 2 ρ 1, 2, and len+ths ℓ 80.8 c and %!. c. 'he ca#acitor ) 4as used 4ith #late distance d 1.019 . 'he 4ire loo#s 4ere connected to the ca#acitor #lates y their s#rin+-like action and su##orted at one #oint 4ith insulation. "t 4as found that the connection #oint of the loo#s to the ca#acitor ) (4hether on the ed+e or close to the centre) had no effect on M. 'he follo4in+ tale sho4s the results. r is the radius of the circles= fro M and fro the self-inductance forula (28)
L =2 ℓ ( lo+ e {8 r / ρ }−2 )
the ca#acitance C is calculated accordin+ to (2) / ℓ
< c 80.8 80.8 %!. %!. %!.
7a#acitor #lates not #olished. 2r 2 ρ L M<2 < c < < c < c 2%.9 1 908 %% 2%.9 1 & 1.% 1 %2 %99 1.% 2 !98 %%9 1.% !%! %&
C
< c &2.9 &2. &.2 &.1 &.%
- E#%9!F 'he ean value is/ C &.1 c
E0.21 #IF
'he 4ires 4ere ent alost e6actly circular. 'his is not critical incidentally, ecause if a circular 4ire of len+th ℓ %!. c 4as ent into an elli#se 4ith the a6is ratio of /!, then M<2 4as reduced y only 0.1C. 3n the other hand, it is very closely de#endent (4ithin 1 ) on the 4ire len+th ℓ . 'he ca#acitor circuit 4as also co#leted y t4o shorter 4ires of ℓ 2% c in len+th. "f the ca#acitance value C &.1 c is used, it follo4s fro (2) that the self-inductance L is ( Lcalc )/ ℓ
< c 2%.0 2%.0
2r < c 8.1 8.1
2 ρ < 1 2
M<2 < c ! !2
L < c L < c
os. 22! 18.%
calc. 22 188
!! 'he oserved and the calculated value of L are thus identical 4ithin 0.%C, i.e., the 4ires 4ere not so thick in co#arison to 2 r that the uneven distriution of current on the surface of the 4ire needed to e considered. Iorula (28) assues that the current flo4s only on the surface of the 4ire, 4ith unifor density aout the 4ire a6is &. hen the 4ire loo# is not surrounded on all sides y air, ut is 4ound on an insulatin+ core of +reater dielectric constant, then this does not chan+e the self-inductance L. - E#%9%F evertheless, the #eriod of the coined ca#acitor and 4ire loo# can e soe4hat lar+er 4hen the dielectric constant of the core is very lar+e. "t 4as found, for a 1 thick co##er 4ire, 4hich 4as ceented onto a 1%.! c 4ide eaker 4ith sealin+ 4a6 in t4o #laces, that M<2 %% c. Distilled 4ater 4as then #oured into the eaker, increasin+ M<2 to %8% c. 3n the other hand an alcohol fillin+ +ave no noticeale increase in M<2, and no 4eakenin+ of the oscillations. 'he increase in M<2 due to the +reater dielectric constant of the core is therefore very sall, and that is understandale for the follo4in+ reason/ "f the len+th ℓ of the 4ire loo# is sall co#ared to the 4avelen+th M, the current in the 4ire loo# is a##ro6iately constant, it decreases only very little at the ends of the 4ire loo#. 'his decrease in current is acco#anied y an electric char+e on the 4ire surface, i.e. it #roduces electric field lines. 'his increases the ca#acitance of the 4hole syste, and therefore M ust e soe4hat lar+er than 2K LC , as forula (2&) deonstrates. "f the electric field lines of the closed loo# run in a ediu of lar+e dielectric constant instead of air, the ca#acitance of the 4hole syste 4ill increase a little ore, i.e., the #eriod continues to increase. But this increase ust e sall since it is oviously in the #ro#ortion of ℓ 2 / M2 as indicated y forula (2). 'herefore, it is safe to use a 4ood core in the study of coils. 'he dielectric constant and dielectric asor#tion of the 4ood has no effect as lon+ as ℓ / M is not si+nificantly +reater than 4as used in the e6#erients here ( ℓ / M m 0.0% )= since not even alcohol fillin+ had any influence on M or the intensity of the oscillations, and yet alcohol has a uch +reater dielectric constant and lar+er dielectric asor#tion, than 4ood has. 3n the other hand, one ust e a4are of another correction, if the 4ire loo# is 4ound on a core. 'he ends of the 4ire have to run out for connection to the ca#acitor #lates in t4o #arallel leads that are #er#endicular to the surface of the coil core. - E#%9&F 'he self-inductance L' of the 4ire ends Esee footnote E299-2F F is (29)
L ' = ! ℓ ' lo+ e
d ' ρ '
,
4here ℓ' is the len+th of each of the t4o 4ire ends, d' is their a6ial distance, and 2 ρ' is the 4ire & E%9!-1F A ra#idly chan+in+ current is al4ays distriuted so that its self-inductance is a iniu= see. (lectrische Schwingung in geraden /eitern Eoscillations in strai+ht conductorsF, . tefan, Ann. Phys. 2(11) (ied. Ann. !1). #!00-!20. 1890. 'he self-inductance circuit for thicker 4ires in 4hich g!r is not ne+li+ile co#ared to 1, 4ere calculated y inchin, 5lectrician 2. #1&8. 189 (see. also :. iedeann, ;ehre von der 5lektricitt, 2. Aufl., !. #8%. 1898). "t ust e soe4hat saller than accordin+ to forula (28). H 'he curvature of the 4ire causes no a##reciale deviation fro the forula, cf. Aeber die Berechnung und #essung $leiner Selbstpotentiale E7alcualation and easureent of sall self-#otentialsF, ien, Ann. Phys 289(1) (ied. Ann. %) 189!. #9289!, see #9%.
!% thickness. ℓ' and d' should of course e ade as sall as #ossile, ecause L' is not co#letely ne+li+ile co#ared to the self-inductance of the rest of the closed loo#. "n one case, for e6a#le, closed loo#s havin+ a len+th ℓ !8 c 4ere 4ra##ed around the +lass eaker, 4hile t4o ! c lon+ 4ire leads havin+ a se#aration of ! led to the ca#acitor (i.e., ℓ' ! c, d' 0.! c, g' 0.0% c) +ivin+ M<2 %9 c. 'he 4ire leads 4ere then shortened y 2.% c (ie, ℓ' 1.% c), resultin+ in M<2 %&% c. "f 4e call the #ercenta+e increase of M, if the self-inductance increases to L' , then/
( )
λ ' ( L + L ' ) C = 2π
( )( 2
LC = λ 2π
2
π2 ℓ 2 1− 2 λ
π 2 ℓ 2 1− 2 λ
)
M M ( 1$ ) 'hus (0)
( )
L ' C =2 ζ λ 2π
2
π 2 ℓ 2 1− 2 λ
"n this case, M' <2 %19 c and M<2 %&% c, i.e., 0.02%. 'herefore, the self-inductance L' for t4o 2.% c lon+ 4ires ( d' ! , g' 0.% ) usin+ the value of C &.1 c is/ L' 2%.% c
4hile accordin+ to forula (29), L' 21 c.
'he self-inductance L of the closed circuit includin+ t4o 1.% c lon+ leads ℓ' , is derived fro M<2 %&% c for L %10 c, 4hile fro e>uations (28) and (29) ( ℓ !8c, 2r 1%.!c, 2 ρ 0.l c, ℓ' 1.% c, d' 0.! c, 2 ρ' 0.1 c) L is calculated to e !91 $ 1 %0! c. - E#%9F A circular loo# on a 4ooden core, 4ith ℓ !.8, 2r 1! c, 2 ρ 0.1 c, ℓ' l c, d' 0. c, and 2 ρ' 0.l, sho4ed/ M<2 %%c. 'herefore, it follo4s fro (2) 4ith C &.1, that L !%& c, 4hile calculated fro (28) and (29) L !%1 c. hen a 0.! thick 4ire 4as used, then L %29 c 4as oserved, co#ared to L %2 c y calculation. 7onse>uently, and fro the tale on #%9, 4hich indicates the values of C , it can e concluded that the observation error for L is not more than "$ according to this method .
!& <. &ectangles
ith rectan+ular closed loo#s, the self-inductance is also calculated, therey findin+ the ca#acitance of the ca#acitor fro the M deterination. Ior a rectan+le 4ith sides a and *, the self-inductance (see . ien, as cited earlier, # 90) is/ (1)
{ (
L =! a lo+ e
2a* ρ( a +√ a 2+* 2)
) ( +* lo+ e
2a* ρ( * +√ a 2+* 2)
)
+2 (√ a 2+ *2− a −* )
"f one side a is uch lon+er than the other side *, and includin+ ters of the result is/ (2)
{ [ ( )]
L = ! a lo+ e
*
ρ
* 1+ a
*!a
}
to second order,
}
2
* * − ( 2− lo+e 2 )− 2 a !a
or, 4hen the len+th ℓ 2 (a $ *) is introduced/
{
()
− * ( 2− lo+e 2 ) + * 2 a a
{
()
* * −1.1 +1.0& a a
L =2 ℓ
lo+ e
*
ρ
2
(
)}
−lo+ 2 e !
i.e. ()
L =2 ℓ
lo+ e
*
ρ
( )} 2
2 c aove a illietre-+raduated 4ooden easurin+ rod, t4o taut 1 thick co##er 4ires 4ere tensioned in #arallel at a se#aration * 2.&% c. - E#%98F At one end, each 4ire 4as affi6ed y a scre4 to a #late of the ca#acitor ) , throu+h 2 c lon+ rass connectors&!= over the other end of the 4ires a 4ire shortin+-stra# B could e oved. De#endin+ on the #osition of B, different side len+ths a 4ere therefore defined for the rectan+ular loo#. 'he loo# 4as stretched-out 10 c - 20 c aove the e6citer, so that it 4as e6cited inductively, and M 4as a+ain deterined fro the resonance. 'he result 4ere/ a
< c 21. 2.2 %.
M<2 < c !! %0 &09
L
C
< c < c &2 &2.& !%1 &2.9 %8 &.% ean C &.0
'his value of C is in a+reeent to 4ithin 0.1C 4ith the value otained usin+ circular loo#s on &! E%98-1F 'hese rass connectors 4ere ! thick. ince they did not have the sae thickness as the 4ires, a sall correction 4as a##lied in the calculation of L in accordance 4ith e>uation ().
! #%9. H 'he ca#acitor 4ith #olished #lates resulted in a !.0, i.e., L %&% c&%, the half 4avelen+th M<2 %98c, +ivin+ C &.% c. o this value is 4ithin 0.1C of the value of C calculated on #%91. 'his #arallel-4ire loo# could no4 e very 4ell used in order to calirate the e6citer for lon+er 4avelen+ths, 4here a direct easureent usin+ the ethod s#ecified on #298 is too inconvenient. 3ne can either #roceed in such a 4ay that the #lates of the ca#acitor C are set in the e6citation circuit to certain se#aration d , and the #osition of the shortin+-stra# B in the #arallel line is adGusted to resonance, i.e., the rectan+le len+ths a are deterined, or alternatively, that B is #laced at a certain #osition a, and the resonance values are deterined fro d . - E#%99F Both ethods lead to the sae #recision. 'he latter ethod 4as chosen, ecause it 4as a little ore convenient. 'he caliration results&& are +iven in the follo4in+ tale. 'he value C &.1 c is used for ca#acitor ) . a < c 21. 2.2 %. !2. %.2 0. 110.
d < %.18 !.02 2.9 2.!% 1.91 1.!!% 0.9
M<2 < c ! %29 &0% &&2 8 8%1 10&&
M<2d 10& 10&0 10!2 10 1021 102 102
M<2d oothed 10& 10% 10!2 10 101 102 2019
Ior the reasons #reviously entioned in footnote E299-2F, M<2d ust decrease soe4hat as d decreases, as the tale sho4s. 'he easiest and ost accurate 4ay to derive the half-4avelen+th M<2 fro the oserved value of d is first to sooth the oserved values of M<2d (colun ! of the tale) +ra#hically (usin+ a curve) ( colun %), thus for any d , the corres#ondin+ value of M<2d , is taken fro the fifth colun, and the value M<2 d is divided y the s>uare root of the oserved d . 'his, ethod is follo4ed fro here on.
&% E%98-2F "n this e6#erient, the rass connectors 4ere scre4ed to the ca#acitor so that the line any4here consisted of only 1 thick co##er 4ire. 5>uation () 4as therefore a##lied 4ithout correction. && E%99-1F 'he ca#acitor C 4as no4 ounted on etal su##orts e e (see #01) to kee# d as steady as #ossile, and one ar h 4as +rounded. 'herefore, the ca#acitance of the ca#acitor 4as sli+htly lar+er than it 4as 4hen usin+ eonite su##orts for e e., and the values of M<2 +iven in the tale, for the s#ecified values of d , are not in a+reeent 4ith those fro the tale on #299, ut are lar+er here y aout 1% c.
!8 ?. 9oils
'he 4ires 4ere fi6ed to 4ooden cores (see #%9%)= the 4ires 4ere fastened at their ends either 4ith 4a6, sealin+ 4a6, or y t4o % lon+, 0. thick iron-4ire #ins. - E#&00F 'his sho4ed no detectale influence on the self-inductance, as deonstrated y control e6#erients that 4ere carried out. "n the follo4in+= n denotes the nuer of turns, 2r the diaeter of the coil, h is the hei+ht of the coil ( h (n-1) g ), g is the #itch of the coil, W is the thickness of the coil 4ire, ℓ is the len+th of the coil 4ire in contact 4ith the core, ℓ' the len+th of the 4ire-ends that #roGect fro the core and lead to the ca#acitor ) (see #%9&), d' is their a6ial distance, L1 is the total self inductance of the closed loo# calculated, usin+ e>uation (2), fro M and the oserved value C &.1 c e.& C &.% c (see # %98), L L1 - L is the self-inductance of the actual coil ℓ , 4here L' is calculated accordin+ to (29). 'he oserved values of L<2ℓ are co#ared 4ith those calculated y tefan &8 for slo4ly chan+in+ current values/ (!)
L 2 ℓ
{(
= n 1+
h
2
) ( )
+ 18 δ2 2
2 r
lo+ e
8 r
2 2 √ h +δ
− + 1+
h
2
. + 2 2 1& r
}
() g
+lo+e δ
?ere, +1 and +2 de#end on Wual to W $ doule insulation thickness), other4ise s#arks Gu# et4een the turns. ?ence, the last ter is fro a64ell&9.
& 'ranslation of the areviation e. is not clear here. Drude +ets the ca#acitance of his 1.019 s#aced ca#acitor as C &.1 c 4ith un-#olished #lates, and &.% c 4ith #olished #lates (see #%92). "t ay e that he re#eated his e6#erients after #olishin+ the ca#acitor, in 4hich case e. could ean res#ectively (eiehun+s4eise), ut he only re#orts his results (tale #&01) 4ith C &.% c, akin+ the first of the t4o ca#acitance values redundant. &8 E&00-1F 'Berechnung der "nductionscoCfficienten von Drahtrollen, Ecalc. of induction coeffs. of 4ire rollsF. . tefan, Ann. Phys 2%8(%) (ied. Ann. 22). #10-11. 188!. "t is assued here that the 4ire thickness W is sall co#ared to the coil diaeter 2r , 4hich 4as the case for the coils investi+ated and 4ill usually e the case as 4ell in #ractice. &9 E&00-2F 7l. a64ell, 5lektricitt und a+netisus 2, deutsch von einstein, 2nd ed. 2. #!0. 188. ee also :. iedeann, ;ehre von der 5lektricitt 2, !th ed. #8&. ection 119. Eote that the section and #a+e nuers of the :eran translation of a64ells treatise do not corres#ond to the ori+inal 5n+lish version.F
!9 - E#&01F -
n
2 ! % & 8 9
2r 2.92 c, W 0.! , g 2 , ℓ ℓ' d' M<2 L1 < c < c < < c < c 18. 1 ! 9 2%0 2.% 1 % %9 !&! &. 1 & &&8 12 !%.8 1 9! 100& %%.1 1.% 8 901 129% &!.& 1.% 9 1020 1&%9 .9 1.% 10 1119 199 8.1 1.% 11 1211 2
g
< c 12 1 1! 1! 22 2 2! 2!
2r
C &.% c. L/2ℓ
L
< c 28 !%1 &98 992 12 1&& 19 21
os. &.%0 8.20 9.%1 10.80 11.%% 12.&% 1.& 1.92
calc. calc - os 8.8 1.88 9.9& 1.& 11.29 1.8 12.!2 1.&2 1.& 1.81 1!.18 1.% 1!.88 1.%2 1%.%% 1.&
'he difference/ p L<2ℓ (calc.) L<2ℓ (os.) therefore decreases sli+htly 4ith increasin+ nuer of turns n. 'his has een oserved in all cases (i.e. for other values of + for n 2), eans that the coil in >uestion has a very unifor #itch ecause the coil 4ire 4as set into a helical +roove that 4as achined y lathe into the coil core. 'he p values 4ithout the => refer to coils for 4hich this 4as not the case, i.e., in 4hich the s#ecified value of #itch g is not as unifor. 'he nuer aove the p value is the ratio g
0 7han+ed to a su#erscri#t in this docuent.
g
%
p 2r
.<< =>
&.%
%0 - E#&02F n
2
!
%
&
8
9
10
p L<2ℓ (calc.) L<2ℓ (os.) 2 .% %
g
1.2
2
2
p 2r
.@
.:7 =>
.77 =>
.!
.<7 =>
.?
1.! 9.1 1.2
!1 &.0
%% &.8 2.2
&1 .2
&. .%
&1 &.2 !.%
.!?
.!
.
.
100 11.%
!2 8.2 2.2
8. .%
!2 &.8 !
.7:
.<@ =>
.<
2% .8 2.2
10.! .%
2% &.1 !
.8
.7! =>
.
2% 8. 2.2
12.2 .%
.%
2% &. !
.8 =>
.<
1.! .%
21 &.% !
.?
21 8.& 2.2 .!
.8
=>
!8 11.0 .%
. => .@7 =>
1%.0
.7
21 &.& !
%
.<< => .:? =>
&.! %
1% .9 %
.: => .<@ =>
8.1 %
100 9.1 %
.8 => .< =>
&0 8.8 %
9.% %
.7 => .: =>
%8 9.8 %
10.8 %
.: => .< =>
!& 9.2 %
11.% %
.7: => .8! =>
21 9.1 2.2
!8 12.0 .%
!& 9.8 %
.8
.< =>
.8<
21 2.2
!8 12.8 .%
21 &. !
.@
.!7 =>
.:@
21 10.1 2.1
!8 1.2 .%
21 &.9 .%
!& 10.8 %
.7?
.?
.! =>
.! =>
21 10.!
21 .!
!8 1.
!& 11.1
12.& %
.!@ => .8 =>
!& 10.! %
1.! %
. => .:! =>
EPrecisely unifor #itch achieved y achinin+ a helical +roove into the coil coreF.
=>
%
1.9
11 .?:
1! .2
%1 - E#&0F Besides the already entioned result that EqF1 sli+htly decreases as n increases E#&01F, the tale also sho4s that at constant n (e+., at n 2), p +ro4s 4ith increasin+ g F in the ran+e g
2 ! % & 8 9 10 1.80 1.8 1.8 1.!9 1.& 1. 1. 1.!1 1.0
oothin+ these values +ra#hically usin+ a curve, for the case g
n
1.2 1. 1.29 H H H H H H H
2 ! % & 8 9 10
'ale of p values g 1.80 => 1.% 1.8 => 1.! 1.! => 1.8 1.&1 => 1.2 1.%0 => 1.29 1.!0 => 1.28 1.% => 1.2 1.2 => 1.2& 1.0 =>
11 1.9& H H H H H H H H
'he values of p for g
(%)
{
2
L h = n 1+ lo+ e 2 2 ℓ 2 r
(
8 r 2 2 √ h +δ
)
2
− +1+
h
2
1& r
+2
}
+lo+ e
(δ)−Δ g
,
1 Drude #uts B here, ut clearly eans q. "t is #roale that he used B ori+inally, ut realised that it had also een used for the transission-line shortin+ +el (fi+. ), and so chan+ed to q ut issed a fe4 instances. 2 'he 4ord used here 4as entladun+en, strictly dischar+es ut this is to e inter#reted in the non-electrical sense= disissed or taken a4ay. E&0!-1F "n this forula, the ter W2 / 2 r 2 is ne+li+ile co#ared to 1 ecause the forula achieves a claied accuracy of only aout 1C, and g
%2 4here +1 and +2 are fro the follo4in+ (tefans) tale/ W
+1
+2
0.%00 0.%!9 0.%92 0.&1 0.&&% 0.&9% 0.22 0.!% 0.&% 0.82 0.9&
0.1 0.1 0.1 0.1! 0.1% 0.1 0.19 0.22 0.2! 0.2 0.1
W
+1
+2
0.808 0.818 0.82& 0.8 0.88 0.8!2 0.8!% 0.8! 0.8!8 0.8!8
0.! 0.8 0.! 0.! 0.%2 0.%8 0.& 0.&9 0.% 0.82
(Ior n 1, h 0, g uation (%) si#lifies to e>uation (28), and in this case q 0.81 ! is used.) 7o#arin+ the oserved => Eunifor #itchF values of L<2ℓ fro the tale on #a+e &02, to the calculated values fro e>uation (%) 4ith soothed p values fro the tale on #a+e &0, there are 11 cases 4here the values deviated y ore than 1C (ut not ore than 2.%C), and 19 cases 4here the values are in a+reeent to 4ithin 1C. Iorula (%) is therefore accurate to 1C for the ran+e g
! E&0!-2F 'his value is not in contradiction 4ith the 'ale of p, since p decreases shar#ly as g
% - E#&0%F @. &igorous testing and application of the formulae in two Tesla transformers a> 'he secondary 4indin+ of a 'esla transforer, consistin+ of 2&8 turns of 1
thick co##er 4ire, 4as 4ound on a hollo4 eonite cylinder of 8 4all thickness and &.! c outer diaeter. 'he coil hei+ht 4as h ! c, the 4ire len+th ℓ %!80 c, and the #itch g 1.& . ince the coil diaeter is e>ual to &.! $ 0.1 &.% c, then h/2r &.&. At this ratio of h<2r and the value g
2r 12.! c, W 1.! , g 1.1 c, ℓ 19% c.
ince Wuation (%) +1 and +2 have the values/ +1 0.%, +2 0.1 (accordin+ to the tale on E#&0!F ). Additionally, g
, i.e., L 8% c E .8% J? F.
'his coil 4as no4 o#ened at t4o facin+ #oints% Eand connectedF to four strai+ht 4ires, t4o of 4hich led to the inc s#ark +a# (len+th of the 4ires ℓ' c, ean distance d' % c, 4ire thickness 2 ρ W 1.! ), 4hile the t4o others led to a sall ;eyden Gar ( ℓ' 9 c, d' 10 c, 2 ρ W 1.! ) Esee illustrationF. 'herefore accordin+ to e>uation (29) of #%9& for L, the t4o values/ L' 120 c and L' 19 c are added, so that the entire self-inductance L of the #riary circuit has a value of L1 !1! c. - E#&0&F 'he ;eyden Gar had an inner Econductive coverin+F hei+ht of 10.2 c, & c inner diaeter, and 2.& c +lass thickness. 'he area of the inner tinfoil coverin+ 4as therefore K (& ^ 10.2 $ 2) 0K c2 , and /!K d 0/1.0! &.% c. % E&0%-1F 'he circuit is accurately descried y e in Zur #essung der Diele$tricit%tsconstante vermittelst ele$trischer Drahtwellen Eeasurin+ the dielectric const. y eans of electric 4ire 4aves (standin+ 4aves)F, Ann. der Phys. 1(&) (!th series vol. 8). #&-!. 1902.
%! 'he ;eyden Gar 4as chosen so that the transforer 4orked 4ell, i.e., resonance et4een the #riary circuit and secondary circuit 4as a##ro6iately& estalished. 'o #roduce the half-4avelen+th M<2 8!0 c on the secondary 4indin+ 4ith a self-inductance L !1!, then accordin+ to forula (2' ) #a+e %9 the ca#acitance is calculated to e C %8 c. ince for the ;eyden Gar C @.S / !Kd , 4here @ is the dielectric constant of the +lass, @ 4as calculated as/ @ %8/&.% %.1 . "n fact, ;V4e, usin+ fast electrical oscillations, has oserved dielectric constants et4een % and ., de#endin+ on the ty#e of +lass. 'hus, this calculation sho4s a+reeent, as far as one can e6#ect, es#ecially since the #riary circuit tunin+ 4as taken >uite crudely8 fro the secondary circuit, and (see footnote E&0&-1F ) #roaly the #riary circuit had a soe4hat lar+er half-4avelen+th than M<2 8!0, hence C 4as #erha#s a little lar+er than %8 c and @ sli+htly +reater than %.1. b> A second saller 'esla transforer, 4ith a secondary 4indin+ of 12 turns
of 1 thick co##er 4ire, 4as also 4ound on a hollo4 eonite cylinder of 8 4all thickness. - E#&0F Iro h 2!.8 c, 2 r &.% c, i.e., h<2r !, g
!.% c, 2r 12. c, W 1.! , g 2.2 c, ℓ 120 c.
'herefore, accordin+ to e>uation (%) (q is assued to e 1.92, as g uency of the oscillations.
%% the oscillations +enerated in the #riary circuit act, 4ith 4eak inductive cou#lin+, on the secondary circuit, descried on #%98, Ei.e.,F a EtransissionF line of t4o 1 thick, #arallel 4ires of 2.&% c distance, over 4hich an adGustale etal shortin+-stra# B could e oved. At the other end, the #arallel 4ires 4ere ent at ri+ht an+les and 4ere, usin+ their sli+htly s#rin+y nature, a##lied to the #lates of a circular-#late ca#acitor of 12.1 c diaeter and 1 #late thickness, the #lates ein+ se#arated y three eonite #lates of (on avera+e) 0.% thickness and 9 2 area. 'he ca#acitance C of this ca#acitor is calculated accordin+ to the forula on #299 to e C 1.& c E19.& #IF. A jehnders vacuu tue 80 4as a##lied to a #late on this ca#acitor, and the racket B oved y hand alon+ the #arallel 4ires. - E#&08F "n a fairly81 shar#ly deterinale resonance #osition of B, the vacuu tue +lo4ed ri+htly. 'he len+th of the rectan+ular secondary line 4as then a 1% c. 'he total len+th of the secondary line 4as ℓ 2.1% $ 2.&% $ 2.% 19 c. 'herefore the self-inductance, usin+ e>uation () on #%9, is L 2%20 c. 'he self-resonance half-4avelen+th of the secondary line, i.e., also of the #riary line, is therefore accordin+ to e>uation (2&) on #%9, M<2 2120 c. o 4e arrive at a value for M<2 that atches very 4ell 4ith the self-resonance half-4avelen+th of the secondary coil of the 'esla transforer as calculated on #&0. H Based on the value M<2 2120 c, the ca#acitance of the ;eyden Gar in the #riary circuit of the 'esla transforer is calculated to e 2!& c, that is, the dielectric constant @ of the +lass of the ;eyden Gar is @ 2!&/!.2 %. .
80 Zur obEectiven Darstellung der Fert,'schen Gersuche *ber Strahlen electrischer 5raft' , ;. jehnder, Ann. Phys. 28(9) (ied. Ann. !) 1892, #-92, see #82 = see also ObEective representation of Fert,'s researches on electrical radiation (astract of the aove), ; jehnder, 'he 5lectrician, Dec. 0, 1892, vol. 0, #2%8. H "n addition to the noral #air of electrodes, a jehnder +as-dischar+e tue has a #air of electrodes that can e used for tri++erin+ or #riin+. A jehnder tue is norally #rovided 4ith D7 ias, across one #air of electrodes, sufficient to ake it strike in the #resence of a tri++erin+ si+nal or soe other electroa+netic disturance. 81 E&08-1F 'he resonance #osition is not deterined as shar#ly as 4hen usin+ a #etroleu or air ca#acitor, ecause of eission of corona dischar+e. Perha#s soethin+ disturs electrical asor#tion in the +lass of the ;eyden Gar. ;V4e (as cited aove) 4as ho4ever not ale to detect electrical asor#tion in the +lass for uch faster virations.
%& Summary of results . The natural period of a coil increases with the dielectric constant of the coil core and its surroundings (e.+., transforer iersed in oil). .
'he dielectric constant of eonite for ?ertian oscillations is @ 2.9. 5onite is electrically isotro#ic. !.
"f the #itch of the turns at the iddle of a coil is saller than that of the end turns, the natural oscillation #eriod of the coil is slo4er than in the o##osite case or at constant #itch. - E#&09F 7. The self-resonance half-wavelength M<2 for a constant coil pitch g depends on the coil wire length ℓ , the coil height h, the coil diameter 2r , the wire thic!ness W such that
M<2 ℓ f (h<2r , g
8. The self-resonance half-wavelength of nearly closed circular loops of thin wire is &.%C larger than the wire length . :. The overtone resonances of a coil are not harmonically related to the fundamental resonance , and the ratio of the fre>uencies of the fundaental and overtone oscillations is soe4hat de#endent on the ratio h<2r . ith the suse>uent #ossile overtones, it is found that
(4ith decreasin+ intensity), the first overtone #roduces t4o current anti-nodes in the coil, the second three current anti-nodes, etc. "n the overtone resonances, the coil does not oscillate in con+ruent #arts. . y applying a capacitance to the free end of a coil, the natural period of the fundamental oscillation of a coil is increased in a calculable way (#) and experimentally confirmed% This increase is always smaller than twice the period of the coil with free ends . <. n p*.# a formula is given for calculating the self-inductance of short coils with fast alternating currents . ?. This formula, and the tables on p/00, /0/ or p/01, give the ability to calculate the correct primary circuit capacitance for every Tesla transformer . "f the 'esla transforer
secondary coil is not free-ended, ut is rather connected to one or t4o ca#acitances, then the est ca#acitance to select for the #riary circuit is the +reater one, and this ay also ay e calculated in advance fro the #recedin+ data. '4o sa#les calculations for t4o different 'esla transforers have confired the a##licaility of the forula and the tales.
% - E#&10F H :roundin+ of one end of the 'esla transforer secondary 4indin+ +ives no definite result82. @. 2econdary windings on wood or cardboard tubes are not as good (due to electrical asor#tion in 4ood or cardoard) as coils on ebonite, or glass, or coreless coils . 'herefore, for
effective construction of 'esla transforers, the forer are less favourale than the latter. H Also, for the ca#acitance of the #riary circuit of the 'esla transforer, it is etter to use etal #lates in a #etroleu ath than a ;eyden Gar, ecause of the corona dischar+es on the tinfoil linin+ (and #erha#s also due to electrical asor#tion of the +lass). H "t is advisale to ake the #riary circuit 4ith a sall nuer (1 - ) turns of thick 4ire (2 - ! ) (so that the self-inductance is as sall as #ossile), 4hile it is advisale to ake the secondary circuit fro thinner 4ire (0.% ) 4ound into a coil 4hose hei+ht aout t4ice its diaeter 8. . y using a liquid-immersed circular plate capacitor of 10 c radius, 4hich is ade to
oscillate 4ith a circular loo# of 21 c in diaeter and thickness, the 4avelen+th can e chan+ed continuously H y varyin+ the #late distance of the ca#acitor, and the li>uid of its ath H over a lar+e interval fro (% c #late distance, air et4een ca#acitor #lates) u# to 1 (1 #late distance, 4ater et4een ca#acitor #lates). ith 4ater fillin+, a 'esla transforer ust e used to initiate the e6citations (this is convenient in any case). :iessen, une 1902.
(eceived 2&th une 1902)
82 E&10-1F ee I. Braun, Ann. d. Phys. 8. #209. 1902. 8 E&10-2F " have considered the theoretical as#ects of these results, 4hich " intend to #ulish later.
%8 +ppendi) . &ecalculation of the table on p!<.
(Added y the translators). Values of f ! / 2ℓ " for #oreless #o$ls. p322 table %/& 1.09 h/2r 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.50 4.00 4.50 5.00 5.50 6.00
h/r 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.30 0.40 0.50 0.60 0.70 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.40 2.80 3.20 3.60 4.00 4.40 4.80 5.20 5.60 6.00 7.00 8.00 9.00 10.00 11.00 12.00
r/h 12.50 10.00 8.33 7.14 6.25 5.56 5.00 3.33 2.50 2.00 1.67 1.43 1.25 1.00 0.83 0.71 0.63 0.56 0.50 0.42 0.36 0.31 0.28 0.25 0.23 0.21 0.19 0.18 0.17 0.14 0.13 0.11 0.10 0.09 0.08
f
2.970 2.980 3.000 2.960 2.930 2.880 2.790 2.520 2.320 2.180 2.080 2.000 1.925 1.790 1.670 1.560 1.470 1.390 1.330 1.225 1.150 1.090 1.040 0.995 0.960 0.925 0.900 0.875 0.845 0.795 0.760 0.735 0.715 0.700 0.685
p327 eq(9) f / √α 7.206 6.430 5.854 5.402 5.035 4.728 4.467 3.557 2.998 2.613 2.333 2.121 1.956 1.720 1.558 1.439 1.348 1.273 1.212 1.113 1.035 0.972 0.920 0.875 0.836 0.801 0.771 0.744 0.719 0.667 0.624 0.589 0.559 0.533 0.511
p328 table √α 0.412 0.463 0.513 0.548 0.582 0.609 0.625 0.708 0.774 0.834 0.892 0.943 0.984 1.041 1.072 1.084 1.091 1.092 1.098 1.101 1.111 1.121 1.131 1.138 1.149 1.154 1.168 1.177 1.175 1.192 1.217 1.248 1.279 1.312 1.341
α 2 √(α ') 0.170 1.461 0.215 1.643 0.263 1.817 0.300 1.942 0.339 2.063 0.371 2.159 0.390 2.214 0.502 2.511 0.599 2.743 0.696 2.957 0.795 3.161 0.889 3.343 0.968 3.488 1.084 3.690 1.149 3.800 1.175 3.842 1.190 3.867 1.191 3.869 1.205 3.891 1.212 3.903 1.234 3.938 1.257 3.974 1.279 4.009 1.294 4.033 1.320 4.073 1.333 4.092 1.363 4.139 1.385 4.172 1.381 4.166 1.422 4.227 1.482 4.315 1.557 4.423 1.635 4.533 1.723 4.653 1.798 4.754
