CHAPTER VIII. VELOCITY OF PROPAGATION OF ELECTRIC FIELD. 67. In the theoretical investigation of electric circuits the velocity of propagation of the electric field through space is usually not considered, but the electric field assumed as instan- taneous throughout space; that is, the electromagnetic com- ponent of the field is considered as in phase with the current, the electrostatic component as in phase with the voltage. In reality, however, the electric field starts at the conductor and propa- gates from there through space with a finite though very high velocity, the velocity of light; that is, at any point in space the electric field at any moment corresponds not to the condi- tion of the electric energy flow at that moment but to that at a moment earlier by the time of propagation from the conductor to the point under consideration, or, in other words, the electric field lags the more, the greater the distance from the conductor. Since the velocity of propagation is very high — about 3 X 1010 centimeters per second — the wave of an alternating or oscillating current even of very high frequency is of considerable length ; at 60 cycles the wave length is 0.5 X 109 centimeters, and even at a million cycles the wave length is 30,000 centimeters, or about 1000 feet, that is, very great compared with the distance to which electric fields usually extend. The important part of the electric field of a conductor extends to the return conductor, which usually is only a few feet distant; beyond this, the field is the differential field of conductor and return conductor. Hence, the intensity of the electric field has usually already become inappreciable at a distance very small compared with the wave length, so that within the range in which an appreciable field exists this field is practically in phase with the flow of energy in the conductor, that is, the velocity of propagation has no appreciable effect. Thus, the finite velocity of propagation of the electric field requires consideration only: 387 388 TRANSIENT PHENOMENA (a) At extremely high frequencies, hundreds of millions of cycles per second, as given by Hertzian resonators. (6) In high frequency discharges having no return circuit or no well defined return circuit, as lightning discharges. In this case the effective resistance of radiation may be so large com- pared with the ohmic resistance, even when considering the unequal current distribution in the conductor (Chapter VII), that the effect of the conductor material practically disappears. In the conductors forming the discharge path of lightning arresters this phenomenon therefore requires serious consideration. (c) With high frequencies, in the case where the field at a considerable distance from the conductor is of importance as in wireless telegraphy. In wireless telegraphy the electric field of the sending antennae propagating through space impinges upon the receiving antennae and there is observed by its electromagnetic and electrostatic effect. 68. The electric field of an infinitely long conductor without return conductor decreases inversely proportionally to the dis- tance, and therefore is represented by ^r #-j, CD where ^ is the intensity of the electric field at unit distance from the conductor. The electric field of a finite conductor of length 1Q decreases inversely proportionally to the distance I and also proportionally to the angle subtended by the conductor 10 from the distance /, and since this angle, for great distances, is inversely proportional to the distance I, the electric field of a finite conductor of length /0 without return conductor is represented by Since the electric field of the return conductor is opposite to that of the conductor, it follows that the electric field of an infinitely long conductor, with the return conductor at distance L, by equation (1) is V V * = - 77- -17 > (3) VELOCITY OF PROPAGATION OF ELECTRIC FIELD 389 where lr = ^ cos r is the projection of the distance lt between the conductors upon the direction I, that is, I' is the difference in the distance of the two conductors from the point I. For large distances I, equation (3), becomes V9 #-£• (4) In the same manner, from equation (2) it follows that the decrease of the electric field of the conductor of finite length 10, with its return conductor at the distance lv that is, of a recti- linear circuit of the dimensions of 1Q and /t : I** V* •(>-&"' >(>+$. hence, , y* ,-v f- -p— (5) 69. Since infinitely long conductors, (1) and (4), are of theoretical interest only, practically available are the cases (2) and (5). The electric field of a closed circuit decreases with the cube of the distance, hence much more rapidly than that of a con- ductor without a return conductor, which decreases only with the square of the distance. Hence, where, as in wireless teleg- raphy, action at great distance is required, only conductors without return conductor can be used. To establish consider- able currents in such open conductors requires high frequen- cies, so that the current is absorbed by the capacity of the conductor or the capacity attached to its end. No conductor f parallel to the ground can be treated as conductor without : return conductor, since secondary currents in the ground and ! also in the higher strata of the atmosphere act as return con- ductor with regard to the electric field. The practical reali- zation of a conductor without return thus requires a vertical position of the conductor, and for this reason in wireless teleg- raphy the vertical sending and receiving antennae are necessary, and the transmission is far more successful across the ocean than across the land, since in the latter case every tree, moun- 390 TRANSIENT PHENOMENA tain, etc., acts inductively as return conductor, and thus increases the rapidity of the decrease of the electric field. In such a case the use of high frequency and of conductors without return conductor, hence with electric fields decreasing relatively slowly with the distance, requires an introduction of the velocity of propagation into the circuit equations. As illustrations will be discussed : (A) The inductance of a finite section of an infinitely long con- ductor without return conductor. (B) The mutual inductance between two finite conductors without return conductors, at considerable distance from each other. ((7) The capacity of a sphere in free space. (D) The capacity of a sphere against ground, in space. Cases A and B deal with the electromagnetic, C and D with the electrostatic component of the electric field. A. Inductance of a length I of an infinitely long conductor without return conductor. 70. The inductance of a length I of a straight conductor is usually given by the equation L = 2Zlog^XlO-9, (6) lr where V = the distance of return conductor, lr = the radius of the conductor, and the total length of the conductor is assumed as infinitely great compared with I and I'. This is approximately the case with the conductors of a long distance transmission line. For infinite distance lf of the return conductor, that is, a conductor without return conductor, equation (6) gives L = oo ; that is, a finite length of an infinitely long conductor without return conductor has an infinite inductance L and inversely, zero capacity C. In equation (6) the magnetic field is assumed as instantaneous, that is, the velocity of propagation of the magnetic field is neglected. With alternating currents traversing the conductor this is permissible when the distance to the return conductor is a negligible fraction of the wave length; that is, if Z' is § negligible compared with -, where S = the speed of light and VELOCITY OF PROPAGATION OF ELECTRIC FIELD 391 / = the frequency of alternating current. It obviously is not permissible in a conductor having no return conductor. If a conductor conveying an alternating current has no return conductor, its circuit is closed by electrostatic capacity, either the distributed capacity of the conductor or capacity connected to the ends of the conductor. To produce in such a case con- siderable currents, either the conductor must be very long or the frequency and e.m.f. very high. No conductor extending parallel to the ground, as a telegraph or transmission wire, can be considered as having no return con- ductor, since even if the conductor is isolated from the ground secondary currents produced in the ground (and in the higher regions of the atmosphere) act inductively as return currents. Hence the case of the conductor without return conductor can physically be realized only by a conductor perpendicular to the ground, as the sending and receiving antennse of a wireless tele- graph station, and even then completely only on the ocean, where there are no other vertical conductors in the space, as trees, mountains, etc., which may act as inductive returns. Since a vertical conductor is limited in length, very high fre- quencies are required, and therefore the wave is of moderate length, that is, the velocity of propagation of the magnetic (and electrostatic) field must be considered when investigating the self-induction and the mutual induction of such a conductor. The magnetic field at a distance I from the conductor and at time t corresponds to the current in the conductor at the time t - t', where if is the time required for the electric field to travel the distance I, that is, t' = -, where $ = the speed of light; o or, the magnetic field at distance I and time t corresponds to the current in the conductor at the time t — - . 71. Representing the time t by angle 6 = 2 nft, where /== the frequency of the alternating current in the conductor, and denoting 2f Q _ TCj A TL ,^_. S lw where a lw = - = the wave length of electric field, 392 TRANSIENT PHENOMENA the field at distance I and time angle 6 corresponds to time angle 6 — al, that is, lags in time behind the current in the conductor by the phase angle al Let i = I cos 6 = current, absolute units. (8) The magnetic induction at distance I then is A =?-^cos(tf -al); (9) i hence, the total magnetic flux surrounding the conductor, from distance I to infinity is r27 — cos (0 - al) dl / cos prm fii j dl cannot be integrated in finite form, but represents i a new function which in its properties is intermediate between the sine function / cos al dl = — sin al a and the logarithmic function and thus may be represented by a new symbol, sine — logarithm = sil. /sin — In the same manner — — dl is related to - cos al and a to log I. Introducing therefore for these two new functions the symbols = Jf 5^U VELOCITY OF PROPAGATION OF ELECTRIC FIELD 393 gives $ = 2 7/0 { cos 0 sil al + sin 0 col a/ } . (13) The e.m.f. consumed by this magnetic flux, or e.m.f. of induc- tance, then is dt dd hence, e = 4 nfll0 { cos 0 col al — sin 0 sil al } ; and since the current is i = I cos 0, (14) the e.m.f. consumed by the magnetic field beyond distance I, or e.m.f. of inductance, contains a component in phase with the current, or power component, e, == 4 TT///O col al cos 0, (15) and a component in quadrature with the current, or reactive com- ponent, e2 = — 4 nfll0 sil a/ sin 0, (16) which latter leads the current by a quarter period. The reactive component e2 is a true self-induction, that is, rep- resents a surging of energy between the conductor and its electric field, but no power consumption. The effective component elt however, represents a power consumption p = eti = 4 nfPl0 col al cos2# (17) by the magnetic field of the conductor, due to its finite velocity; that is, it represents the power radiated into space by the conductor. The energy component et gives rise to an effective resistance, r = % = 4 ;r/70 col al, (18) ^ and the reactive component gives rise to a reactance, 4K/4 sil al, (19) 394 TRANSIENT PHENOMENA When considering the finite velocity of propagation of the electric field, self-inductance thus is not wattless, but contains an energy component, and so can be represented by an impe- dance, Z = r - jx = 4 TT/TO (col al - j sil al) 1(T9 ohms. (20) The inductance would be given by I j = — - = 2 10 { sil al + j col al } 10~9 henrys, (21) and the power radiated by the conductor is p = i*r. 72. The functions Ji I and col al = Jt ^p dl can in general not be expressed in finite form, and so have to be recorded in tables.* Close approximations can, however, be derived for the two cases where / is" very small and where I is very large compared with the wave length lw of the electric field, and these two cases are of special interest, since the former rep- resents the total magnetic field of the conductor, that is, its self- inductance, and the latter the magnetic field interlinked with a distant receiving conductor, that is, the mutual inductance between sending and receiving conductor. It is silO =00, (22) sil oo = 0, col 00= 0. Tables of these and related functions are given in the appendix, page 545. VELOCITY OF PROPAGATION OF ELECTRIC FIELD 395 And it can be shown that for small values of al, that is, such values of I as are only a small fraction of a wave length, the approximations hold: = log - 0-5772, col al = - , (23) and for large values of I, that is, values of I which make al equal to a considerable number of wave lengths, we have 2 sin al ., 7 2 sin al - < sil al < - T-I TTTi! 7T^ 2 cos al i 7 2 cos al - < colal < - — , TLUi XH2 where n1 and n2 are the two successive quadrants between which al lies. For instance, for al = 40, since 40 = 25.5 X ^ , 2i n, - 25, ^ = 26, sin a£ = sin 1.5 X \ = + 0.707, J cos al = cos 1.5 X \ = - 0.707, 2 and 0.01725 < sil 40^ < 0.01805, - 0.01805 < col 40 < -- 0.01725. As seen, for larger values of al sil al has the same sign as the sine function, col al the same sign as the cosine function. 73. From equations (20) and (21) then follow, for I = lr, the self-inductive impedance and the self-inductance of the con- ductor, where lr = the radius of sending conductor, and since lr 396 ' TRANSIENT PHENOMENA is very small compared with the wave length lw, the values (23) can be used, and give Self -inductive impedance : Z = 4 7r//0 ^ - j (log 4 - 0.5772) j 1(T9 ohms, (25) ( A alr and effective self-inductance: L = 2 I, \ log-i- - 0.5772 + j^ \ 1(T9 henrys. (26) ( alr 2 ) As an example let a current of i= 100 amperes be impressed upon a sending antenna of /0 = 100 feet = 3 X 103 centimeters, consisting of a cylindrical conductor of radius lr = 0.4 inch = 1 centimeter, at a frequency of /= 200,000 cycles, then lw = 1.5 X 105 = 0.94 mile, a = 4.19 X 10~5; hence, L= (57.2 + 9.4 j) 10~6 henrys, Z= (11.8 - 71.8 j) ohms, or, absolute, z = 72.8 ohms. Hence, the voltage required by i = IjOO amperes is e = 7280 volts, and the power radiated into space during the oscillation is p = tfr = 118 kilowatts. 74. Since the effective resistance of the total electromagnetic radiation, from the conductor surface to infinity, is, by (25), -9, (27) it follows that the effective resistance, of electromagnetic radia- tion of a conductor is proportional to the frequency and to the length of the conductor, but independent of its size or shape, and the radiated power is p = 2 Tr2/^ 10-», (28) VELOCITY OF PROPAGATION OF ELECTRIC FIELD 397 or proportional to the frequency. Thus while the radiated power is moderate at commercial frequencies, it becomes considerable at very high frequencies, and then requires consideration. For instance, at i = 100 amperes per 100 feet = 3000 centi- meters of conductor, the radiated power is At 60 cycles ............. 3.5 watts; At 10,000 cycles .......... 5.9 kilowatts; At 106 cycles ............ 5900 kilowatts. The imaginary component of self-inductance L, that is, the term in L which represents the power radiation, is Z0?r 10~9 henrys; (29) hence independent of conductor size, shape, and material, of fre- quency, current, etc. The imaginary or reactive component of the impedance, x = 4 nfllog- - 0.5772 10~9 ohms, \ Cwj* / is approximately, neglecting 0.5772 against log — , and substitut- ed,. ing equation (7), o x = 4 7r/70log — ^— 10~9 ohms j- -log/) nir i 10~9 ohms. (30) Hence, with increasing frequency /, the reactance x increases, but less than proportional to the frequency, due to the appearance of the term — log /in equation (30). For instance, with the constants Z0 = 100 feet = 3 X 103, lr = 0.4 inch = 1, at the speed of light, S = 3 X 1010, we have /= 102 104 106 108, x = 0.0667 4.94 319 14,550. B. Mutual inductance of two conductors of finite length at con- siderable distance from each other. 398 TRANSIENT PHENOMENA 75. Let Zj and 12 be the length of the sending and of the receiv- ing conductor respectively. By equation (2), the electric field of a conductor of length lv at a considerable distance I, is given by hence, for current i = I cos 6 CB = IJ C°S/ - gQ (32) is the electromagnetic component of the field at distance I. The magnetic flux intercepted by the receiving conductor of length 12J at distance ld from the sending conductor, and assumed to be inductively parallel thereto, then is p.!./ eoe(fl -a!) J^ Z2 (34) n cos a^ 77 r°° sin al „ ) cos 0 ^ -y- dl + sin 0 J^ -y- dl j . (33) By partial integration, rcos a£ . r°° . 1 cos aZ — - — dl = — / cos al a- = — - -- a col al, I II rsin al _ T00 1 sin a^ — — dl = — J sin al dj = — - -- f- a sil al; hence, 7 7 T ( /i /cos a^ i 7 \ n /sin ald _ \ ) $ = Z^27 ) cos ^ f-j- -a col ay + sm ^ f— p^ + a sil ald\ i (35) and the mutual inductance is _ \ ./sin aL \ ) - a col aU + jf — — - -f asil ald\ ( 10~9 henrys; (36) cos VELOCITY OF PROPAGATION OF ELECTRIC FIELD 399 the mutual impedance, * - a col ald)\ * ' X 10~9 ohms, (37) or, absolute, sin ald . ) 2 ( cos ald + col a?>= 1.661X10-' =1.665X10- Ld X 10~9 ohms. (38) 76. As an example, let ^ = 12 = 100 feet - 3 X 103, ld - 100 miles = 15.9 X 106, / = 200,000 cycles; hence, a = 4.19 X 10-5, ald = 666 = 424.5 xj; and sin ald = sin 0.5 *- = 0.707, Zi 71 cos ald = cos 0.5— = 0.707, then, by (24), = 1.661 X 10~3 = 1.665 X 10-3, 400 TRANSIENT PHENOMENA hence, approximately, sil ald = 0.1663 X 10~3, col ald = 0.1663 X 10~3, and LM = (- 0.227 + 1.026 j) 10~9 henrys, Z = (0.387 + 0.086 /) 10~3 ohms, or, absolute, Lm = 1.051 X 10-' henrys, z = 0.3964 X 10~3 ohms. Hence, with an oscillating current of 100 amperes in the sending antenna, the oscillating voltage generated in the receiving an- tenna, 100 miles distant, is e = iz = 0.03964 volts. C. Capacity of a sphere in space. 77. The electrostatic field of a sphere in free space decreases with the square of the distance Z; hence, * = y> (39) where lr = the radius of the sphere and e = the voltage of the sphere. Therefore, if e0 = E cos 0 (40) is the potential or voltage of the sphere, the electrostatic field at distance I is P or, neglecting lr compared with Z, y_lrEax(0-al)^ VELOCITY OF PROPAGATION OF ELECTRIC FIELD 401 and the potential, or voltage at distance I, is e = J Vdl _^£~^4 (42) hence, expanded, e = 1$ jeostf J/-^ dl + sintf f^<« }; (43) by equations (34) this gives i ( * /cos °l \ * /sm al i\ 1 = lrE \ cosdl— -- acolan + smflf— _ -fasilaH | ; (44) e = hence, for I = lr, e = E cos 6. The inductive capacity thus is e . ( /cos al A . /sin a£ A ) K = w^Tff = M h— a co1 az) + H-T-+ a Sl1 a9 1 ; (45) hence, at distance ^0 , ( /cos aZ _ \ /sin aL _ \ ) K = lr] (— =— ° - a col a/0 ) + j (-7— ° + a sil aU ; (46) ( ^ ^o / ^ ^o /•) or the absolute value is (47) ^<+ osil< J / ^0 78. For instance, let lr = 10 feet = 300 centimeters; Z0 = 100 miles = 15.9 X 106 centimeters, and / = 200,000 cycles per second; then (see example in section B) we have k = 10.5 X 10-«; hence, with e0 = 10,000 volts impressed upon the sending sphere, the voltage induced statically in the receiving sphere, at 100 miles distance, is e = ke0 = 0.105 volts. D. Sphere at a distance l^ from ground. 402 TRANSIENT PHENOMENA 79. If lr is the radius of a sphere, at a distance ^ from ground and at a potential difference e from ground, the ground, as zero potential surface, can be replaced by the image of the sphere, that is, by a sphere of radius lr) elevation — Z, and potential difference —e. The electrostatic field of such a system of spheres, at dis- tance Z and elevation Z2, then is the difference of the fields of the two spheres, thus : f== elr ___ elr J 2 hence, if Z is large compared with ll and Z2, 4 ILLe (48) and the induced potential, at distance Z and elevation Z2, is 2dl. (49) The potential difference of the sending sphere at elevation Zt is e0 = E cos 0. (50) This voltage e, at considerable distances Z, is very small; that is, the purpose of the condenser at the top of wireless sending and receiving antennae seems not so much to send out or receive elec- trostatic fields, as to afford a capacity return for the oscillating current in the conductor, and thus produce a large current, hence a powerful electromagnetic field.