OHM S LAW.] E L E C T K I 1 T Y 49 The above results seem to compel us to one or other of two conclusions, either that Ohm s law does not apply to rapidly alternating currents, where the maximum of polar ization is not reached, or else that the electromotive force of exceedingly small deposits of the ions must be very considerable. The fact that, under certain conditions, the electrolyte is apparently a better, and under others, appa rently a worse conductor than a certain metal wire, seems at first sight rather to point to the former conclusion. On the other hand, the result with the 29 cm. electrodes, is a direct verification of Ohm s law. Kohlrausch, therefore, adopted the latter conclusion, and justified his doing so by special researches on the electromotive force of small gas deposits. He showed that, with the currents he used, the electromotive force is proportional to the surface density of the deposit, and estimated that the products of decom position of ^ ing. of water per square metre would gene rate an electromotive force equal to that of a DanielPs cell. It is of the greatest importance to remark that the polarization effects, from which this result is deduced, must have arisen and disappeared in some cases in much less than n.^ of a second. The anomalous behaviour of the electrolyte with small electrodes is explained by Kohlrausch by taking into account the self-induction of the circuit. A little consideration will show that the electromotive force due to this cause always opposes the electromotive force of polarization, when the current" strength is a simple harmonic function of the time. Let i denote the current strength, reckoned positive in a given direction, then, according to Kohlrausch s law, the electro motive force of polarization at time t is -ft idt , where p is the o electromotive force generated by the passage of a unit of electricity ; its value depends on the electrolyte and on the electrode being, ccetervs paribus, nearly inversely proportional to the surface cf the latter. Let n be the number of revolutions of the machine per second, and let 2r = ; then we may represent the electromotive force of the n k TT machine at time t by sin t, and the electromotive force of self-induction by - y , dt where k and q are constants, the latter being the coefficient of self-induction of the circuit (see Electroinagnetism, p. 76 ). If w be the whole resistance of the circuit, we may write di , cos-< . Neglecting disturbances that die away very soon after starting the machine, we get for the value of i, k . 1C ~ sin t T rr i= 7= where the origin of time has been thrown back by T . i I/ T -to* Zp--q-) IT W TT f J The deflection a of the dynamometer is proportional to - / i 2 and may be written * ATI* 2 / P _ V m imq ) Kohlrausch found that this formula completely accounted for all the peculiarities in the behaviour of the electrolyte (for the numerical verifications see the papers quoted). We see that the deflection is increased or diminished by the insertion of the electrolyte, according as n is greater or less than / -? -, and, if n - / , the insertion of 7T/ y <? "A/ ^? the electrolyte makes no difference. Again, if n = / -. 27rV J the deflection will be the same as if there were no extra current and no polarization. So that, for any given electromagnetic machine, working at any given speed, a certain electrolytic arrangement can be found, which will exactly eliminate the effect of self-induction, and thereby render the efficiency of the machine a maximum. It is obvious too that, with a given electrolytic cell, the deflec tion reaches a maximum when P --- * this maximum was actually observed by Kohlrausch (I.e.). Having due regard to these circumstances, Kohlrausch Test of and Nippoldt found that Ohm s law was applicable to Ohm s their alternating currents, for electromotive forces varying law - from over to under T * T of a Grove s cell. By using the constant current of an iron-copper thermo-electric pair, they found Ohm s law applicable to zinc sulphate with amalgamated zinc electrodes, when the electromotive force was reduced to ^^-ginro" of a Grove s cell. It is important to remark that the fact that the electro lyte behaves like a metallic conductor through a consider able range of velocities of the sine inductor, is not a conclusive proof that the last trace of polarization has been eliminated. In fact, let x be the resistance of the electrolyte, W that of the rest of the circuit, and w the metallic resistance that gives the same electrodynamometer deflection for n revolutions of the inductoj pc-r second, then the above formula gives 1 p- since we suppose x very nearly = w. If now p be reduced to a very small value, it may happen, especially for tolerably high speeds, V* that ., 2 is very small compared with pq, in which case x - w will OTT^ft" be independent of n through a considerable range of speed, and the electrolyte will be replaceable by a wire whose resistance is less than the real resistance of the electrolyte by a small constant quantity. The earlier results cf Kohlrausch and Nippoldt for sulphuric acid, in which, they used 29 cm. electrodes, were affected with an- error due to this ca use, amounting to about 4 per cent. In the later experiments of Kohlrausch and Grotrian, 1 this error was finally eliminated by " pla^ tinizing" the platinum electrodes. Kohlrausch had found that, with " platinized" electrodes of only 1 sq. cm. sur face, the polarization of the currents of his sine-inductoi 1 was insensible ; he therefore concluded that, with 25 sq. cm. platinized electrodes, the residual polarization would be finally eliminated. To make quite certain, he instituted three tests, which were carried out on the method used in all the later experiments on this subject. 2 The AVheatstone s bridge arrangement was adopted. Fig. 20 gives a scheme of the arrangement. The fluid and a rheostat occupy two arms of the bridge, the remaining two con tain each 100 Siemens units ; A is the fixed and B the suspended coil of the electrodyna mometer, and S the sine-inductor. In this way, (1) the resistance of a receiver with 25 cm. Fi &- 26> platinized electrodes was found, when filled first witn. H.,S0 4 of maximum conductivity, and secondly, with NaCl, the driving weight of the inductor being varied, so as to give speeds of 10 to 100 revolutions. The results, reduced 1 I ogg. Ann., cliv., 1875.
- KoLlruuscb and Grotriau,
Ibid., clii., 1873. . Ann., cliv., 3875 ; Kohlvaiiicli
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