I. Note on specific inductive capacity

Abstract

Consider a condenser formed of two parallel plates at distance x from each other, their area A being so great, or the distance x so small, that the whole of the lines of force may be considered to be uniformly distributed perpendicular to the plates. The space between the plates is occupied by air, or by any insulating fluid. Let e be the charge of the condenser and V the difference of potential between the plates. If the dielectric be air, there is every reason to believe that V ∞ e , that is, there is for air a constant of specific inductive capacity. My own experiments ([1880] ‘Phil. Trans,’ vol. 172, p. 355) show that in the case of flint-glass the ratio of V to e is sensibly constant over a range of values of V from 200 volts per cm. to 50,000 volts per cm. From experiments in which the dielectric is one or other of a number of fluids and values of V upwards of 30,000 volts per cm. are used, Professor Quincke concludes (‘Wiedemann, Annalen,’ vol. 28, 1886, p. 549) that the value of e /V is somewhat less for great electric forces than for small. From the experiments described in that paper, and from his previous experiments (‘Wiedemann, Annalen,’ vol. 19, 1883, p. 705, et seq .) he also concludes that the specific inductive capacity determined from the mechanical force resisting separation of the plates is 10 per cent. to 50 cent. greater than that determined by the actual charge of the condenser. The purpose of the present note is to examine the relations of these important conclusions, making as few assumptions as possible. The potential difference V is a function of the charge e and distance x , and if the dielectric be given of nothing else. The work done in charging the condenser with charge e is ∫ e 0 e V de . If the distance of the plates be changed to x + dx , the work done in giving the same charge is ∫ e 0 (V + d V/ dx dx ) de , hence the mechanical force resisting separation of the plates is ∫ e 0 d V/ dx de . If the dielectric be air, A V/ x =4π e , and the attractive force between the plates is 2π e 2 /A or AV 2 /8π x 2 . If K p be the dielectric constant as determined by an experiment on the force between the plates when the potential difference is V and distance is x , K p = ∫ e 0 d V/ dx de /AV 2 8 πx 2 . . . . (1) If K be the dielectric constant obtained by direct comparisons of charge and potential, K = 4 πxe /AV, . . . . . . . (2) whence K p K = ∫ e 0 d V/ dx de /V e /2 x . . . . . . . (3)