William Thomson to Faraday   6 August 1845

St Peter's College, August 6th 1845

Dear Sir,

I beg to thank you for your kindness in sending me the Italian memoir¹, which you mentioned to me when I saw you here². I have to apologise for not acknowledging it before, but I did not wish to write till I could say something on the contents, as you were so good as to ask for my opinion. I shall return it to you almost without delay.

The memoir is entirely occupied with the determination of the distribution of electricity on two equal spheres, in contact, which had been examined experimentally by Coulomb³, calculated mathematically by Poisson⁴, according to the general theory of Coulomb. The hypothesis which M. Avogadro makes is that the intensity of electricity at any point of the surface of an electrified conductor is proportional to the portion of sky which can be seen from the point, projected orthographically upon the tangent plane at the point, or, as he states it, to the sum of all the portions of a large spherical surface, described round the two spheres, each multiplied by the cosine of the obliquity of the direction in which it is seen. Thus if the two spheres were black and were exposed to a sky uniformly bright in every direction, above and below (as might be produced by laying them on a perfect mirror, placed horizontally) the intensity of electricity at any point of either would be proportional to the quantity of light which would be received by a small white piece of paper laid upon the surface at the point. You will readily from this conceive whether the hypothesis is even analogous in any respect to the actual physical conditions of the problem. The numerical determinations differ very widely indeed from the measurements of Coulomb, but the differences are attributed by M. Avogadro to his having neglected the curvature of the lines of inductive action. For an experimental investigation of the curved line of induction, he refers to your eleventh series⁵. The only numbers which he gives are the ratios of the intensities at 30˚ and 60˚ from the point of contact to the intensity at 90˚, (which latter he of course on account of his hypothesis finds to be the same for all of the unopposed hemispheres). For these ratios he finds .6 and .95. Coulomb's measurements give .21 and .80; Poisson's calculations .17 and .74. The measurements are as Coulomb himself considers, very uncertain, and may differ considerably from the truth on account of the excessively delicate and precarious nature of his most difficult experiments.

I am at present engaged in preparing a paper of which I read an abstract at the late meeting of the British Assn⁶, for the first Number of the "Cambridge and Dublin Mathl Journal" (a continuation of the "Cambridge Mathl Journal["]) of which a principal object is to show that in all ultimate results relative to distribution, and to attraction or repulsion, it agrees identically with a complete theory based on your views⁷. If my ideas are correct, the mathematical definition, and condition for determining the curved lines of induction in every possible combination of electrified bodies are very readily expressible. The distribution of force (or in Coulomb's language, of electrical intensity) on a conductor of any form may be determined by purely geometrical considerations, after the form of the curved lines has been found. Thus let A be an electrified conductor, placed in the interior of a chamber, and let S be the interior surface of the walls, which we may consider as conducting. The lines of induction will of course be curves leaving the surface of A at right angles, and terminating at S, cutting it also at right angles. By means of these lines let any portion a of A be projected on S, giving a corresponding portion s. The quantity of electricity produced by induction on s will be exactly equal in amount, but of the opposite kind, to that on a, according to your theory (or according to Coulomb's, as follows from a general theorem on attraction)[.] If now we suppose S to be a very large sphere, having A at its centre (and it may be shewn that the distribn on A will be very nearly the same as if this were the case, provided every portion of S be very far from A, whatever the form of S) the distribution of the induced electricity on A will be very nearly uniform. Hence the problem of the determination of the distribution on A is reduced to the purely geometrical problem of the determination of the ratio of the s to a. The great mathematical difficulty is the determination of the form of the lines, when curved, as they will always be, except when A is a sphere. In some cases, as for instance when A is an ellipsoid, then the curved lines are found with great ease. In other cases, such as that of the mutually influencing spheres, the problem admits of an exceedingly simple solution, if attacked from another direction.

It was from the connection with the mathematical theory of heat (Mathematical Journal, Vol III, p.75)⁸ that I was first able to perceive the relation which the lines of inductive action have to the mathematical theory.

I have long wished to know whether any experiments have been made relative to the action of electrified bodies on the dielectrics themselves, in attracting them, or repelling them, but I have never seen any described. Any attraction which may have been perceived to be exercised upon a non-conductor, such as sulphur has always been ascribed to a slight degree of conducting power. A mathematical theory based on the analogy of dielectrics to soft iron would indicate attraction, quite independently of any induced charge (such for instance as would be found by breaking a dielectric, and examining the parts separately)[.] Another important question is whether the air in the neighbourhood of an electrified body, if acted upon by a force of attraction or repulsion, shows any signs of such forces by a change of density, which, however, appears to me highly improbable. A third question which I think has never been investigated is relative to the action of a transparent dielectric on polarized light. Thus it is known that a very well defined action, analogous to that of a transparent crystal, is produced upon polarized light, when transmitted through glass in any ordinary state of violent constraint. If the constraint, which may be elevated to be on the point of breaking the glass, be produced by electricity, it seems probable that a similar action might be observed⁹.

I remain, with great respect, | Your's faithfully | William Thomson