Fig. 26.
A great irregularity is seen between the 10th and 12th miles, which is due, undoubtedly, to a deficiency of accuracy in the weighing apparatus. I take pleasure in sending you the following calculation of the law of the conducting power of wires, for which I am indebted to my friend Prof. Draper, of the New York City University.
On the Law of the Conducting Power of Wires.
By John W. Draper, M. D. &c. &c.
It has been objected, that if the conducting power of wires, for electricity was inversely as their length, and directly as their section, the transmission of telegraphic signals, through long wires, could not be carried into effect, and even the galvanic multiplier, which consists, essentially, of a wire making several convolutions round a needle, could have no existence. This last objection was first brought forward by Prof. Ritchie, of the University of London, as an absolute proof, that the law referred to is incorrect. There is, however, an exceedingly simple method of proving that signals may be despatched through very long wires, and that the galvanic multiplier, so far from controverting the law in question, depends for its very existence upon it.
Assuming the truth of the law of Lenz, the quantities of electricity which can be urged by a constant electromotoric source through a series of wires, the lengths of which constitute an arithmetical ratio, will always be in a geometrical ratio. Now the curve whose ordinates and abscissas bear this relation to each other, is the logarithmic curve whose equation is aʸ = x.
1st. If we suppose the base of the system, which the curve under discussion represents, be greater than unity, the values of y taken between x = 0, and x = 1, must be all negative.
2d. By taking y = 0, we find that the curve will intersect the axis of the x’s, at a distance from the origin, equal to unity.
3d. By making x = 0, we find y to be infinite and negative. Now, these are the properties of the logarithmic curve, which furnish an explanation of the case in hand. Assuming that the x’s represent the quantities of electricity, and the y’s the lengths of the wires, we perceive at once, that those parts of the curve which we have to consider, lie wholly in the fourth quadrant, where the abscissas are positive and the ordinates negative. When, therefore, the battery current passes without the intervention of any obstructing wire, its value is equal to unity. But, as successive lengths of wire are continually added, the quantities of electricity passing, undergo a diminution, at first rapid, and then more and more slow. And it is not until the wire becomes infinitely long that it ceases to conduct at all; for the ordinate y, when x = 0, is an asymptote to the curve. In point of practice, therefore, when a certain limit is reached, the diminution of the intensity of the forces becomes very small, whilst the increase in the lengths of the wire is vastly great. It is, therefore, possible to conceive a wire to be a million times as long as another, and yet, the two shall transmit quantities of electricity not perceptibly different, when measured by a delicate galvanometer. But, under these circumstances, if the long wire be coiled, so as to act as a multiplier, its influence on the needle will be inexpressibly greater than the one so much shorter than it. Further, from this we gather that for telegraphic despatches, with a battery of given electromotoric power, when a certain distance is reached, the diminution of effect for an increased distance becomes inappreciable.