(ii.) 'A conductor of capacity 10 is charged with 4 units. What is the energy of discharge?' (ii.) 'A sphere of radius 4 cm. is charged with 12 units. Find the energy of discharge.' Two harder examples are worked out in § 35. $30. Energy of Discharge in the Cascade Arrangement of Leyden Jars. We must now ask the reader to turn back to Chapter VI. § 9, and to notice the arrangement there described and the notation employed. We found that if there were n equal jars in cascade, the difference of potential between the inside and outside coatings of each jar was th that of a single jar charged from the same source and I n having its outside coating to earth, the charge of each jar being It follows that, in the case of the single jar thus fully charged, we have for the energy of discharge E the value while, for all the n jars of the cascade, we have for the total energy of discharge E' the value This result may be better understood if we give an analogy. (ii.) Analogy. If instead of one tower 100 feet in height we build 10 towers each having th of the height, and th of the number of bricks of the large one, a little thought will show us that the total work done in the second case will be th of that done in the first case; for in the second case the same total weight of bricks will have been raised through th of the former average height. § 31. Electroscopes and Electrometers. In general these instruments give their indications by movements. These movements are due to the action of a field of force on electrical quantity, the extent of the movement being such that equilibrium is finally arrived at between electrical forces on the one hand and gravitation-, magnetic-, or torsional-forces on the other hand. We can have movements when there is only one charged body, e.g. when this body is situated nearer to one wall of a room than to the other. Here there is a field of force between the body and the walls, and this field is strongest in one direction. Now, besides the effect of a field on a charged body situated in it, we know that the two sides of a field tend to close in ; or we may say 'the lines of force tend to shorten.' Hence the charged body will move so as to close up the strongest portion of the field due to it. (a) In a gold-leaf electroscope each leaf is screened from one wall by the other leaf. Hence, there is no wavering between two walls about equally distant, but each leaf moves off towards the wall from which it is not screened. This view, or one like it, is more in accordance with present ideas than is the view of 'repulsion between the two leaves.' (b) Bohnenberger's electroscope.-Here a single charged leaf moves along the field of force that lies between two brass knobs ; these brass knobs being maintained at different potentials by being connected with the two poles respectively of a dry pile. In both cases here given we indicate differences of potential by the extent of the movement; since in case (a) the field is stronger according as the leaves are at a larger or smaller difference of potential from the zero potential of the walls, and in case (b) the movement of the single leaf in the fixed field of force depends on the potential up to which this single leaf is charged. § 32. Electrometers. The 'Attracted-Disc 'Form.-In electrometers we aim at measuring differences of potential, or absolute potentials with respect to the earth as zero. Consider the plate condenser of § 26, and make the same assumptions as to uniformity of field and of p as were there made. It can be shown that the total stress F between the plates is given by the formula F= S (V1 - V2)2 This expression therefore gives in dynes the total force with which one plate is urged towards the other; and, if we measure F by counterbalancing 'weights,' each gramme weight being about 981 dynes force, we can express (V, V2) in absolute units. But in the above apparatus there is error; for the assumptions made do not hold near the edges of the plates, and this error is not one to be readily allowed for. Hence, in the actual attracted-disc electrometer, Sir W. Thomson adopted a contrivance called a guard-ring. The moveable disc was cut out in the centre of a larger disc. Then, since the fixed and moveable portions were connected and at one potential, and since there was no perceptible break between them, it followed that with respect to the moveable portion of the one disc, and the central portion of the other fixed disc that was opposite and parallel to the former, the assumptions as to uniformity did hold. In fact, only the central portions of the discs were used. In the figure we have the arrangement indicated. The two plates are hh and g; and the formula applies to the moveable The force required to keep fin its place gives us F. Whence, by knowing the dimensions, &c., of the instrument, we measure (V1 - V2). $ 33. Sir William Thomson's Quadrant Electrometer.-The principle of the quadrant electrometer can be illustrated by the Bohnenberger's electroscope. If the gold-leaf were kept at a constant high potential, while the difference of potential between the two knobs was the variable quantity, then the amount of deflexion of the gold-leaf would indicate the difference of potential between the two knobs. The next figure gives a sketch, taken looking down directly from above, of the essential parts of the quadrant electrometer. Let us take a cylindrical box of thin brass about one inch high and about five inches in diameter. Let us cut this into four sectors as indicated in the figure, and let us connect opposite sectors respectively by wire, and support all four sectors on insulating glass legs. We thus have four hollow brass sectors; in the interior of each of which, as an approximately closed vessel,' there is, excepting near the edges, a constant potential. The opposite pairs of these are connected. + Wire Wire FIG. i. By u is represented a light aluminium needle suspended by two parallel silk fibres-this being called bi-filar suspension. Hence there will be called into play, when the needle is deflected from its position of rest, a couple tending to restore it. The value of this couple for any given angle of deflexion can be calculated. This needle swings horizontally in the interior of the sectors; and matters must be so arranged that it may come to rest exactly along the line of one of the slits. The diagram represents the needle thus at rest, unacted upon by any electrical forces. This needle is maintained at some constant potential V, by being connected with the inside coating of a charged Leyden jar. If the one pair of sectors and b be at a potential V1, and the other pair a and dat some lower potential V2, there will be a field of force running across the gap or slit from c to a; and from symmetry, an equal field will run across the gap from b to d. Moreover, this field will be confined to a region only somewhat wider than these gaps; the region in the interior of each sector will be practically of a uniform potential, or will not be a field of force. From the symmetry of the whole, the needle will be acted upon by a pure couple due to the electrical field. And, from its peculiar shape, that amount of the needle which is in the field of force (ie. the portion lying under the slit) will remain constant, since only very small deflexions are employed. We thus contrive that there shall act on the needle, for any given values of V1, V2, and V, a constant electrostatic couple whatever be its deflexion ; provided that this deflexion do not exceed a certain maximum depending on the construction of the instrument. This constant couple will deflect the needle until the restoring couple, due to the twisting of the two parallel silk fibres, is sufficiently great to give equilibrium. If the potential V of the needle be known, and if the 'constants' of the instrument be known, then the difference of potential (V, V2) can be calculated from the observed deflexion. ་ In the next figure we have a sketch of the Elliott-pattern' quadrant electrometer; a comparatively simple form of instrument. One of the quadrants is represented as removed, so that the needle may be seen. This needle is maintained at a high potential in somewhat the following manner. From it there hangs a platinum wire, dipping into a glass vessel that contains strong sulphuric acid. Outside this vessel is a coating of tin foil, so that it is in fact a Leyden jar. Its capacity being very great as compared with the capacity of an isolated body, it serves to maintain the potential of the needle approximately constant for a considerable time. FIG. ii. From the needle rises a light stem, bearing a small mirror. |