Here D is the total distance between either pair of large plates. The variation in the value obtained for K when d varies from its true value is the error. Call the error E. Substitute this value of d in (11), and we obtain (11) (12) The error is least when E=0. Differentiating (12) we obtain This says that a variation of d3 from its true value produces the minimum error in the value deduced for K when the plate N is halfway between the two large plates A and C. To have the plate in this position the following relations must hold : CASE II.- Where two equal Slabs of Dielectric are used. In this case plate A (fig. 4) is placed exactly midway between the plates B and C. Two equal slabs of dielectric are used, one being placed upon C, the other upon A. The thickness of the blocks of dielectric and the distances between BA and CA are determined by considerations respecting the approximate value of K for the dielectric. If these distances are correctly taken, positions may be found for plates N and M where they are always at the same potential, and where they are approximately in the centre of the spaces between the plates and the dielectric. The plate N is supported on three small ebonite posts upon the slab D. Or, in the case of any other substance than glass, a hole may be bored through the dielectric, through which the ebonite rod which carries N may pass. The capacities of the two sides of the apparatus will now always remain equal, for if the field alternates at different rates, both dielectrics will change their specific inductive capacities equally. The formula giving the value of K is then obtained as follows: Use of Apparatus according to First Method. According to the theory given under Case I., the value of K may be obtained when the capacities of the two sides of the apparatus are unlike and unknown. It was found, however, unless the apparatus is so disposed that the capacities of the two sides are nearly equal, that the value of K obtained is not quite exact in the case of slowly varying fields, and that with oscillating fields no result whatever can be obtained. The formula is based upon the assumption that static conditions hold. Now the formula assumes that, when the centre plate has a given potential, the surface-density of each side of As the plate is proportional to the capacity of the corresponding side of the apparatus. This, however, is not strictly true. the alternations grow more rapid, the average surface-densities upon the two sides of the middle plate tend to become equal, though the capacities of the two sides of the apparatus are unlike. Moreover, if the capacities of the two sides differ largely, there is such an interference of waves, when rapid oscillations are taking place, that no position can be found for the movable plates where their potentials in reference to each other will always be equal, and consequently the sparks in the detector spark-gap cannot be made to disappear. This difficulty is not observable if the capacities are made nearly the same. It becomes necessary then, in using the apparatus with one slab of dielectric, to have the capacities of the two sides approximately equal. This adjustment can be made in a manner to be shown presently. In taking observations by this method we proceed thus: The plates M and N are first brought into contact with the centre plate and the zero positions upon the scales are determined. The block of dielectric to be tested is then placed upon the plate A, and plate M is brought to rest upon it. The reading upon the vernier scale gives the thickness of the block. Great care, of course, is taken to cut this block so that its two faces are parallel. We may now leave plate M resting upon the dielectric or give it any desired position above it. The wires leading from N and M are first connected directly to the spark-gap and the balls, p and n, are so far separated that no spark can pass between them. In the experiment as tried, two large coils were connected in series so as to make the period of the slowly changing field as long as possible. The coil being set in action, the sparks at the detector spark-gap are observed, and the plate N is slowly moved up and down until a point is found where the sparks entirely disappear. There is a distance of about 2 millim. through which the plate may be moved without the sparks appearing. A reading is taken at the upper and at the lower limit of this space, and the mean taken as the true reading. The limits of accuracy of the method are confined to the accuracy with which this setting of the plate N can be made. Any other error should be attributed to crudeness in the construction of the apparatus, and not be considered as intrinsic to the method. These readings so taken, give by applying formula (8) an approximate value of K for slowly changing fields. The capacities of the two sides may now be made nearly equal by approaching the two lower plates until they are the proper distance apart. This distance is determined. as follows:-Using the nearly correct value of K just obtained, we have for the capacity of the side AB, taking d1=0, Equating the right-hand members of (18) and (19), we obtain This value of D, then, makes the capacities of the two sides. the same, within the limits in which K is known. A still closer value of K may now be obtained as before. Two adjustments of the apparatus are sufficient for an accurate determination. If the adjustment is exactly right σ and σ' cancel out in expression (3), and hence Readings should now be taken with the different values given to d. The closeness of the agreement of the values of K then obtained furnish a test of their accuracy. To obtain, next, the value of K under oscillating fields we proceed as follows:-The two balls, n and p, are approached so that sparks can pass between them, the distance being chosen such that the best conditions may exist for oscillations taking place. The movable plates are then connected to the primary of the transformer, and the spark-gap to the secondary. Great care should be taken that the primary leads are symmetrically arranged and of the same length, and that no conductor be near them. Carelessness on this point led for some time to conflicting results. Waves of energy should reach the transformer along both leads at the same instant, which they will not do if the leads are of unequal length. The plates are set by means of the spark-gap, as in the case of slowly changing fields. If the value of K comes out much smaller than in the case of slow fields, another adjustment of the apparatus may be necessary. This being made, another determination can be made for slowly varying fields, using formula (8), which for small differences in the capacities of the two sides gives results correct within the limits of other necessary errors. Thus, after the apparatus has been correctly arranged, a comparison by this method of the values of K under slowly changing and oscillating fields is possible, without changing the plates. However, if the capacities are equal for the two sides when slowly changing fields are used, it was found by a careful test that di the simple formula, K= da-da, may still be used when the field is oscillating, although the capacities of the two sides are now slightly different. The reason for this is that with oscillating fields the average surface-densities upon the two sides of the centre plate are the same, and hence cancel out of formula (3). The proof of this rests upon the fact that K was found to be the same for all positions of d1, and this would not be the case if the formula were incorrect. Thus, as shown, Let oσ'Ao', then do' K= od2-o'ds di d2-d Now whenever d, is changed d2 must change. Hence if Ao has any value, K must change with d if calculated by the formula K = but as a series of tests showed, using the simple formula to calculate the value of K, that its value did. not change, Ao'=0, σ=o'; and hence under the conditions. stated, the simple formula, K= d dad, may be used. Use of the Apparatus according to the Second Method. The disposition of the apparatus discussed under Case II. admits of results being obtained more quickly and satisfactorily than in the one just described, and should be preferred to the other if two equal slabs of dielectric are obtainable. Readings are taken in the same manner as with the other disposition of the apparatus. The results may be tested in a variety of ways which are readily suggested by the above discussion. wax. Observations upon Paraffin. The first substance experimented upon was hard paraffinThe block used was a little larger than the plates and 4:34 centim. thick. Its melting-point was determined to be 5440 Centigrade. Only one suitable block was readily obtainable, and so the first method was employed. Previously |