FAILURE OF OHm's LAW AT HIGH CURRENT DENSITIES. 157 



Now/ at one extreme, for very slow rates of flow, contains an impor- 

 tant term which is the reciprocal of its argument, whereas at high rates 

 of flow we have seen that experiment suggests that it tends to become 

 constant. Hence, keeping all the other arguments constant except 

 b, the variation of r may be expected to be between l/b and l/b^. 



Experimentally the three narrowest breadths were 0.090, 0.155, 

 and 0.305 mm. The shift of D.C. setting (proportional to temperature 

 rise) for 0.2 amp. was respectively 8.5, 3.4, and 0.2 cm. Variation 

 as 1/P would have given 8.5, 1.7, 0.3; as l/b^ 8.5, 2.9, 0.9; and as 

 l/b, 8.5, 5.0, and 2.7. The variation is within the limits set and proba- 

 bly nearly as 1/6-, within the limits of error. 



This check is very rough, but does bear out the paradoxical variation 

 with b given by the dimensional argument. The difficulties of the 

 experiment are great, it being impossible to exactly reproduce the 

 conditions of flow when the sample is removed from position, its 

 breadth cut down, and then replaced. In the actual measurements 

 of the departures from Ohm's law, this source of irregularity was of 

 course not present, a complete series of measurements at different 

 D.C. strengths being made on the sample unchanged in position and 

 with the same flow. 



These measurements also showed the same fact as that mentioned in 

 the preceding section, namely that the relation of proportionality 

 between heat input and temperature rise ceases at high rates of heat 

 input, and the point of break in the linear relation comes at lower 

 values of heat input per unit area for the large than the small breadths. 

 This set of measurements with changing breadth of the same sample 

 showed that the current at the point of break is roughly proportional 

 to the square root of the width, which means that the break occurs 

 when the generation of heat per unit length reaches a fixed value, 

 independent of the breadth. 



Dependence of Results on Position of Electrodes. There is another 

 sort of check of entirely different character which may be made by 

 changing the position of the electrodes on the sheet of metal leaf. 

 There is an essential difference between the resistance of a two and a 

 three dimensional mass of conducting material. If two electrodes are 

 immersed in a three dimensional conducting medium, and the distance 

 between them is increased indefinitely, the resistances between them 

 will approach a finite value, the "sum of the so-called electrode resist- 

 ances. On the other hand in a two dimensional medium, the total 

 resistance between electrodes of definite shape increases indefinitely 

 as the distance between the electrodes is increased indefinitely. One 



