FAILURE OF OHM's LAW AT HIGH CURRENT DENSITIES. 149 



r 

 resistance to heat out flow is in the thin layer of cooHng water immedi- 

 ately in contact with the surface of the metal. Our previous estimate 

 of the maximum current density that a wire can carry must be cut 

 down many fold. A 0.001 inch wire cannot carry 10'^ amp/cm^ under 

 practical conditions. It still remains true under these new conditions, 

 however, that the only change of dimensions of the specimen which 

 will increase the maximum obtainable density is a decrease of diameter. 

 Our numerical example shows that the body of the metal can be 

 regarded as approximately at a single constant temperature, both for 

 the steady rise of temperature and for the alternating fluctuations. 

 In the mathematical discussion above it was assumed that the tem- 

 perature of the metal could be specified by a single number; the 

 numerical discussion just given constitutes the justification of this. 



Dimensional Discussion of the Cooling Process. 



In order to get further in our understanding of the phenomena we 

 must now consider in some detail the steady and alternating changes 

 of temperature ro and ri, remembering that practically all the resist- 

 ance to heat outflow is in the cooling water. It is of course not possible 

 to give an exact solution; the best that we can do is to give a dimen- 

 sional discussion. Let us consider in the first place the ecjuations of 

 heat transfer in a fluid that is in motion. The equations may be 

 obtained by a slight generalization of the process by which the equa- 

 tion of heat transfer is deduced for a medium at rest. Let us suppose 

 that the medium is homogeneous except for temperature differences, 

 that its specific heat per unit volume is c and its velocity of motion at 

 any point v. Consider a small closed surface S at any point in the 

 liquid. The rate of rise of temperature of the matter within this 

 surface is the total heat input divided by the heat capacity. The heat 

 input consists of two parts. The first is the ordinary conduction 



across the boundary, and is I I k — dS, where k is the thermal con- 

 ductivity. This assumes that the velocity v is so small compared with 

 the velocities of molecular motion within the liquid that the ordinary 

 process of conduction takes place independent of the motion. The 

 second part of the heat input is that which is convected, and is 

 — f/cTPndS. From these two expressions we get the equation 



