FAILURE OF OHM's LAW AT HIGH CURRENT DENSITIES. 167 



where ^ is a known function. Without essential restriction we may 

 put 



M X - 



The sohition is 



= 1. We now have to solve 



dy 



dx X 



y 



^(.v) 



y 



f fJo 



<p 



(.r) d. 



+ const. \ ■ 



Figure 13. Final results, showing the departure from Ohm's law in per 

 cent (ordinates) against current density (abscissae) in 10^ amp/cm^. Curve 1 

 is for silver 2X10"* cm. thick, curve 2 for gold 8X10"^ cm. thick, and curve 3 

 for gold 1.67X10-5 cm. thick. 



The value of the constant is unity in order to satisfy the condition 



/dy\ . y 



-;- I ^ = 1. Now the departure from Ohm's law is - — 1, or 

 \d.v/x = X 



f 



^(.r) dx 



X 



This may be determined graphically from the experimental curve for 

 (p. The departure from Ohm's law has been computed from the 

 graphs according to this formula for the two thicknesses of gold and for 

 silver, and is reproduced in Figure 13. 



