FAILURE OF OHM's LAW AT HIGH CURRENT DENSITIES. 137 



' Now conditions (3) and (4) are incompatible unless tan 6' = tan 6. 

 Except for singular points, this means that the relation between 

 current and E.M.F. in the arm x must be linear, or Ohm's law is satis- 

 fied. Conversely if Ohm's law is not satisfied, the setting for balance 

 will not be the same for D.C. and small A.C., tan 6 may be called the 

 direct current resistance, and tan 6' the alternating resistance. They 

 may both be determined by the ordinary bridge formulas by first 

 adjusting i?3 and Ri for D.C. balance, and then readjusting them for 

 A.C. balance. The departure from Ohm's law at a given current 

 density, which I denote by D, is the fractional dift'erence between 

 tan 6 and tan ^o, the tangent to the curve at the origin, that is, the re- 

 sistance under small currents. This definition gives the equation for D : 



tan 6 — tan ^o 

 tan ^0 



It is now obvious that if we measure 6 and 6' at all points of the curve 

 we can find the curve itself by an integration, hence the tangent at 

 the origin, and so the departure from Ohm's law at any given current 

 density. The mathematical details of this deduction will be given 

 later. 



It is evident that the method in simple outline, as given above, 

 avoids the difficulty of the unknown temperature correction because 

 both currents are fiowing simultaneously, and hence the temperature of 

 the wire is the same to both. There is, however, a temperature effect 

 of a different kind from that usually met in this sort of experiment 

 which arises as follows. The total rate of heat input under the cur- 

 rent is proportional to the square of the total current, that is to 

 (/i + ii sin oj/)-. The 27i ii sin cot term in this expression denotes 

 an alternate heating and cooling, so that superposed on the large 

 steady temperature increase there is a small sinusoidal fluctuation of 

 temperature whose average is zero. But this small fluctuation of 

 temperature produces a small fluctuation of resistance, and a heavy 

 current flowing through a fluctuating resistance gives rise to a fluctu- 

 ating difference of potential at the terminals of the resistance. There 

 is, therefore, effectively introduced into the .r arm of the bridge a 

 spurious additional sinusoidal E.M.F. which changes the A.C. balance. 

 The action is similar to that of a microphone. 



We now discuss mathematically this spurious E.M.F. and the experi- 

 mental means taken to eliminate its effects. We are for the present 

 concerned solely with this effect, and in the following treat the resist- 

 ance as ohmic. Any residual effect left after the elimination of this 



