FAILURE OF OHM's LAW AT HIGH CURRENT DENSITIES. 139 



Using the same notation as before for the current in the other arms of 

 the bridge, we have at D.C. balance 



and 



Roh il-\-aTo-\-^aTij 



hRo = hRi. 



= ^3 ^3, 



Dividing to ehminate the currents, we obtain 



i?fl 



hi 



1 + ocTo + lari - r 



= jRo 



Rs 



This shows that in general, even neglecting the cos^ term in the 

 heat input as we have above, the D.C. balance will depend on the A.C. 

 But this effect is doubly small, since ri is small compared with to and ii 

 small compared with /i, and hence the effect may be neglected. 

 The correctness of this assumption was checked experimentally. 



\Yith regard to the A.C, the expression above shows that there 

 cannot be complete balance. There will always be higher harmonics, 

 and there will be an out-of-phase component (in sin co<). These terms 

 are small, as examination of the coefficients shows, but may neverthe- 

 less be perceptible. The ear can set on the fundamental alone, and so 

 eliminate the higher harmonics. The out-of-phase component gives 

 rise to a smearing out of the sharpness of the minimum. This can be 

 corrected by introducing another out-of-phase component to neutral- 

 ize it by a variable mutual inductance between input and detecting 

 circuits. 



The equilibrium conditions for the in-phase component are 



Roil 

 and 



Eliminating the current, 



R, 



= is Rs 



i\Ri — iz Ri- 



h' 



1 + a I To + n T 

 V '^1 



i?2 



Rz 

 Rd 



The condition for A.C. balance is therefore different from that for 

 D.C. balance, the large term Ii/ii occurring in the expression for A.C. 

 balance against the small term ii/Ii in the expression for D.C. balance. 



As the experiment was actually performed, Ro was kept constant, 

 and Rs and Rt were varied. R3 and Ri consisted of extension coils 



