[sHAw] POTENTIAL ENERGIES 157 



arrangement is to place the coils so that they have a common axis and 

 are at a distance apart equal to half the sum of their radii, and to put 

 the magnet with its centre at the origin, the whole being arranged so 

 that the magnet and the planes of the coils lie in the meridian. If the 

 currents ii and 4 in the two coils are adjusted so that the magnet is 

 balanced, we have 



dWi ^ dWi 

 dd dd 



where 6 is equal to 7r/2, and Wi, and W2 are respectively the mutual 

 potential energies of each coil and the magnet. We have from 

 equation (5) which holds for this arrangement, 

 Wi = ti(MbiPi + MLShPf,) 

 and W2 = i2(MciPi-{-MLSc,P,) 

 where Ci and C2 for the second coil, correspond to bi and b^, for the first. 

 Hence 



'^J^^Uuh'-^+MLSh"-^) (6) 



do \ d9 dd / 



and similarly for W2, 



dP dP 1 ^ 



but ^^ =sin d, and "^^ = ir sin 6(21 cos-'e-U cos-^+1) 

 dd dd 8 



therefore when 6 = — 



il fMbi+ - MLSb^ =i2 (mci-\- - ML\c'^ =K 



(7) 



15 

 therefore LS = {i2C\—i]b]) / — {i\bi — i2C^ (8) 



8 



Now i], to, bi, &5, Ci, C5, can be determined with great accuracy, but 

 unfortunately the quantities separated by the minus signs are usually 

 very nearly equal. It is, however, possible to determine the couple K 

 (equation (7)), mechanically and also to use more coils, and with the 

 aid of these extra quantities the accuracy can be improved. 



It will be seen that M can be determined at the same time from 

 equation (7) if K is known, and if we move our coils to new positions 

 such that the bs and C3 terms are not zero, we can obtain Li from the 

 equations which contain these terms. 



It will be apparent from considerations based on equations (3) and 

 (7) that the use of a null method in magnetometric tests does not 

 necessarily eliminate the constants of the magnetometer from the 

 required calculations. 



