430 Mr. A. llusseil on the Magnetic 



Hence, on our assumptions 



This integral can be easily found. Since, however, we 

 are supposing that r and £ are very small, we may write 

 Z = 2i/i; simply, and thus <3> 2 =7ra, approximately. 



I£ L, therefore, denote the self-inductance, we have 



L^^ + ^2, 



= 87ra(F-E) + 7ra. 



Substituting for F and E their approximate values 

 (1/2) log (Sa/r) and 1, we get 



L = 47ra{log(8a/r)-l-75}, . . . (22) 

 approximately. 



The formula given by Lord Rayleigh * and Sir W. D. 

 Niven is 



L=47ra|log (Sa/r) — 1*75} +■ {-irr 2 / 2a) {log (Sa/r) + 1/3}, 

 -and by Max Wien + 

 L=4flra{log {Sa/r) -1*75} + (W 2 /2a){log (Sa/r) -0'0664}. 



When r/a = 1/100 we should expect (22) to give L with 

 considerable accuracy. We see that in this case, the value 

 of L found by it differs from that given by either Rayleigh's 

 or Wien's formula by less than the five-hundredth part of 

 one per cent. 



If we suppose that the conductivity of the ring is infinite 

 so that the current flows entirely on the surface,, then our 

 method of proof shows that 



L= <S>x = Sira(F - E) = 47ra{log (Sa/r) - 2}, 



approximately. When r/a — 1/100, the value of L given by 

 this formula is more than five per cent, lower than that 

 given by any of the preceding formulae. It will be seen, 

 therefore, that even with a thin wire the manner in which 

 the current is distributed over the cross section has an 

 appreciable effect on the inductance. It must also be noticed 

 that when r/a is small, dlj/dr(= — knra/r) is large, and hence 

 L varies rapidly with r and a small error made in measuring 

 r may lead to a large error in the calculated value of L. 



It is interesting to compare the formula for the self- 

 inductance of a ring with the formula for its electrostatic 



* ' Scientific Papers/ vol. ii. p. L5. 



t Wiedemann's Annalen, liii. p. 934 (1894). 



