108
ON THE SELF-INDUCTION OF
I *
The leading term is, of course, «0. Relatively to this, aa is of order p, a., order p\ and so on. Accordingly, a2, «3, etc., may be omitted entirely fro: (34), which is only expected to be accurate up to />2 inclusive. Also, in only the leading term need be retained.
The ratio of ^ to «0 is to be found by elimination of H between (3£ (36). We get
.(38)
8a 15 *0 oi I°g7~~2
Substituting this in (34), we find as the coefficient of self-induction T , f, 8a /, p2\ 0 , p2 /0, Sa 15
iy = 47TO lOOf — 1 + T1- —' ^ + 7—i O lOg —
|_ & p \ 4aV 4a- V ° p
The approximate value of a0 in terms of H is
Ha
, 8ft 17\ I /£,r\\
2A51og---2-)|.(39)
.(40)
A closer approximation can be found by elimination of o^ between (35),
In (39) the currents are supposed to be induced by the variation (in tirm of an unlimited uniform magnetic field. A problem, simpler from, tl theoretical point of viewj arises if we suppose the uniform field to be lirnitc to a cylindrical space co-axial with the ring, and of diameter less than tl smallest diameter of the ring (2a — 2p). Such a field may be supposed to I due to a cylindrical current sheet, the length of the cylinder being infinit The ring currents to be investigated are those arising from the iristantaneoi abolition of the current sheet and its conductor.
If Tri2 be the area of the cylinder, (33) is replaced simply by
The expression (34) remains unaltered and the equations replacing (3^ (36) are thus
J?62 + 4a«0 llog— fl + £) - 2J +pai (log —- i) = 0, ....(42) ( p \ 4>a~/ I \ p z/
The introduction of (43) into (34) gives for the coefficient of self-inductic in this case —
It will be observed that the sign of c^/a,, is different in (38) and (43). require is the corresponding value of T', formed from T by omission of the terms containing x^.