106
ON THE SELF-INDUCTION OF
We now proceed to consider the solid ring. By (22), (23) the terms additional to those previously obtained on the supposition that the currert was uniformly distributed, are
1 f f i T [""smaller of pj2 and p22 — dp^dpo- I ---------
p4.'.'
4a2
2a2 °2-2 jlog 8a - 1 - greater of log pl and log p2} . ... (26)
02 The first part of this is p2/6a2, and the second is £- [log 8a -1 - log p -f £}
The additional terms are accordingly
57-5}..............................(27)
These multiplied by 4nra are to be added to (1). We thus obtain
7 3p3, 8a~] /oav
_ _)—£_ log — ..................(zo)
for the self-induction of the solid ring when currents are slowly generated in it by uniform magnetic forces parallel to the axis of symmetry. Ir Wien's result for this case there appears an additional term within the brackei equal to - 0'092 p2/a2.
A more interesting problem is that which arises when the alternations ir the magnetic field are rapid instead of slow. Ultimately the distribution o: current becomes independent of resistance, and is determined by indnctior alone. A leading feature is that the currents are superficial, although the ring itself may be solid. They remain, of course, symmetrical with reaped to the straight axis, and to the plane which contains the circular axis.
The magnetic field may be supposed to be due to a current x± in a circuit at a distance, and the whole energy of the field may be represented by
-}-M xx +.........(29)
oc2, x.i} etc., being currents in other circuits where no independent electromotive force acts. If #x be regarded as given, the corresponding values of Xz, x3i ... are to be found by making T a minimum. Thus
!3+...=0,
•3+... = o,
and so on, are the equations by which xz, etc., are to be found in terms of ^ What we require is the corresponding value of T', formed from T by omission of the terms containing x^.
The method here sketched is general. It is not necessary that «2, etc. be currents in particular circuits. They may be regarded as generalizec
.(30)for a shell (of uniform infinitesimal thickness) in the form of an anchor-ring, the currents being excited in the manner supposed. The result is