1912]
ELECTRIC CURRENTS IN A THIN ANCHOR-RING
105
To this are to be added the leading terms (17) ; whence, introducing 4?ra; we get finally the expression for L already stated in (1).
It must be clearly understood that the above result, and the corresponding one for a hollow anchor-ring, depend upon the assumption of a uniform distribution of current, such as is approximated to when the coil consists of a great number of windings of wire insulated from one another. If the conductor be solid and the currents due to induction, the distribution will, in general, not be uniform. Under this head Wien considers the case where the currents are due to the variation of a homogeneous magnetic field, parallel to the axis of symmetry, and where the distribution of currents is governed by resistance, as will happen in practice when the variations are slow enough. In an elementary circuit the electromotive force varies as the square of the radius and the resistance as the first power. Assuming as before that the whole current is unity, we have merely to introduce into (4) the factors
(a + PJ cos </>i) (a + p2 cos <£2) _
M1Z retaining the value given in (2).
The leading term in (21) is unity, and this, when carried into (14), will reproduce the former result. The term of the first order in p in (21) is (pi cos $! + p2 cos <j6a)/a, and this must be combined with the terms of order p° and p1 in (2). The former, however, contributes nothing to the integral. The latter yield in (4)
n o -1 4. M 11 i ,
{log 8a - 1 - greater of log Pl and log p2} +
smaller of pf and p22 -^
/00v
(22)
The term of the second order in (21), viz., p1pa/aa.cos ^cos^g, needs to be combined only with the leading term in (2). It yields in (4)
smaller of p^ and p22 /OON ---------------7-5--------------...........................(*<*)
If pi and p2 are equal (p), the additional terms expressed by (22), (23) become
£l ^
2a2 °g P ............................
.(24)
If (24), multiplied by 4nra, be added to (16), we shall obtain the self-induction 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
.(25). When there is no refraction, there is also no reflexion. By introducing a mutual reaction between the resonators, and probably in other ways, it would be possible to modify the situation in such a manner that the boundary conditions would involve higher derivatives, as in the case of the stiff string, and thus to allow reflexion in spite of equality of wave- velocities for a given ray.