THE CORRECTION TO THE LENGTH OF TERMINATED RODS IN ELECTRICAL PROBLEMS.
[Philosophical Magazine, Vol. xxv. pp. 1—9, 1913.]
IN a short paper " On the Electrical Vibrations associated with thin terminated Conducting Rods"* I endeavoured to show that the difference between the half wave-length of the gravest vibration and the length (I) of the rod (of uniform section) tends to vanish relatively when the section is reduced without limit, in opposition to the theory of Macdonald which makes X = 2*53 I. Understanding that the argument there put forward is not considered conclusive, I have tried to treat the question more rigorously, but the difficulties in the way are rafcher formidable. And this is not surprising in view of the discontinuities presented at the edges where the flat ends meet the cylindrical surface.
The problem assumes a shape simpler in some respects if we suppose that 'the rod of length I and radius a surrounded by a cylindrical coaxial conducting case of radius b extending to infinity in both directions. One advantage is that the vibrations are now permanently maintained, for no waves can escape to infinity along the tunnel, seeing that I'is supposed great compared with b~f. The greatness of I secures also the independence of the two ends, so that the whole correction to the length, whatever it is, may be regarded as simply the double of that due to the end of a rod infinitely long.
At an interior node of an infinitely long rod the electric forces, giving rise (we may suppose) to potential energy, are a maximum, while the magnetic forces representing kinetic energy are evanescent. The end of a terminated rod corresponds, approximately at any rate, to a node. The complications
* Phil. Mag. Vol. vm. p. 105 (1904); Scientific Papers, Vol. v. p. 198. t Phil. Mag. Vol. XLIII. p. 125 (1897); Scientific Papers, Vol. iv. p. 276. The conductors are supposed to be perfect.
R. VI. 10ediate region, its shape is such as to entail natural resonances of frequency agreeing with that of the principal waves. The method would then fail.