The time-constant of circumferential, electric currents in an infinitely long conducting cylinder is as the square of the diam eter.
The amplitude of a periodic current (frequency constant = p) in a circuit of resistance R and self-inductance L must be given by = where E is the maximum electromotive • R R/ force.
Although many valuable hints may be obtained in this way the chief problem in electricity is the logic of the method in which certain quantities have their dimensions suppressed. If the time comes when electric charges, magnetic poles etc. are analyzed into more fundamental entities a greater logical sim plicity will be obtained. But when that time comes it is possible, or even probable, that the notion of mass will also have been further analyzed.
General.—Let z1, z2, z3, etc. be any terms of zero dimensions each consisting of powers and products of physical elements (such as length, velocity, force, viscosity, electric charge, etc.) some of which may be ratios of like elements (such as a ratio of two masses or of two lengths). Let Q be any term, completely defined by them so that Q=f( z1, z2, z3, etc). From the principle of dimensional homogeneity it follows that Q, which may con sist of powers and products of physical elements, must also be of zero dimensions.
If zi, z2, z3, etc. are all constant during any phenomenon, any systems satisfying the equation are said to be dynamically similar. For such systems Q is necessarily a constant.
In practice it is usually sufficient to retain only two or three of the z-terms it being assumed that the more " remote " causes of an event have so little influence that they can be ignored or treated as constant. Thus, in considering the motion of the earth round the sun we may at first ignore the influence of the other planets, and the possible influence of magnetic forces. These secondary effects may subsequently be treated as perturbations. Again, in considering the flow of liquids in pipes, any possible electrification has not been considered. One of the most impor tant considerations in the design of an experimental investiga tion is the exclusion of such extraneous effects in order thereby to reduce the number of the 2-terms that need to be taken into account.
Any or all of the z-terms may be changed by multiplying by powers of Q or of each other for they will still remain terms of zero dimensions. In this way they may be selected so as best to
suit particular problems. For example, any one of them may so be changed that its elements depend upon the intrinsic proper ties of the medium under examination (density, viscosity, etc.) and not at all on induced properties (such as velocity or accelera tion). If this is so, the particular z-term will be a constant so long as the same medium is dealt with and thereby drops out of the list automatically.
If the retained terms are Q, zl, z2 and their experimental values are plotted as the x y z co-ordinates of points in space any one such point represents all the members of a group of cases which are dynamically similar to one another. The behaviour of any one member of the group can be determined by measure ments made on any other member of the same group. If the curve through the points is nearly parallel to one of the planes of co-ordinates say x y it shows that the third co-ordinate, z, is nearly constant. Hence, for approximate purposes it may be treated as a constant and if it be a z-term it can be ignored.
BIBLIOGRAPHY.-MUCh use has been made of the above principles by Lord Rayleigh (Collected papers) ; by A. H. Gibson in The mechanical properties of fluids (Blackie, 1923). The fullest treatment of the logic of the process is by E. Buckingham (Physical Riview, 1954) applications are given in article, Dynamic Similarity, by H. Levy, in the Dictionary of Applied Physics [Macmillan (1922)] and by Karman, Zeitschr. f. angew. Math. u. Mech. Band 1, p. 233 (1921) and in the publications of the Aerodynamic Institute at Aachen ; also in the reports of the Advisory Committee for Aeronautics (London).
(A. W. Po.)