OHM S LAW.] ELECTRICITY 41 it. is easy, by applying the law of continuity to multiple circuits, to verify that the measure of current intensity thus got is proportional to the electrostatic measure. Thus let AB (fig. 18) lie a circuit splitting up into two exactly similar branches BCDG, BEFG, and uniting again at G. Then, since elec tricity behaves like an in compressible fluid, it is obvious that any current of intensity C in AB will split up into two currents each of intensity ^C in CD p^,, -jg and EF. By placing a magnet in similar positions at the s<ime distance with respect to AB, CD, and EF, it will be found that the magnetic action in the last two positions is just half that in the first. ectro- The appropriate unit in magnetic measurements of ignetic current intensity is that current which, when flowing in a circular arc of unit radius and unit length, exerts unit of force on a unit north pole placed at the centre of the arc, the unit north pole being such that it repels another equal north pole at unit distance with unit force. This is called the electromagnetic unit of current intensity. Unless thf contrary is stated, all our formulae are stated in terms of this unit. To facilitate the detection and measurement of currents by mag netic means, an instrument called a galvanometer is used. It con sists of a coil of wire, of rectangular, elliptical, or circular section, inside which is suspended a magnetic needle, so as to be in equi librium parallel to the coil windings under the magnetic action of the earth, or of the earth and other fixed magnets. When a current passes through the coil a great extent of the circuit is in the imme diate neighbourhood of the magnet, and the magnetic action is thus greatly accumulated. See article GALVANOMETER. If we connect two points A and B of a homogeneous linear conductor, every point of which is at the same temperature, by two wires of the same metal to the elec trodes of a quadrant electrometer, then, if a steady current (measured in electrostatic units) be flowing from A to B, we shall find that the potential at A is higher than that at B by a certain quantity E, which we may call the electromotive force between A and B, and we may suppose E for the present to be measured in electrostatic units. Tjl If we examine the value of the ratio ,, for different posi O tions of the points AB, we shall find that it varies directly as the length of linear conductor between A and B, provided the section of the conductor is everywhere the same. If we try wires of different section, but of T71 the same length and the same material, we find that ^ is j inversely proportional to the sectional area ; in fact we may write ilvano- Jter. ectro- >tive ce, iv- tanut-, J cur- nt engtli. where / denotes the length of the wire, w its section, and k a constant depending on its material, temperature, and physical condition generally. This is Ohm s law. In whatever unit measured, R is called the resistance of the conductor. The unit of resistance can always be con ceived as established by means of a certain standard wire. The unit of electromotive force is then such that if applied at the end of the standard wire it would generate a unit current in the wire. The constant k is called the specific resistance of the material of which the wire is made ; it is obviously the resistance of a wire of the material of unit length and unit section. mien- j n t i le electrostatic system of unitation the unit of E is the work i done by a unit particle of -(-electricity in passing to infinity from ""- the surface of an isolated sphere of radius unity charged with an electrostatic unit of + electricity. The dimension of E is [QL" 1 ] > where [Q] is the dimension of the electrostatic unit of quantity (see p. 22), [Q] = [iJllsT" 1 ]. Hence the dimension of E is LL M^T J. The unit of C we have already discussed; its dimension is [<~>T J = [lv*M T T J. From these results, and equa tion (1), it follows that the dimension of R is [ L~ Tj, i.e., that of the reciprocal of a velocity. We shall show hereafter that, if C be measured in electromagnetic units, its dimension is[_L^M^T~ J J; hence that of Q is |_LSM 5 J, the unit of Q being the quantity of electricity conveyed across any section by the unit current. Also ECT = work done in time T in conveying C units of + electricity from potential V + E to potential V, whence [ECT] = dimension of energy =[l/MT~~]. Hence [E] = [lAM^Y" 2 ]- In this case then [_Rj = [_LT J > so that in electromagnetic measure R has the dimension of a velocity. We can put the equation (1) into another form, which suggests Ohm * at once the generalization of Ohm s law for any conductor. Con lawgei.e- sider two points P and Q on a linear conductor, at a distance dx ralized, from each other, x being measured in the direction of the current. Let the potentials at P and Q be V and V + dV, then E dV. If u denote the current per unit of area of the section, then C = , Icdx, and since. 1 = dx we have R = . Substituting these values in (1) we get 1 rfV X (2), IdV k dx where X is the component electric force at P in the direction of the current. Since the electric current is of the nature of a flux, it is determined at any point of a conductor by the flux components uvw, representing the quantities of electricity which in unit of time cross three unit areas perpendicular to three rectangular axes drawn through P. If X,Y,Z be the components of the electric force at P, then the general statement of Ohm s law for a homogeneous isotropic conductor is Y V 7 In such a conductor the resistance of a small linear portion of given dimensions, cut out of the substance any where or any how, will be the same. It is conceivable, however, that the resistance of such a small portion would be different if cut in different directions at any point, in which case the conductor would be seolotropic. The most general statement of Ohm s law would then be (4), where r lt &c., p lt &c., q lt &c., are constants for any one point. If they are the same for all points, the body is said to be homogeneous ; if they vary from point to point, the body is said to be hetero geneous. If we may liken our conductor to an arrangement of linear conductors (see Maxwell, 297, 324, vol. i.), then it may be shown that the skew system of (4) becomes symmetrical, inasmuch as P!=*qi, P-2 = <1 Pa^Qz- The great majority of the substances with which the electrician has to deal are, however, isotropic ; and unless the experiments of Wiedemann on certain crystals point to ceolotropic conduction, we do not know of any case which has been experimentally examined. The reader will find interesting deve lopments of the subject in Maxwell, vol. i. 297 sqq. A very important remark to be made with regard to the equa tions (4) is that, being linear, the principle of superposition applies. Thus, if u,v,w be the current components due to electric forces X,Y, Z and u v ,w similar components for X ,Y , Z , then the current for X + X , Y + Y , Z + Z is given by u + u , v + v ,w + u/. It is obvious, moreover, that (4) are the most general equations that can be written down to connect current with electromotive force, subject to the condition that the currents due to superposed elec tric forces are to be found by the superposition of the currents due to the separate forces. Besides the equations (4), u,v,w are subject like any other flux components to an equation of continuity. This equation, investi gated in the usual manner, is Equa tions of eonduc- tion. } a _ dx dy dz dt where p is the electric volume density at the time t. At a surface of discontinuity (5) must be replaced by (u-u )l + (v-i- )m + (w-w )n-^ = Q . . (6), where u,v,w, and u ,v ,w are components of flux on the first and second sides of the surface, Lm.n the direction cosines of the normal
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