Page:EB1911 - Volume 09.djvu/233

Fig. 7. we have formed the cycle equation $x(a+b+c)-by-cz{=}{\text{E}}.$ For each mesh a similar equation may be formed. Hence we have as many linear equations as there are meshes, and we can obtain the solution for each cycle symbol, and therefore for the current in each branch. The solution giving the current in such branch of the network is therefore always in the form of the quotient of two determinants. The solution of the well-known problem of finding the current in the galvanometer circuit of the arrangement of linear conductors called Wheatstone’s Bridge is thus easily obtained. For if we call the cycles (see fig. 7) $(x+y),$ $y,$ and $z,$ and the resistances ${\text{P, Q, R, S, G}}$ and ${\text{B}},$ and if ${\text{E}}$ be the electromotive force in the battery circuit, we have the cycle equations

$({\text{P}}+{\text{G}}+{\text{R}})(x+y)-{\text{G}}y-{\text{R}}z{=}0,$ $({\text{Q}}+{\text{G}}+{\text{S}})y-{\text{G}}(x+y)-{\text{S}}z{=}0,$ $({\text{R}}+{\text{S}}+{\text{B}})z-{\text{R}}(x+y)-{\text{S}}y{=}{\text{E}}.$

From these we can easily obtain the solution for $(x+y)-y{=}x,$ which is the current through the galvanometer circuit in the form

$x{=}{\text{E}}({\text{PS}}-{\text{RQ}})\Delta ,$

where $\Delta$ is a certain function of ${\text{P, Q, R, S, B}}$ and ${\text{G}}.$

Currents in Sheets.—In the case of current flow in plane sheets, we have to consider certain points called sources at which the current flows into the sheet, and certain points called sinks at which it leaves. We may investigate, first, the simple case of one source and one sink in an infinite plane sheet of thickness $\delta$ and conductivity $k.$ Take any point ${\text{P}}$ in the plane at distances ${\text{R}}$ and $r$ from the source and sink respectively. The potential ${\text{V}}$ at ${\text{P}}$ is obviously given by

${\text{V}}{=}{\frac {\text{Q}}{2\pi k\delta }}\log {\frac {r_{1}}{r_{2},}}$

where ${\text{Q}}$ is the quantity of electricity supplied by the source per second. Hence the equation to the equipotential curve is $r_{1}r_{2}{=}$ a constant.

If we take a point half-way between the sink and the source as the origin of a system of rectangular co-ordinates, and if the distance between sink and source is equal to $p,$ and the line joining them is taken as the axis of $x,$ then the equation to the equipotential line is

${\frac {y^{2}+(x+p)^{2}}{y^{2}+(x-p)^{2}}}{=}$ a constant.

This is the equation of a family of circles having the axis of $y$ for a common radical axis, one set of circles surrounding the sink and another set of circles surrounding the source. In order to discover the form of the stream of current lines we have to determine the orthogonal trajectories to this family of coaxial circles. It is easy to show that the orthogonal trajectory of the system of circles is another system of circles all passing through the sink and the source, and as a corollary of this fact, that the electric resistance of a circular disk of uniform thickness is the same between any two points taken anywhere on its circumference as sink and source. These equipotential lines may be delineated experimentally by attaching the terminals of a battery or batteries to small wires which touch at various places a sheet of tinfoil. Two wires attached to a galvanometer may then be placed on the tinfoil, and one may be kept stationary and the other may be moved about, so that the galvanometer is not traversed by any current. The moving terminal then traces out an equipotential curve. If there are $n$ sinks and sources in a plane conducting sheet, and if ${r,r',r''}$ be the distances of any point from the sinks, and ${t,t',t''}$ the distances of the sources, then

${\frac {{rr'r''}{\text{ . . . }}}{{tt't''}{\text{ . . . }}}}{=}$a constant.

is the equation to the equipotential lines. The orthogonal trajectories or stream lines have the equation

$\textstyle \sum \displaystyle (\theta -\theta '){=}$ a constant,

where $\theta$ and $\theta '$ are the angles which the lines drawn from any point in the plane to the sink and corresponding source make with the line joining that sink and source. Generally it may be shown that if there are any number of sinks and sources in an infinite plane conducting sheet, and if $r,\theta$ are the polar co-ordinates of any one, then the equation to the equipotential surfaces is given by the equation

$\textstyle \sum \displaystyle ({\text{A}}\log _{e}r){=}$ a constant,

where ${\text{A}}$ is a constant; and the equation to the stream or current lines is

$\textstyle \sum \displaystyle (\theta ){=}$ a constant.</math>

In the case of electric flow in three dimensions the electric potential must satisfy Laplace’s equation, and a solution is therefore found in the form $\textstyle \sum \displaystyle ({\text{A}}/r){=}$ a constant, as the equation to an equipotential surface, where $r$ is the distance of any point on that surface from a source or sink.

Convection Currents.—The subject of convection electric currents has risen to great importance in connexion with modern electrical investigations. The question whether a statically electrified body in motion creates a magnetic field is of fundamental importance. Experiments to settle it were first undertaken in the year 1876 by H. A. Rowland, at a suggestion of H. von Helmholtz.^[1] After preliminary experiments, Rowland’s first apparatus for testing this hypothesis was constructed, as follows:—An ebonite disk was covered with radial strips of gold-leaf and placed between two other metal plates which acted as screens. The disk was then charged with electricity and set in rapid rotation. It was found to affect a delicately suspended pair of astatic magnetic needles hung in proximity to the disk just as would, by Oersted’s rule, a circular electric current coincident with the periphery of the disk. Hence the statically-charged but rotating disk becomes in effect a circular electric current.

The experiments were repeated and confirmed by W. C. Röntgen (Wied. Ann., 1888, 35, p. 264; 1890, 40, p. 93) and by F. Himstedt (Wied. Ann., 1889, 38, p. 560). Later V. Crémieu again repeated them and obtained negative results (Com. rend., 1900, 130, p. 1544, and 131, pp. 578 and 797; 1901, 132, pp. 327 and 1108). They were again very carefully reconducted by H. Pender (Phil. Mag., 1901, 2, p. 179) and by E. P. Adams (id. ib., 285). Pender’s work showed beyond any doubt that electric convection does produce a magnetic effect. Adams employed charged copper spheres rotating at a high speed in place of a disk, and was able to prove that the rotation of such spheres produced a magnetic field similar to that due to a circular current and agreeing numerically with the theoretical value. It has been shown by J. J. Thomson (Phil. Mag., 1881, 2, p. 236) and O. Heaviside (Electrical Papers, vol. ii. p. 205) that an electrified sphere, moving with a velocity $v$ and carrying a quantity of electricity $q,$ should produce a magnetic force ${\text{H}},$ at a point at a distance $p$ from the centre of the sphere, equal to $qv\sin \theta /\rho ^{2},$ where $\theta$ is the angle between the direction of $\rho$ and the motion of the sphere. Adams found the field produced by a known electric charge rotating at a known speed had a strength not very different from that predetermined by the above formula. An observation recorded by R. W. Wood (Phil. Mag., 1902, 2, p. 659) provides a confirmatory fact. He noticed that if carbon-dioxide strongly compressed in a steel bottle is allowed to escape suddenly the cold produced solidifies some part of the gas, and the issuing jet is full of particles of carbon-dioxide snow. These by friction against the nozzle are electrified positively. Wood caused the jet of gas to pass through a glass tube 2.5 mm. in diameter, and found that these particles of electrified snow were blown through it with a velocity of 2000 ft. a second. Moreover, he found that a magnetic needle hung near the tube was deflected as if held near an electric current. Hence the positively electrified particles in motion in the tube create a magnetic field round it.

Nature of an Electric Current.—The question, What is an electric current? is involved in the larger question of the nature of electricity. Modern investigations have shown that negative electricity is identical with the electrons or corpuscles which are components of the chemical atom (see Matter and Electricity). Certain lines of argument lead to the conclusion that a solid conductor is not only composed of chemical atoms, but that there is a certain proportion of free electrons present in it, the electronic density or number per unit of volume being determined by the material, its temperature and other physical conditions. If any cause operates to add or remove electrons at one point there is an immediate diffusion of electrons to re-establish equilibrium, and this electronic movement constitutes an electric current. This hypothesis explains the reason for the identity between the laws of diffusion of matter, of heat and of electricity. Electromotive force is then any cause making or tending to make an inequality of electronic density in conductors, and may arise from differences of temperature, i.e. thermoelectromotive force

1. See Berl. Acad. Ber., 1876, p. 211; also H. A. Rowland and C. T. Hutchinson, "On the Electromagnetic Effect of Convection Currents," Phil. Mag., 1889, 27, p. 445.