| r r′ r″ ... | = a constant, |
| t t′ t″ ... |
is the equation to the equipotential lines. The orthogonal trajectories or stream lines have the equation
Σ (θ − θ′) = a constant,
where θ and θ′ 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, θ are the polar co-ordinates of any one, then the equation to the equipotential surfaces is given by the equation
Σ (A log e r) = a constant,
where A is a constant; and the equation to the stream of current lines is
Σ (θ) = a constant.
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 Σ (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.[8] 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 H, at a point at a distance ρ from the centre of the sphere, equal to qv sin θ/ρ², where θ is the angle between the direction of ρ 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.