Oliver Heaviside

“ We may therefore regard convection-current as impressed electric current. From this comparison also, we may see that an infinite plane sheet of electrification of uniform density cannot produce magnetic force by motion perpendicular to its plane. Also we see that the sources of disturbances when ρ is moved are the places where ρu has curl; for example, a dielectric sphere uniformly filled with electrification (which is imaginable) , when moved, starts the magnetic force solely upon its boundary. ”

“ In the extreme case of u=v, the spheroid reduces to a flat circular disc, with a single circular line of electrification on its edge. It would seem, however, to be a matter of indifference, in this extreme case, whether the conductor be a disc or a solid sphere. Bearing in mind the conditions assumed to prevail in the problem of motion of sources of displacement in a uniform medium, we see that if we introduce conductors, say by filling up spaces void of electric force with conducting matter, this should not interfere with the assumed motions. ”

“ To further realize, let the field terminate internally at r=a, giving a cylindrical-surface distribution of electrification, and terminate the tubes of displacement externally upon a coaxial cylindrical surface; we then produce a real electromagnetic plane wave with electrification, and of finite energy. We have supposed the electrification to be carried through the dielectric at speed ν, to keep up with the wave, which would of course break up if the charge were stopped. ”

“ It may be remarked that (if my calculations are correct) equation (7) or its equivalents expresses the mutual energy of any two rational current-elements (see § 1) in a medium of uniform inductivity, of moments q1u1, and q2u2, whether the currents be of displacement, or conduction, or convection, or all mixed, it being in fact the mutual energy of a pair of definite magnetic fields. But, since the hypothesis of instantaneous action is expressly involved in the above, the application of (7) is of a limited nature. ”

“ The transverse concentration of the displacement is obvious. Now the function P (equation (14) , p. 499) is the electrostatic potential in the stationary eolotropic problem, so that its slope , which call F, is the electric force, and the displacement D is a linear function thereof, say D=λF, where λ is the permittivity operator. The condition of equilibrium is that F is perpendicular to the surface where it terminates, this being required to make curl F=0, or the voltage zero in every circuit. ”

“ Let the axis of z be the line of motion of the charge q at the speed u; then the lines of magnetic force H are circles centred upon the axis, in planes perpendicular to it, and its tensor H at distance r from the charge, the line r making an angle θ with the axis, is given by
where v=sin θ, E the intensify of the radial electric force, c the permittivity such that μ0cv2=1, if μ0 is the other specific quality of the medium, its inductivity, and v is the speed of propagation. ”

“ When u=v, with accumulation of the charge at the equator of the sphere, we have infinite values, and it appears to be only possible to have finite values by making a zone at the equator cylindrical instead of spherical. ”

“ There is H, and the convection-current is not now cancelled by coexistent displacement-current.
The existence of displacement-current, or changing displacement, was the basis of the conclusion that moving electrification constitutes a part of the true current. Now in the problem (48) the displacement-current has gone, so that the existence of H appears to rest merely upon the assumption that moving electrification is true current. But if the plane be not infinite, though large, we shall have (48) nearly true near it, and away from the edges ”

“ Only when u>v is there any difficulty. We must first settle upon what basis to work. First the Faraday-law (p standing for d/dt) ,
requires no change when there is moving electrification. But the analogous law of Maxwell, which I understand to be really a definition of electric current in terms of magnetic force, (or a doctrine) , requires modification if the true current is to be
viz. the sum of conduction-current, displacement-current, and convection-current ρu, where ρ is the volume-density of electrification. ”

“ It gives a real solution for the hinder cone, a real solution for the forward cone, and an unreal solution in the rest of space, but we have no instruction to reject the part for the forward cone and the unreal part, nor have we any means of testing that the remainder, confined to the hinder cone, is the proper solution, viz., by the test of divergence, to give the right amount of electrification. The integral displacement comes to -∞. Now this may require to be supplemented by +∞+q on the boundary of the cone, but we have no way of testing it. ”

“ When the speed is very slow we may regard the electric field as given by plus a small correcting vector, which we may call the electric force of inertia. But to show the physical inanity of Ψ, go to the other extreme, and let u nearly equal v. It is now the electric force of inertia (supposed) that equals nearly (except about the equatorial plane) , and its sole utility or function is to cancel the other of the (supposed) electrostatic field. ”

“ And since the moving charge is a convection-current qu, the displacement-current should be towards the apex in the axial portion of the cone, and change sign at some unknown distance, so as to be away from the apex either in the outer part of the cone or else upon its boundary. The pulling back of the charge by the electric stress would require the continued application of impressed force to keep up the motion, and its activity would be accounted for by the continuous addition made to the energy in the cone ”

“ Let
and calculate A, which may be any vector satisfying (15) . Its characteristic is
The divergence of A is of no moment, and it is only vexatious complication to introduce ψ. The time-rate of decrease of A is not the real distribution of electric force, which has to be found by the additional datum
where E is the real force.
9. "Symbolically" expressed, the solution of (16) is
Here the numerator of the fraction to the right is the vector-potential of the convection-current. ”

“ The clearest course to pursue appears to me to invariably make E and H the primary objects of attention, and only use potentials when they naturally suggest themselves as labour-saving appliances.
Special Tests. The Connecting Equations. ”

“ Whilst, however, agreeing with J. J. Thomson in the fundamentals, I have been unable to corroborate some of his details; and since some of his results have been recently repeated by him in another place, [2] it may be desirable to state the changes I propose, before proceeding to the case of a charge moving at any speed.
The Energy and Forces in the Case of Slow Motion.
3. First, as regards the magnetic energy, say T. This is the space-summation Σμ0H2/8π; or, by (1) , [3] ,
The limits are such as include all space outside the sphere r=a. ”

“ A Charged Plane moving Transversely.
18. Coming next to a plane distribution of electrification, let q be the surface-density, and the plane be moving perpendicularly to itself. Let it be of finite breadth and of infinite length, so that we may calculate H from (43) . The result at P is
When P is equidistant from the edges, H is zero. ”