Lord Kelvin: An account of his scientific life and work

by Andrew Gray

In the thermal analogy explained by Thomson in his first paper, the positive point-charges are point-sources of heat, which is there poured at constant rate into the medium (supposed of uniform quality) to be drawn off in part from the medium at constant rate where there are sinks (or negative sources),—the negative point-charges in the electrical case,—while the remainder is conducted away to more and more distant parts of the conducting medium supposed infinitely extended. Whenever a point-source, or a point-sink, exists at a distance from other sources or sinks, the flow in the vicinity is in straight lines from or to the point, and these straight lines would be indefinitely extended if either source or sink existed by itself. As it is, the direction and amount of flow everywhere depends on the flow resulting from the whole arrangement of sources and sinks. Lines can be drawn in the medium which show the direction of the resultant flow from point to point, and these lines of flow can be so spaced as to indicate, by their closeness together or their distance apart, where the rate of flow is greater or smaller; and such lines start from sources, and either end in sinks or continue their course to infinity. In the electrical case these lines are the analogues of the lines of electric force (or field-intensity) in the insulating medium, which start from positive charges and end in negative, or are prolonged to infinity.

Across such lines of flow can be drawn a family of surfaces, to each of which the lines met by the surface are perpendicular. These surfaces are the equitemperature surfaces, or, as they are usually called, the isothermal surfaces. They can be drawn more closely crowded together, or more widely separated, so as to indicate where the rate of falling off of temperature (the "temperature slope") is greater or less, just as the contour lines in a map show the slopes on a hill-side.

Instead of the thermal analogy might have been used equally well that of steady flow in an indefinitely extended mass of homogeneous frictionless and incompressible fluid, into which fluid is being poured at a constant rate by sources and withdrawn by sinks. The isothermal surfaces are replaced by surfaces of equal pressure, while lines of flow in one are also lines of flow in the other.

Fig. 1.

Now let heat be poured into the medium at constant rate by a single point-source P (Fig. [1]), and drawn off at a smaller rate by a single point-sink P', while the remainder flows to more and more remote parts of the medium, supposed infinite in extent in every direction. After a sufficient time from the beginning of the flow a definite system of lines of flow and isothermal surfaces can be traced for this case in the manner described above. One of the isothermal surfaces will be a sphere S surrounding the sink, which, however, will not be at the centre of the sphere, but so situated that the source, sink, and centre are in line, and that the radius of the sphere is a mean proportional between the distances of the source and sink from the centre. If a be the radius of the sphere and f the distance of the source from the centre of the sphere, the heat carried off by the sink is the fraction a ⁄ f of that given out by the source.

In the electrical analogue, the source and sink are respectively a point-charge and what is called the "electric image" of that charge with respect to the sphere, which is in this case an equipotential surface. And just as the lines of flow of heat meet the spherical isothermal surface at right angles, so the lines of force in the electrical case meet the equipotential surface also at right angles. Now obviously in the thermal case a spherical sink could be arranged coinciding with the spherical surface so as to receive the flow there arriving and carry off the heat from the medium, without in the least disturbing the flow outside the sphere. The whole amount of heat arriving would be the same: the amount received per unit area at any point on the sphere would evidently be proportional to the gradient of temperature there towards the surface. Of course the same thing could be done at any isothermal surface, and the same proportionality would hold in that case.

Similarly the source could be replaced by a surface-distribution of sources over any surrounding isothermal surface; and the condition to be fulfilled in that case would be that the amount of heat given out per unit area anywhere should be exactly that which flows out along the lines of flow there in the actual case. Outside the surface the field of flow would not be affected by this replacement. It is obvious that in this case the outflow per unit area must be proportional to the temperature slope outward from the surface.

The same statements hold for any complex system of sources and sinks. There must be the same outflow from the isothermal surface or inflow towards it, as there is in the actual case, and the proportionality to temperature slope must hold.