[Footnote: From the Philosophical Magazine for December, 1880.]

Any portion of non-conducting space disturbed by electricity is called an electric field. At every point of this field, if a small electrified body were placed there, there would be a certain resultant force experienced by it dependent upon the distribution of electricity producing the field. When we know the strength and direction of this resultant force, we know all the properties of the field, and we can express them numerically or delineate them graphically, Faraday (Exp. Res., § 3122 et seq.) showed how the distribution of the forces in any electric field can be graphically depicted by drawing lines (which he called lines of force) whose direction at every point coincides with the direction of the resultant force at that point; and Clerk-Maxwell (Camb. Phil. Trans., 1857) showed how the magnitude of the forces can be indicated by the way in which the lines of force are drawn. The magnitude of the resultant force at any point of the field is a function of the potential at that point; and this potential is measured by the work done in producing the field. The potential at any point is, in fact, measured by the work done in moving a unit of electricity from the point to an infinite distance. Indeed the resultant force at any point is directly proportional to the rate of fall of potential per unit length along the line of force passing through that point. If there be no fall of potential there can be no resultant force; hence if we take any surface in the field such that the potential is the same at every point of the surface, we have what is called an equipotential surface. The difference of potential between any two points is called an electromotive force. The lines of force are necessarily perpendicular to the surface. When the lines of force and the equipotential surfaces are straight, parallel, and equidistant, we have a uniform field. The intensity of the field is shown by the number of lines passing through unit area, and the rate of variation of potential by the number of equipotential surfaces cutting unit length of each line of force. Hence the distances separating the equipotential surfaces are a measure of the electromotive force present. Thus an electric field can be mapped or plotted out so that its properties can be indicated graphically.

On The Space Protected By A Lightning Conductor 288 14a

Fig. 1

The air in an electric field is in a state of tension or strain; and this strain increases along the lines of force with the electromotive force producing it until a limit is reached, when a rent or split occurs in the air along the line of least resistance--which is disruptive discharge, or lightning.

On The Space Protected By A Lightning Conductor 288 14b

Fig. 2

Since the resistance which the air or any other dielectric opposes to this breaking strain is thus limited, there must be a certain rate of fall of potential per unit length which corresponds to this resistance. It follows, therefore, that the number of equipotential surfaces per unit length can represent this limit, or rather the stress which leads to disruptive discharge. Hence we can represent this limit by a length. We can produce disruptive discharge either by approaching the electrified surfaces producing the electric field near to each other, or by increasing the quantity of electricity present upon them; for in each case we should increase the electromotive force and close up, as it were, the equipotential surfaces beyond the limit of resistance. Of course this limit of resistance varies with every dielectric; but we are now dealing only with air at ordinary pressures. It appears from the experiments of Drs. Warren De La Rue and Hugo Muller that the electromotive force determining disruptive discharge in air is about 40,000 volts per centimeter, except for very thin layers of air.

On The Space Protected By A Lightning Conductor 288 14c

Fig. 3

If we take into consideration a flat portion of the earth's surface, A B (fig. 1), and assume a highly charged thunder-cloud, C D, floating at some finite distance above it, they would, together with the air, form an electrified system. There would be an electric field; and if we take a small portion of this system, it would be uniform. The lines, a b, a' b'...would be lines of force; and cd, c' d', c" d' ...would be equipotential planes. If the cloud gradually approached the earth's surface (Fig. 2), the field would become more intense, the equipotential surfaces would gradually close up, the tension of the air would increase until at last the limit of resistance of the air, e f, would be reached; disruptive discharge would take place, with its attendant thunder and lightning. We can let the line, e f, represent the limit of resistance of the air if the field be drawn to scale; and we can thus trace the conditions that determine disruptive discharge.

On The Space Protected By A Lightning Conductor 288 14d

Fig. 4

If the earth-surface be not flat, but have a hill or a building, as H or L, upon it, then the lines of force and the equipotential planes will be distorted, as shown in Fig. 3. If the hill or building be so high as to make the distance H h or L l equal to e f (Fig. 2), then we shall again have disruptive discharge.

If instead of a hill or building we erect a solid rod of metal, G H, then the field will be distorted as shown in Fig. 4. Now, it is quite evident that whatever be the relative distance of the cloud and earth, or whatever be the motion of the cloud, there must be a space, g g', along which the lines of force must be longer than a' a or H H'; and hence there must be a circle described around G as a center which is less subject to disruptive discharge than the space outside the circle; and hence this area may be said to be protected by the rod, G H. The same reasoning applies to each equipotential plane; and as each circle diminishes in radius as we ascend, it follows that the rod virtually protects a cone of space whose height is the rod, and whose base is the circle described by the radius, G a. It is important to find out what this radius is.

On The Space Protected By A Lightning Conductor 288 14e

Fig. 5

Let us assume that a thunder-cloud is approaching the rod, A B (Fig. 5), from above, and that it has reached a point, D', where the distance. D' B, is equal to the perpendicular height, D' C'. It is evident that, if the potential at D be increased until the striking-distance be attained, the line of discharge will be along D' C or D' B, and that the length, A C', is under protection. Now the nearer the point D' is to D the shorter will be the length A C' under protection; but the minimum length will be A C, since the cloud would never descend lower than the perpendicular distance D C.

Supposing, however, that the cloud had actually descended to D when the discharge took place. Then the latter would strike to the nearest point; and any point within the circumference of the portion of the circle, B C (whose radius is D B), would be at a less distance from D than either the point B or the point C.

Hence a lightning-rod protects a conic space whose height is the length of the rod, whose base is a circle having its radius equal to the height of the rod, and whose side is the quadrant of a circle whose radius is equal to the height of the rod.

I have carefully examined every record of accident that was available, and I have not yet found one case where damage was inflicted inside this cone when the building was properly protected. There are many cases where the pinnacles of the same turret of a church have been struck where one has had a rod attached to it; but it is clear that the other pinnacles were outside the cone; and therefore, for protection, each pinnacle should have had its own rod. It is evident also that every prominent point of a building should have its rod, and that the higher the rod the greater is the space protected.