By WILLIAM HENRY PREECE.
[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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
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