96 
MR. C. T. R. WILSON: INVESTIGATIONS ON LIGHTNING DISCHARGES 
section. The electric field at the ground due to a cloud of this kind will be the 
resultant of the fields of the upper and lower charges. 
In the ordinary thunder-cloud or cumulo-nimbus cloud we are concerned with 
rapidly ascending air currents of comparatively small horizontal dimensions. The 
heads of such clouds generally reach to heights of several kilometres : according 
to Wegener* the top of a thunder-cloud may reach almost to the upper limit of 
the troposphere (about 10 km.). The average height of the bases is probably about 
1 km. 
If we suppose a cumulo-nimbus cloud to have charges Q 2 and Qi of opposite sign 
in its upper and lower portions, we may, for the purpose of calculating its electric 
field at a distance, treat these charges as if they were concentrated at definite “ poles ” 
at heights H 2 and H,. The effect of the charges induced on the surface of the 
ground is the same as if they were replaced by charges equal and opposite to 
Q 2 and Qi and at depths H 2 and Hi below the surface. The problem is then the same 
as that of finding the magnetic field due to two bar magnets of lengths 2Hi and 2H 2 
and moments 2(^11!, 2Q 2 H 2 , placed so that their centres coincide, the axes being 
vertical and their polarities opposed. 
The vertical electric force due to the cloud at a point on the ground at a distance 
L from the axis is 
F = 
2Q 2 
2Q, 
Ho 2 ( 1 
L 2 \i 
H 2 
Hd 1 
2 / 
L 2 \*' 
Hf 
- Immediately below the cloud, where L = 0, the second term (representing the effect 
of the lower charge) will be the greater unless the ratio of Q 2 to Q x exceeds H^/Eb 2 , 
and for distant points the first term (representing the vertical force due to the upper 
charge) will be the greater unless QJQj exceeds Eb/Hj. Thus the surface of the 
ground may generally be divided into two areas, an inner and outer, in which the 
electric field due to the cloud has opposite signs ; in the central area the effect of the 
lower pole of the cloud predominates and determines the sign of the potential gradient 
and of the charge on the ground, while in the outer area the effect of the upper pole 
of the cloud is the greater. 
The maximum intensity of the resultant field anywhere near the centre of the 
inner area will generally greatly exceed the maximum reached in the outer area. The 
curve III., fig. 3, represents the resultant potential gradient produced at the ground 
by equal and opposite charges of 20 coulombs at heights of 1 and 2 km. The inner 
area has a radius of approximately 2 km., the maximum potential gradient at the 
centre amounts to 270,000 volts per metre, while the maximum reached by the 
potential gradient of opposite sign in the outer area is less than 10,000 volts per 
metre. Greater differences in heights of the two poles are probable in actual 
* Wegener, 1 Thermodynamik der Atmosphare,’ p. 210. 
