25. If V be the potential, ρ the density of free electricity at a point in the atmosphere, at a distance r from the earth’s centre, then assuming statical conditions and neglecting variation of V in horizontal directions, we have
r−2(d/dr)(r² dV/dr) + 4πρ = 0.
For practical purposes we may treat r² as constant, and replace d/dr by d/dh, where h is height in centimetres above the ground.
We thus find
ρ = −(1/4π) d²V/dh².
If we take a tube of force 1 sq. cm. in section, and suppose it cut by equipotential surfaces at heights h1 and h2 above the ground, we have for the total charge M included in the specified portion of the tube
4πM = (dV/dh)h1 − (dV/dh)h2.
Taking Linke’s (28) figures as given in § 10, and supposing h1 = 0, h2 = 15 × 104, we find for the charge in the unit tube between the ground and 1500 metres level, remembering that the centimetre is now the unit of length, M = (1/4π) (125 − 25)/100. Taking 1 volt equal 1⁄300 of an electrostatic unit, we find M = 0.000265. Between 1500 and 4000 metres the charge inside the unit tube is much less, only 0.000040. The charge on the earth itself has its surface density given by σ = −(1/4π) × 125 volts per metre, = 0.000331 in e ectrostatic units. Thus, on the view now generally current, in the circumstances answering to Linke’s experiments we have on the ground a charge of −331 × 10−6 C.G.S. units per sq. cm. Of the corresponding positive charge, 265 × 10−6 lies below the 1500 metres level, 40 × 10−6 between this and the 4000 metres level, and only 26 × 10−6 above 4000 metres.
There is a difficulty in reconciling observed values of the ionization with the results obtained from balloon ascents as to the variation of the potential with altitude. According to H. Gerdien (61), near the ground a mean value for d²V/dh² is −(1⁄10) volt/(metre)². From this we deduce for the charge ρ per cubic centimetre (1/4π) × 10−5 (volt/cm²), or 2.7 × 10−9 electrostatic units. But taking, for example, Simpson’s mean values at Karasjok, we have observed
ρ ≡ I+ − I1 = 0.05 × (cm./metre)3 = 5 × 10−8,