How much am I RADIATING at VLF?

As a matter of interest, this afternoon I worked out how much power I am actually radiating when carrying out my earth-mode tests. The main transmission mode is conduction through the soil/rocks and buried utilities, but an earth-electrode antenna will produce some very very small amount of radiation too.

The first thing is the effective area of the "loop in the ground" and based on a guestimate of 40 ohm metre soil resisitivity (could be somewhere between 10-100 ohm metres) my calculations give me an effective loop area of 600m sq at 8.97kHz - i.e. the signal current flows quite deeply into the ground.

The second figure is the current flowing in the loop (I) which I measure as 0.2A using a current transformer to sense the current.

Rrad = (31171 x Ae^2)/lambda^4 = 0.94 *10^-8

Lambda = 33km
Ae = 600 sq m
I = 0.2 amps

Rrad = 0.94 * 10^-8 ohms
ERP = I^2 x Rrad

So, plugging in the numbers:

ERP = 0.2 x 0.2 x 0.94 x 10^-8

ERP = 37nW

There may well be errors in my sums and in the assumptions made, but clearly 37nW is a tiny amount of radiated power and (almost) negligible. To get to a level where the radiated signal is detectable over 100km away, I would need to radiate around 4-8uW, i.e. several orders of magnitude more. Radiation resistance is proportional to the loop area squared so increasing the baseline by a factor of 10 increases the radiation resistance by 100 times. This could be helped with a much longer baseline (200m long rather than 20m) and increasing the power into the earth-electrode loop to 500W and elevating the loop part that feeds the far earth electrode with current. Such changes could result in a radiated power level of around 4-8uW based on the assumptions about soil/rock resistivity and skin depth. A 200m long piece of wire (e.g. along a field edge) and 500W of audio power are not that hard to envisage and a LOT easier than winding a huge loading coil and raising a kite supported antenna several hundreds of metres high.