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The circuit below contains a battery light-bulb circuit connected to Earth.

enter image description here

If the distance between points A and B where relatively small, then one could see that the earth would be acting as a sort of resistor (practically speaking, as soil has some resistance) and the lightbulb could be potentially lit.

If we however put A and B at large seperation, is current flow still possible? I say this because if the earth is considered an infinite electron source/sink, there would then be reason for current to flow with a battery as a voltage potential, thus lighting our light bulb

Reasons I don't think it will flow. The potential over the battery is 12 volts, but as far as I know there isn't a relationship between any terminal and the earth, as voltage is relative.There doesn't appear to be a voltage difference, yet I still can't reconcile this thought with earth as an electron source/sink

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In the circuit you've drawn, you've accepted that there's a resistance between A and B through the soil, so current will flow. We have a complete circuit.

So the question is, how does the resistance between A and B change as we increase their separation?

It turns out that as we increase the distance between A and B, the resistance increases (no surprises there) but very, very slowly.

In fact, the resistance between A and B is dominated by the ground very close to A, and very close to B. When electricity codes tell you to set a ground connection, they'll tell you what length of rod, what minimum diameter, and to set it in soil rather than rock. The assumption is that the conductivity to any remote earth is dominated by the conditions around that electrode.

The reason is the scale, and the way the path changes as we increase the distance. Let's say we use a 1m long rod for A, 20mm diameter, so its surface area is about 0.06m2. Let's follow the current out 10mm, to a cylinder of soil 40mm diameter. To a crude approximation (I'm not going to do integral calculus here) we have 0.01m length of soil, 0.06m2 area of soil, so the resistance, length/area, is proportional to 0.01/0.06 = 0.16.

When we take the next step, we notice the surface area we are interested in has increased. So the extra resistance of the next 10mm out will be less than for the first 10mm. As we go further away from the rod, we have more surface area, more parallel paths to carry the current.

Once we are as far away from the rod as its length, we notice that instead of the current just spreading out sideways, we need to consider the down spread as well. The surface we are interested in is now getting to be a hemisphere, centred on our ground rod. Now the surface area is growing as distance squared. The resistance is increasing even more slowly than it was before.

Eventually, these surfaces will intercept subterranean watercourses, and the resistance will increase yet more slowly.

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