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The Earth's surface has a negative surface charge density of $1/10^9\,\mathrm{C/m}^2$. The potential difference of $400\,\mathrm{kV}$ between the top of the atmosphere and the surface results in a current of only $1800\,\mathrm{A}$ over the entire globe. So a naive calculation of the rough time required to neutralize the earth's surface comes out to $283\,\mathrm{s}$ which seems too small.

Why doesn't this happen in reality?

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The time comes out to be 283 seconds because you assume that there is no mechanism of sustaining the atmospheric electric field. According to the generally accepted global atmospheric electrical circuit concept, this never happens in reality because continual thunderstorms and lightning replenish electric charges.

https://en.wikipedia.org/wiki/Global_atmospheric_electrical_circuit

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