We consider the Earth to be a spherical conductor of radius 6370 km and to be initially uncharged. A metal sphere, having a radius 13 cm and carrying a charge is earthed, that is, put into electrical contact with the Earth. We have to show that this process effectively discharges the sphere. We will calculate the fraction of the excess electrons originally present on the sphere that remain after the sphere is earthed.

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Let the charge on the after it has been connected to a charged conducting sphere of radius 13 cm and charge be Q. If the Earth is considered to be a conducting sphere, then after the conducting sphere has been earthed, the Earth and the sphere will become equipotential. That is we have the following condition on the distributed charge

where radius of the Earth is

and radius of the conducting sphere is

.

Therefore,

and

Charge on the conducting sphere after it has been earthed will be

Therefore,

The charge left on the conducting sphere is about that of

electrons. And it is effectively discharged.

The fraction of the electrons that remain on the conducting sphere after it has been earthed will be