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The diagram below represents two metal spheres, and one of them is connected to the earth so is at $0 \; \text V$:

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The answer says that the charge of the earthed sphere is negative, and less than the positively charged ball. They explain that if the sphere has zero potential, the charge in it must cancel out the effect if the positive charge on the other ball, and since the earthed sphere is some distance away from the positive sphere, it will not need so much charge to cancel out the potential.

I do not get why this is the case. Shouldn’t an earthed sphere have no charge? Moreover, if potential is the amount of work done per unit charge in taking a positive charge from infinity to that point, wouldn’t the charge have to be exactly $-Q$?

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  • $\begingroup$ Missing from what "they explained": what effect must be cancelled out? $\endgroup$
    – garyp
    Jan 1 '20 at 17:05
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Shouldn’t an earthed sphere have no charge?

This is absolutely incorrect. When they say that an object is connected to ground, it means that the electric potential of the object is zero. Charge will flow from or to ground from the conductor in order to make this happen. Hence net charge will be negative as it has to cancel out the positive charge from the other sphere.

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The positive charge has field lines going to the earth. These push negative charge to the grounded sphere until the resultant field along the grounding wire is zero. So the grounded sphere is at ground potential but does have a negative charge. Since this needs to cancel only a small part of the field from the positive charge, its charge will be smaller. (If the negative sphere surrounded the positive sphere, then their charges wold be equal. Electric field lines must start and end on charges.)

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