Final answer:
The charge that will flow from the conducting shell to the earth when it is earthed, given charges placed near it, is 4q. This ensures that the potential of the earthed shell is zero, maintaining electrostatic equilibrium.
Step-by-step explanation:
When a thin conducting shell is earthed, the charges within and outside influence the charge distribution. Placing two point charges q and 3q near the shell causes induction, resulting in charge rearrangements. If the shell is then earthed, it allows charges to flow until the shell's potential equals that of the Earth (zero potential). The charge on the surface of the conducting shell will redistribute so that the net potential due to the induced charges and the point charges outside is zero. Since the charge inside is 2q, earth connection will cause a charge to flow, ensuring the charge inside is neutralized.
The charge flowing will be 4q, considering the shell initially had 2q and to neutralize the effects due to point charges q and 3q. The charge redistribution process in conductors and the grounding concept are accordance with electrostatic principles. It should be noted that the interior of a conductor must have a net charge of zero when in electrostatic equilibrium. Therefore, the earthed shell must lose a charge of 4q to maintain this equilibrium state when influenced by the given point charges.