Eventually, the process approaches a steady state. In that steady state, the charge of the capacitor is not changing. What is the current in the circuit in the steady state?

The current in the circuit must be zero.

Explanation:

In a RC circuit, the steady state is reached when either the capacitor is fully charged or fully discharged. In either case, there must not be any current through the circuit because if it exists, it will deliver charge to the capacitor and thus change its charge, which is not a steady state.

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