Final answer:
Long after the switch is closed, the current through the capacitor becomes zero as it reaches a steady state with a constant voltage equal to the battery's emf.
Step-by-step explanation:
The voltage and current through the capacitor in the circuit a long time after the switch is closed can be determined by understanding the behavior of the circuit as it reaches a steady state. During the charging phase, the voltage on the capacitor initially rises rapidly, and the current through the capacitor is at its maximum. However, as the capacitor's voltage approaches the emf of the battery, the current decreases. According to the formula V = emf(1 - e-t/RC) for the charging of a capacitor, we can see that as time extends to infinity (a long time after the switch is closed), the current through the capacitor reduces to zero, and the voltage on the capacitor equals the battery's emf (constant voltage). Therefore, in the steady state, the voltage across the capacitor will be the same as the voltage of the battery, and there will be no current flow through the capacitor.
So, the answer to the question is that a long time after the switch is closed, the voltage across the capacitor is constant, and the current through the capacitor is zero.