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I had the following question on a quiz recently:

A capacitor consisting of two parallel plates, separated by a distance $d$, is connected to a battery of EMF ε. What happens if the separation is doubled while the battery remains connected?

The correct answer was "The electric charge on the plates is halved." However, we were previously given a question in which a circuit consists of a capacitor of 6 μF and a resistor of 300,000 Ω connected to a battery, and the separation between the plates of the capacitor is quadrupled. However, in this circumstance, the charge on the capacitor immediately after the plates are separated remains the same, and the potential difference across the capacitor increases.

I am confused as to why these questions have different answers to them, despite having an extremely similar setup. My best guess is that the circuit in the second question also has a resistor, but I am not sure, and help would be greatly appreciated.

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the charge on the capacitor immediately after the plates are separated remains the same

The key word here is immediately. Since there is a $300\mathrm{k\Omega}$ series resistor between the capacitor and the battery, the series current is limited and thus, some time must elapse for the system to reach steady state.

Since the separation between the plates is quadrupled, the capacitance is reduced by a factor of 4 so, eventually, the capacitor discharges through the resistor to a charge 1/4 of the initial charge.

But in the first case, the system (effectively) reaches steady state immediately since there is no series resistance. This is best seen by adding a series resistance and taking the limit as the resistance goes to zero. You will then see that, in the limit, there is an infinitesimally brief, infinitely large discharge current that immediately brings the system into steady state.

Of course, for a physical battery, there is non-zero internal resistance and, further, the plate separation of a physical capacitor cannot be change instantaneously.

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The new question asks about the change in capacitance, which is a property of the geometric arrangement of the plates and the material enclosed between them. You will have a formula with capacitance proportional to the area divided by the distance between the parallel plates: divide by 2d instead of d and your answer is halved.

In the earlier case the question concerns the charge on the plates for a specific circuit, and an instantaneous (very rapid) change in the geometry. Since DC current doesn't flow through a capacitor, the charge is not subject to rapid changes; and the resistor adds to the "inertia" of the system. In fact, you have what is called an RC circuit, which has a decay time.

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