Final answer:
In the context of capacitors in a circuit, increasing the plate area of a capacitor with a fixed voltage, adding capacitors in parallel, or doubling the voltage will all increase capacitance and stored energy. However, increasing the distance between the plates of a capacitor reduces its capacitance.
Step-by-step explanation:
The student's question pertains to the behavior of capacitors and how certain changes in a circuit affect capacitance and voltage. When considering a parallel plate capacitor where the plate area increases while voltage is held constant, the capacitance and the stored energy both increase. Adding capacitors in parallel to a circuit increases the total capacitance, therefore increasing the energy stored in the circuit, while adding capacitors in series decreases the total capacitance, which in turn decreases the energy stored.
If you double the voltage across a capacitor, the stored energy will increase four times because energy stored in a capacitor is proportional to the square of the voltage. When increasing the distance between the plates of a capacitor, the capacitance decreases. Specifically, doubling the distance between the plates reduces the capacitance to one-half; it does not reduce it fourfold nor does it increase the capacitance.
In a circuit consisting of a battery and a capacitor, if you were to alter the distance between the plates without changing the applied voltage, the voltage across the capacitor would remain constant, because the voltage across a capacitor is determined by the charge on it and the capacitance. Altering the distance -- and thus the capacitance -- would affect how much charge can be stored at that voltage, but not the voltage itself.