Animated Solution for Physics - Electrostatics: A parallel plate capacitor is charged and the charging battery is then disconnected. If the plates of the capacitor are moved farther apart by means of insulating handles

Trapped Charges

Imagine a parallel plate capacitor that has been fully charged by a battery. The top plate is brimming with positive charges, and the bottom plate is packed with negative charges. Now, we disconnect the battery.

What does this mean physically? It means the charges are completely trapped! They have no conducting path to flow back or neutralize. Therefore, the total charge $Q$ on the capacitor must remain absolutely constant. This is our first and most crucial constraint: $Q=\text{constant}$.

Now, the problem states that we move the plates farther apart using insulating handles. The handles ensure no charge leaks onto our hands. As we pull the plates apart, the distance $d$ between them increases.

Let's recall the geometric formula for the capacitance of a parallel plate capacitor:

Since the distance $d$ is in the denominator, increasing $d$ will cause the capacitance $C$ to decrease. The capacitor's ability to hold charge for a given voltage has dropped because the plates are further apart and their mutual electrostatic attraction is less effective at "holding" the charge.

Next, let's investigate what happens to the voltage $V$ across the plates. The fundamental relationship between charge, capacitance, and voltage is:

We have already established that $Q$ is constant, but $C$ has decreased. Mathematically, dividing a constant numerator by a smaller denominator yields a larger result. Therefore, the voltage $V$ across the plates must increase!

Finally, let's analyze the electrostatic potential energy stored in the capacitor. We have a few formulas for energy, but the best one to use here is the one involving our constant $Q$:

Again, with $Q$ being constant and $C$ decreasing, the energy $U$ must increase.

But wait, where did this extra energy come from? Energy cannot be created out of nowhere! The answer lies in the mechanical work we did. The positive and negative plates attract each other strongly. To pull them apart, we had to exert a force and do mechanical work against this electrostatic attraction. This external work we performed is exactly what gets stored as the additional electrostatic potential energy in the capacitor.

Therefore, both the voltage and the stored electrostatic energy increase, making options (b) and (d) the correct choices.