91Ó°ÊÓ

Capacitance is a measure of a capacitor's ability to store electric charge. The capacitance of a parallel-plate capacitor is particularly easy to calculate based on its physical dimensions and the properties of the material between the plates—the dielectric. For such a capacitor, the formula is
\( C = \frac{\varepsilon_0 A}{d} \),
where

• \( \varepsilon_0 \) represents the permittivity of free space,
• \( A \) is the area of one plate, and
• \( d \) is the separation distance between the two plates.

This equation shows that the capacitance is directly proportional to the plate area and inversely proportional to the plate separation. Understanding this relationship is crucial, as changes to the plate area or separation will impact the capacitance, which in turn affects other properties of the capacitor, such as the stored energy and the electric potential difference.

A capacitor stores electrical energy when electric charges of equal magnitude, but opposite polarity, are separated across its plates. The energy stored in a capacitor can be expressed by the formula:
\( U = \frac{1}{2}CV^2 \),
where \( U \) is the stored energy, \( C \) is the capacitance, and \( V \) is the electric potential difference across the capacitor's plates. When the plate separation of a disconnected parallel-plate capacitor is doubled, the charge \( Q \) remains constant because it's disconnected from the battery that provided the charge. However, because capacitance depends on the plate separation, the act of doubling the separation will halve the capacitance (\( C' = \frac{C}{2} \)).

As a result, the potential difference across the plates must increase to maintain the constant charge (since \( Q = CV \)), specifically it will double (\( V' = 2V \)). When these new values are placed into the stored energy formula, it can be seen that the stored energy, surprisingly, doubles. This is a non-intuitive outcome that highlights the importance of considering all aspects of the capacitor's relationships—capacitance, charge, potential difference, and stored energy—to understand how they influence each other.

The electric potential difference, or voltage, between two points is a measure of the work done to move a unit charge from one point to the other. In the context of capacitors, the electric potential difference across the capacitor's plates is crucial because it determines how much charge is stored for a given capacitance.
The relationship between charge (\( Q \)), capacitance (\( C \)), and voltage (\( V \)) is given by the simple equation \( Q = CV \). Therefore, if the amount of stored charge on a capacitor remains constant while the capacitance is reduced (for example by increasing the distance between plates), the voltage across the capacitor must increase accordingly.

This effect is precisely what happens when the plate separation of a disconnected parallel-plate capacitor is doubled: the capacitance is halved, but since the charge cannot go anywhere (the capacitor is disconnected), the electric potential difference must double to maintain the equation balance. This is why the stored energy, contrary to what might be expected, increases when the plate separation increases in a charged, disconnected capacitor.