91Ó°ÊÓ

Capacitance is a measure of a capacitor's ability to store charge per unit of electric potential difference between its plates. It is defined by the equation, \( C = \frac{Q}{V} \) where \( C \) is the capacitance, \( Q \) is the charge stored, and \( V \) is the potential difference across the plates. Capacitors are commonly used in electronic circuits to store energy and filter signals.

In the case of a parallel-plate capacitor, the capacitance is also affected by the area of the plates and the distance between them, given by \( C = \frac{\varepsilon_0 A}{d} \) where \( \varepsilon_0 \) is the permittivity of free space, \( A \) is the area of one plate, and \( d \) is the separation between the plates. This is crucial as it means that if we change the separation while keeping the charge constant, the capacitance will change. In our example, doubling the separation of the plates halves the capacitance, which in turn affects the potential difference and the energy stored in the capacitor.

Understanding Potential Difference

Electric potential difference, often called voltage, is the amount of electric potential energy per unit charge at one point in space compared to another point. In a capacitor, it is the energy required to move a charge from one plate to another.

Using the exercise as an example, the initial potential difference can be found by dividing the charge on the plates by the capacitance. When the plate separation is doubled, the capacitance is halved, so the potential difference is doubled under the condition that the charge remains the same, as shown in the step-by-step solution. It is essential to note that this relationship holds due to the inverse proportionality between capacitance and the separation between the plates in a parallel-plate capacitor.

Calculating Work in a Capacitor

When discussing work done on capacitors, we are generally referring to the energy required to move charge within the electric field between the plates, which in turn changes the electric potential energy of the system. The formula to calculate the work done to change the capacitance (or the separation of the plates) is \( W = \frac{1}{2} C V^2 - \frac{1}{2} C' V'^2 \) where \( W \) is the work done, \( C \) and \( C' \) are the initial and final capacitance values, and \( V \) and \( V' \) are the initial and final voltages. This equation shows that the work done is equal to the change in stored energy.

In our scenario, the work required to double the separation is calculated using the change in capacitance and voltage due to the increased distance. This reflects the energy needed to increase the separation against the attractive force between the oppositely charged plates. Knowing how to calculate this work is critical in designing circuits and understanding the energy dynamics within them.