Question

A parallel plate capacitor is charged with a battery and then disconnected from the battery, leaving a certain amount of energy stored in the capacitor. The separation between the plates is then increased. What happens to the energy stored in the capacitor? Discuss your answer in terms of energy conservation.

Short answer

Answer: When the separation between the plates of a charged parallel plate capacitor is increased, the energy stored in the capacitor increases due to the work performed against the electric field. This change in energy is consistent with the principle of energy conservation, as the external work done contributes to the increase in potential energy of the capacitor.

Step-by-step solution

Understand the Initial Situation

Initially, a battery charges a parallel plate capacitor, and the capacitor stores a certain amount of energy. We can denote the initial capacitance (before separation) as C1 and the initial voltage across the capacitor as V. The energy stored in the capacitor can be calculated using the formula: E1 = 1/2 * C1 * V^2

Increasing the Plate Separation

Now, suppose the separation between the plates is increased. When we increase the distance between the plates, the capacitance of the capacitor decreases, resulting in a new capacitance denoted as C2. Since the capacitor is disconnected from the battery, the electric charge (Q) on the capacitor plates remains the same. We can use the formula for the capacitance as: C = Q/V From this, we can infer: C1 * V = C2 * V2 where V2 is the new voltage across the capacitor after increasing the plate separation.

Calculate the New Energy Stored

Let's find the new energy stored in the capacitor, denoted as E2. The energy can be calculated using the same formula as before, but we'll substitute C2, and V2: E2 = 1/2 * C2 * V2^2 We can rewrite the V2 in terms of V and the capacitance ratios: V2 = (C1 * V) / C2 Now, we can substitute this expression of V2 into the formula for E2: E2 = 1/2 * C2 * ((C1 * V)/C2)^2 = 1/2 * ((C1^2 * V^2)/C2)

Comparing the Initial and Final Energies

Let's compare the initial energy E1 and the final energy E2: E1 = 1/2 * C1 * V^2 E2 = 1/2 * ((C1^2 * V^2)/C2) Notice that E2 > E1, which means the energy has increased. However, we know that energy must be conserved.

Energy Conservation Considerations

In this case, energy conservation does hold, as the process of increasing plate separation requires work to be performed against the electric field generated by the charged capacitor plates. This external work increases the potential energy of the capacitor, resulting in the increase in energy stored. In conclusion, when the separation between the plates of a charged parallel plate capacitor is increased, the energy stored in the capacitor increases due to the work performed against the electric field. This change in energy is consistent with the principle of energy conservation.