Problem 68 - Master Resource Book in JEE Main Physics

Question

A parallel plate capacitor of capacitance \(C\) is connected to a battery and is charged to a potential difference \(V\). Another capacitor of capacitance \(2 C\) is connected to another battery and is charged to potential difference \(2 V\). The charging batteries are now disconnected and the capacitors are connected in parallel to each other in such a way that the positive terminal of one is connected to the negative terminal of the other. The final energy of the configuration is (a) infinite (b) zero (c) \(\frac{3 C v^{2}}{2}\) (d) \(\frac{6 \mathrm{C} \mathrm{V}^{2}}{2}\)

Step-by-Step Solution

Answer

The final energy of the configuration is zero.

Step 1: Calculate Initial Energy of Each Capacitor

The energy stored in a capacitor is given by the formula: \( U = \frac{1}{2} C V^2 \). For the first capacitor with capacitance \(C\) and potential difference \(V\), its energy \(U_1\) is \( \frac{1}{2} C V^2 \). For the second capacitor with capacitance \(2C\) and potential difference \(2V\), its energy \(U_2\) is \( \frac{1}{2} \times 2C \times (2V)^2 = 4CV^2 \).

Step 2: Calculate Total Initial Energy

The total initial energy when both capacitors are charged by their respective batteries is the sum of both energies: \( U_{\text{initial}} = U_1 + U_2 = \frac{1}{2} C V^2 + 4 C V^2 = \frac{1}{2} C V^2 + 4 C V^2 = \frac{9}{2} C V^2 \).

Step 3: Connect Capacitors in Parallel

When the two capacitors are connected in parallel with opposite polarity (the positive terminal to the negative), the total capacitance is \(C + 2C = 3C\), but the effective voltage difference will be zero because they are connected with opposite charge polarities.

Step 4: Calculate Final Energy

Once the two capacitors are connected, they achieve equilibrium with zero net potential difference due to opposite charge connection. The final energy can be calculated as follows: \( U_{\text{final}} = \frac{1}{2} (3C) (0)^2 = 0 \).

Step 5: Conclusion on the Final Energy

Based on the calculation from Step 4, the final stored energy in the system is zero.

Key Concepts

Parallel Plate Capacitor

A parallel plate capacitor consists of two conductive plates separated by a distance. This simple device can store electric charge when connected to a power source. Here are key points about parallel plate capacitors to help you understand how they work:

• Each plate builds up opposite charges, creating an electric field between them.
• The ability of the capacitor to hold charge is represented by its capacitance, denoted as \( C \).
• Capacitance depends on the surface area of the plates, the separation between them, and the permittivity of the material in between.
• A higher surface area and smaller distance between plates typically result in greater capacitance.

In this exercise, we consider two capacitors with different charges and their interactions when connected in parallel. Remember, a parallel connection changes the total capacitance and charge distribution.

Potential Difference

The potential difference, often referred to as voltage, across a capacitor's plates is critical in determining the energy stored. Here’s what you need to know about potential difference:

• It is the energy per unit charge and measures the difference in electric potential between two points.
• A potential difference \( V \) across a capacitor indicates the work done to move charge between the plates.
• In the exercise, one capacitor is charged to \( V \) and the other to \( 2V \), impacting the energy calculations.
• The greater the potential difference, the more energy is stored in the capacitor.

After disconnection from the battery, if capacitors are connected with opposite terminals, they reach equilibrium with no potential difference, leading to no net stored energy.

Capacitance

Capacitance is a measure of a capacitor's ability to store charge per unit voltage. Here's what you should know about it:

• It is defined as \( C = \frac{Q}{V} \), where \( Q \) is the charge stored and \( V \) is the potential difference.
• Units of capacitance are farads \( (F) \), where 1 farad equals 1 coulomb per volt.
• In parallel plate capacitors, capacitance is influenced by plate area, separation, and dielectric material.
• A larger capacitance means more charge storage at a given potential difference.

In the problem, capacitors of different capacitances were considered. Connecting them in parallel affected total capacitance and charge equilibrium, leading to a zero final energy.