Question

2-24 An uncharged capacitor having capacitance $C$ is connected across a battery of voltage $V$. Now the capacitor is disconnected and then reconnected across the same battery but with reversed polarity. Then which of the statement is INCORRECT

• A. After reconnecting, heat energy produced in the circuit will be equal to two-third of the total energy supplied by battery
• B. After reconnecting, no energy is supplied by battery
• C. After reconnecting, whole of the energy supplied by the battery is converted into heat
• D. After reconnecting, thermal energy produced in the circuit will be equal to $2CV^2$

Answer 1

Answer: (B)

After reconnecting, no energy is supplied by battery

Answer 2

To solve this problem, we need to analyze the energy transfers during two phases:

1. Initial charging of the capacitor.
2. Reconnection of the capacitor with reversed polarity.

Let the capacitance be $C$ and the battery voltage be $V$.

Phase 1: Initial Charging

An uncharged capacitor is connected across a battery of voltage $V$.

• Initial charge on capacitor $Q_i = 0$.
• Final charge on capacitor $Q_f = CV$.
• Energy stored in the capacitor $U_{stored,1} = \frac{1}{2}CV^2$.
• Charge supplied by the battery $\Delta Q_1 = CV$.
• Work done by the battery $W_{battery,1} = V \times \Delta Q_1 = V(CV) = CV^2$.
• Heat produced during initial charging $H_1 = W_{battery,1} - U_{stored,1} = CV^2 - \frac{1}{2}CV^2 = \frac{1}{2}CV^2$.

Phase 2: Reconnection with Reversed Polarity

The capacitor is initially charged to $V$ (from Phase 1). Let's assume one plate has charge $+CV$ and the other has $-CV$. The energy stored is $U_{initial,2} = \frac{1}{2}CV^2$.

Now, it's disconnected and reconnected across the same battery but with reversed polarity. This means the capacitor will eventually be charged to voltage $V$ again, but with the opposite polarity. So, the plate that had $+CV$ will now have $-CV$, and the plate that had $-CV$ will now have $+CV$.

• Final energy stored in the capacitor $U_{final,2} = \frac{1}{2}CV^2$.

• Change in stored energy $\Delta U_2 = U_{final,2} - U_{initial,2} = \frac{1}{2}CV^2 - \frac{1}{2}CV^2 = 0$.

• To reverse the charge on a plate from $+CV$ to $-CV$, a total charge of $2CV$ must flow through the circuit. For example, if the plate connected to the positive terminal of the battery initially had charge $+CV$, after reversal it will be connected to the negative terminal and will acquire charge $-CV$. The battery effectively draws $CV$ charge from this plate and then supplies $CV$ charge to it in the opposite direction.

  Alternatively, consider the plate that was initially at $-CV$. When reconnected to the positive terminal, it will acquire $+CV$. The charge supplied by the battery to this plate is $(+CV) - (-CV) = 2CV$.

• Charge supplied by the battery during reconnection $\Delta Q_2 = 2CV$.

• Work done by the battery during reconnection $W_{battery,2} = V \times \Delta Q_2 = V(2CV) = 2CV^2$.

• Heat produced during reconnection $H_2 = W_{battery,2} - \Delta U_2 = 2CV^2 - 0 = 2CV^2$.

Now let's evaluate each statement based on the calculations for "After reconnecting" (Phase 2):

(A) After reconnecting, heat energy produced in the circuit will be equal to two-third of the total energy supplied by battery

• Heat energy produced after reconnecting ($H_2$) = $2CV^2$.
• "Total energy supplied by battery" in this context can be interpreted as the total energy supplied by the battery throughout the entire process described in the problem (initial charging + reconnection).
  • Total energy supplied by battery $W_{total} = W_{battery,1} + W_{battery,2} = CV^2 + 2CV^2 = 3CV^2$.
• Check the statement: Is $H_2 = \frac{2}{3} W_{total}$? Is $2CV^2 = \frac{2}{3} (3CV^2)$? $2CV^2 = 2CV^2$. This statement is CORRECT.

(B) After reconnecting, no energy is supplied by battery

• We calculated that energy supplied by battery during reconnection ($W_{battery,2}$) is $2CV^2$. Since $C$ and $V$ are non-zero, $2CV^2$ is not zero.
• This statement is INCORRECT.

(C) After reconnecting, whole of the energy supplied by the battery is converted into heat

• Energy supplied by battery during reconnection ($W_{battery,2}$) = $2CV^2$.
• Heat produced during reconnection ($H_2$) = $2CV^2$.
• Since $W_{battery,2} = H_2$, this statement is CORRECT.

(D) After reconnecting, thermal energy produced in the circuit will be equal to $2CV^2$

• We calculated that thermal energy produced during reconnection ($H_2$) is $2CV^2$.
• This statement is CORRECT.

The question asks for the INCORRECT statement. Based on our analysis, statement (B) is the only incorrect one.