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A condenser of capacity C(1) is charged ...

A condenser of capacity `C_(1)` is charged to a potential `V_(0)`. The electrostatic energy stored in it is `U_(0)`. It is connected to another uncharged condenser of capacity `C_(2)` in parallel. The energy dissipated in the process is

A

`(C_(2))/(C_(1)+C_(2))U_(0)`

B

`(C_(1))/(C_(1)+C_(2))U_(0)`

C

`((C_(1)-C_(2))/(C_(1)+C_(2)))U_(0)`

D

`(C_(1)C_(2))/(2(C_(1)+C_(2)))U_(0)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, let's break it down step by step. ### Step 1: Determine Initial Energy Stored in the Capacitor The initial energy stored in the capacitor \( U_0 \) can be calculated using the formula for the energy stored in a capacitor: \[ U_0 = \frac{1}{2} C_1 V_0^2 \] ### Step 2: Calculate Initial Charge on the Capacitor The initial charge \( Q_i \) on the charged capacitor can be calculated as: \[ Q_i = C_1 V_0 \] ### Step 3: Determine the Final Common Potential When the charged capacitor is connected in parallel to an uncharged capacitor, the total charge will redistribute. The final common potential \( V_c \) can be found using charge conservation: \[ Q_i = Q_f \] Where \( Q_f \) is the final charge on the combined capacitors. The final charge can be expressed as: \[ Q_f = (C_1 + C_2) V_c \] Setting the initial charge equal to the final charge gives: \[ C_1 V_0 = (C_1 + C_2) V_c \] From this, we can solve for \( V_c \): \[ V_c = \frac{C_1 V_0}{C_1 + C_2} \] ### Step 4: Calculate Final Energy Stored in the Combined Capacitors The final energy \( U_f \) stored in the system after connecting the capacitors can be calculated as: \[ U_f = \frac{1}{2} (C_1 + C_2) V_c^2 \] Substituting \( V_c \) into the equation gives: \[ U_f = \frac{1}{2} (C_1 + C_2) \left( \frac{C_1 V_0}{C_1 + C_2} \right)^2 \] This simplifies to: \[ U_f = \frac{1}{2} \frac{C_1^2 V_0^2}{C_1 + C_2} \] ### Step 5: Calculate Energy Dissipated The energy dissipated during the process can be calculated as the difference between the initial and final energies: \[ \text{Energy Dissipated} = U_i - U_f \] Substituting the expressions for \( U_0 \) and \( U_f \): \[ \text{Energy Dissipated} = \frac{1}{2} C_1 V_0^2 - \frac{1}{2} \frac{C_1^2 V_0^2}{C_1 + C_2} \] Factoring out \( \frac{1}{2} V_0^2 \): \[ \text{Energy Dissipated} = \frac{1}{2} V_0^2 \left( C_1 - \frac{C_1^2}{C_1 + C_2} \right) \] Finding a common denominator: \[ = \frac{1}{2} V_0^2 \left( \frac{C_1(C_1 + C_2) - C_1^2}{C_1 + C_2} \right) \] This simplifies to: \[ = \frac{1}{2} V_0^2 \left( \frac{C_1 C_2}{C_1 + C_2} \right) \] Thus, the energy dissipated can be expressed as: \[ \text{Energy Dissipated} = \frac{C_2 U_0}{C_1 + C_2} \] ### Final Answer The energy dissipated in the process is: \[ \frac{C_2 U_0}{C_1 + C_2} \] ---
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