Two capacitors `C_(1)` and `C_(2)` are charged separately to potentials `20 V and 10V`, respectively. The terminals of capacitors `C_(1) and C_(2)` are marked as (A-B) and (C-D), respectively. A is connected with `C and B` is connected with D.
i. Find the final potential difference across each capacitors.
ii. Find the final charge in both capacitors
iii. How much heat is produced in the circuit.

Options

A. `128muC` `142muC` `60 uJ`
B. `28muC` `42muC` `60 uJ`
C. `28muC` `42muC` `160 uJ`
D. `2.8muC` `4.2muC` `60 uJ`

Explanation

To solve the problem involving two capacitors \( C_1 \) and \( C_2 \) charged to different potentials and then connected, we will follow these steps:

### Step 1: Understand the Initial Conditions
- Capacitor \( C_1 \) is charged to \( 20 \, V \).
- Capacitor \( C_2 \) is charged to \( 10 \, V \).
- The capacitors are connected such that terminal A of \( C_1 \) is connected to terminal C of \( C_2 \), and terminal B of \( C_1 \) is connected to terminal D of \( C_2 \).

### Step 2: Calculate the Equivalent Capacitance
When the capacitors are connected in parallel, the equivalent capacitance \( C_{eq} \) is given by:
\[ C_{eq} = C_1 + C_2 \]
However, we need to find the final potential difference across each capacitor, which requires us to consider the charge on each capacitor.

### Step 3: Calculate Initial Charges
The initial charge on each capacitor can be calculated using the formula:
\[ Q = C \cdot V \]
Let \( Q_1 \) be the charge on \( C_1 \) and \( Q_2 \) be the charge on \( C_2 \):
\[ Q_1 = C_1 \cdot 20 \, V \]
\[ Q_2 = C_2 \cdot 10 \, V \]

### Step 4: Find the Final Voltage Across Each Capacitor
When the capacitors are connected, they will share the total charge. The final voltage \( V_f \) across both capacitors can be found using the conservation of charge:
\[ Q_{total} = Q_1 + Q_2 \]
The final voltage across both capacitors will be the same:
\[ V_f = \frac{Q_{total}}{C_{eq}} \]

### Step 5: Calculate Final Charges
The final charge on each capacitor can be calculated using:
\[ Q_{final} = C \cdot V_f \]
Thus, for each capacitor:
\[ Q_{1,final} = C_1 \cdot V_f \]
\[ Q_{2,final} = C_2 \cdot V_f \]

### Step 6: Calculate Heat Produced
The heat produced in the circuit can be calculated from the energy stored in the capacitors before and after they are connected. The energy stored in a capacitor is given by:
\[ U = \frac{1}{2} C V^2 \]
Calculate the initial energy \( U_{initial} \) and the final energy \( U_{final} \):
\[ U_{initial} = \frac{1}{2} C_1 (20^2) + \frac{1}{2} C_2 (10^2) \]
\[ U_{final} = \frac{1}{2} C_1 V_f^2 + \frac{1}{2} C_2 V_f^2 \]
The heat produced \( Q_{heat} \) is:
\[ Q_{heat} = U_{initial} - U_{final} \]

### Final Answers
1. **Final potential difference across each capacitor**: \( V_f \) (calculated from the above steps).
2. **Final charge in both capacitors**: \( Q_{1,final} \) and \( Q_{2,final} \) (calculated from the above steps).
3. **Heat produced in the circuit**: \( Q_{heat} \) (calculated from the above steps).