Two parallel plate condensers of capacity `20 mF` and `30 mF` are charged to the potentials of 30 V and 20 V respectively. If likely charged plates are connected together then the common potential difference be

Options

A. 100 V
B. 50 V
C. 24 V
D. 10 V

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the charge on each capacitor
The charge \( Q \) on a capacitor is given by the formula:
\[ Q = C \times V \]
where \( C \) is the capacitance and \( V \) is the voltage.

For the first capacitor:
- Capacitance \( C_1 = 20 \, \text{mF} = 20 \times 10^{-3} \, \text{F} \)
- Voltage \( V_1 = 30 \, \text{V} \)

Calculating the charge \( Q_1 \):
\[ Q_1 = C_1 \times V_1 = (20 \times 10^{-3}) \times 30 = 600 \, \text{mC} \]

For the second capacitor:
- Capacitance \( C_2 = 30 \, \text{mF} = 30 \times 10^{-3} \, \text{F} \)
- Voltage \( V_2 = 20 \, \text{V} \)

Calculating the charge \( Q_2 \):
\[ Q_2 = C_2 \times V_2 = (30 \times 10^{-3}) \times 20 = 600 \, \text{mC} \]

### Step 2: Determine the total charge when connected
When the like plates of the capacitors are connected together, the total charge \( Q_{\text{total}} \) is the sum of the charges on both capacitors:
\[ Q_{\text{total}} = Q_1 + Q_2 = 600 \, \text{mC} + 600 \, \text{mC} = 1200 \, \text{mC} \]

### Step 3: Calculate the equivalent capacitance
Since the capacitors are connected in parallel, the equivalent capacitance \( C_{\text{eq}} \) is given by:
\[ C_{\text{eq}} = C_1 + C_2 = 20 \, \text{mF} + 30 \, \text{mF} = 50 \, \text{mF} \]

### Step 4: Calculate the common potential difference
The common potential difference \( V \) across the equivalent capacitance can be calculated using the formula:
\[ V = \frac{Q_{\text{total}}}{C_{\text{eq}}} \]
Substituting the values:
\[ V = \frac{1200 \, \text{mC}}{50 \, \text{mF}} = \frac{1200 \times 10^{-3} \, \text{C}}{50 \times 10^{-3} \, \text{F}} = \frac{1200}{50} = 24 \, \text{V} \]

### Final Answer
The common potential difference when the like plates are connected together is **24 V**.

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