A battery of 10 cells each of e.m.f. E=1.5V and internal resistance `0.5Omega` has 1 cell wrongly connected. It is being charged by 220V power supply with an external resistance of `47Omega` in series. The potential difference across the battery.

Answer

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

### Step 1: Calculate the total EMF of the battery
Given that there are 10 cells, each with an EMF of \( E = 1.5 \, \text{V} \), and one cell is wrongly connected, the effective number of cells contributing to the EMF is \( 10 - 1 = 9 \).

\[ \text{Total EMF} = 9 \times 1.5 \, \text{V} = 13.5 \, \text{V} \]

### Step 2: Calculate the internal resistance of the battery
Each cell has an internal resistance of \( r = 0.5 \, \Omega \). Since one cell is wrongly connected, the total internal resistance contributed by the remaining 9 cells is:

\[ \text{Total Internal Resistance} = 9 \times 0.5 \, \Omega = 4.5 \, \Omega \]

### Step 3: Calculate the total resistance in the circuit
The total resistance in the circuit includes the external resistance and the internal resistance of the battery:

\[ \text{Total Resistance} = R_{\text{external}} + R_{\text{internal}} = 47 \, \Omega + 4.5 \, \Omega = 51.5 \, \Omega \]

### Step 4: Calculate the total voltage supplied by the power supply
The power supply voltage is given as \( V_{\text{power supply}} = 220 \, \text{V} \).

### Step 5: Calculate the current flowing through the circuit
Using Ohm's law, the current \( I \) flowing through the circuit can be calculated as:

\[ I = \frac{V_{\text{power supply}} - \text{Total EMF}}{\text{Total Resistance}} = \frac{220 \, \text{V} - 13.5 \, \text{V}}{51.5 \, \Omega} \]

Calculating the numerator:

\[ 220 - 13.5 = 206.5 \, \text{V} \]

Now substituting this value into the current formula:

\[ I = \frac{206.5 \, \text{V}}{51.5 \, \Omega} \approx 4.01 \, \text{A} \]

### Step 6: Calculate the potential difference across the battery
The potential difference \( V \) across the battery can be calculated using the formula:

\[ V = \text{Total EMF} + I \times R_{\text{internal}} \]

Substituting the values we have:

\[ V = 13.5 \, \text{V} + (4.01 \, \text{A} \times 4.5 \, \Omega) \]

Calculating the second term:

\[ 4.01 \times 4.5 \approx 18.045 \, \text{V} \]

Now substituting this back into the equation for \( V \):

\[ V \approx 13.5 \, \text{V} + 18.045 \, \text{V} \approx 31.545 \, \text{V} \]

### Final Answer
The potential difference across the battery is approximately \( 31.55 \, \text{V} \).

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