`n` identical cells are joined in series with two cells A and B with reversed polarities. EMF of each cell is E and internal resistance is r. Potential difference across cell A or B is (n > 4)

Options

A. `(2E)/(n)`
B. `2E(1-(1)/(n))`
C. `(4E)/(n)`
D. `2E(1-(2)/(n))`

Answer

To solve the problem, we will follow these steps systematically:

### Step 1: Understand the configuration
We have `n` identical cells connected in series, along with two additional cells A and B that are connected in reverse polarity. Each cell has an EMF of `E` and an internal resistance of `r`.

### Step 2: Calculate the total EMF
When cells A and B are connected in reverse polarity, they effectively reduce the total EMF contributed by the other `n` cells. The total EMF from the `n` cells is `nE`. Since cells A and B are connected in reverse, they will subtract their EMF from the total:

\[ \text{Total EMF} = nE - E - E = nE - 2E = (n - 2)E \]

### Step 3: Calculate the total internal resistance
The internal resistance of each cell is `r`, and since all cells are in series, the total internal resistance is:

\[ \text{Total internal resistance} = nr + r + r = nr + 2r = (n + 2)r \]

### Step 4: Calculate the current in the circuit
Using Ohm's law, the current `I` in the circuit can be calculated as:

\[ I = \frac{\text{Total EMF}}{\text{Total internal resistance}} = \frac{(n - 2)E}{(n + 2)r} \]

### Step 5: Calculate the potential difference across cells A and B
The potential difference \( \Delta V \) across cells A and B can be calculated by considering the drop in potential across the internal resistances of cells A and B. The potential drop across the internal resistance of cells A and B is given by:

\[ \Delta V = \text{Total EMF} - \text{Potential drop across internal resistance} \]

The potential drop across the internal resistance of cells A and B is:

\[ \text{Potential drop} = I \cdot (2r) = 2Ir = 2 \cdot \frac{(n - 2)E}{(n + 2)r} \cdot r = \frac{2(n - 2)E}{(n + 2)} \]

Thus, the potential difference across cells A and B is:

\[ \Delta V = (n - 2)E - \frac{2(n - 2)E}{(n + 2)} \]

### Step 6: Simplify the expression
To simplify this expression, we can find a common denominator:

\[ \Delta V = \frac{(n - 2)E(n + 2) - 2(n - 2)E}{(n + 2)} = \frac{(n - 2)E(n + 2 - 2)}{(n + 2)} = \frac{(n - 2)E(n)}{(n + 2)} \]

### Step 7: Final expression
Thus, the potential difference across cells A and B is:

\[ \Delta V = \frac{(n - 2)E \cdot n}{(n + 2)} \]

### Conclusion
The potential difference across cells A and B is given by the expression derived above.

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