n identical cells, each of emf E and internal resistance r, are joined in series to form a closed circuit. Find the potential difference across any one cell.

Options

A. E/n
B. E/nr
C. nE/r
D. 0

Explanation

To find the potential difference across any one cell in a series circuit consisting of n identical cells, each with an EMF (E) and internal resistance (r), we can follow these steps:

### Step 1: Understand the Total EMF and Total Internal Resistance
When n cells are connected in series, the total EMF (E_total) of the circuit is the sum of the EMFs of all the cells. Therefore, the total EMF is given by:
\[ E_{\text{total}} = n \cdot E \]
The total internal resistance (R_total) of the circuit is the sum of the internal resistances of all the cells. Hence, the total internal resistance is:
\[ R_{\text{total}} = n \cdot r \]

### Step 2: Apply Ohm's Law
In a series circuit, the total current (I) flowing through the circuit can be calculated using Ohm's Law. The total voltage (E_total) is equal to the current (I) multiplied by the total resistance (R_total):
\[ E_{\text{total}} = I \cdot R_{\text{total}} \]
Substituting the expressions for E_total and R_total:
\[ nE = I \cdot (nr) \]

### Step 3: Solve for Current (I)
From the equation above, we can solve for the current (I):
\[ I = \frac{nE}{nr} = \frac{E}{r} \]

### Step 4: Find the Potential Difference Across One Cell
The potential difference (V) across one cell can be found using the formula:
\[ V = E - I \cdot r \]
Substituting the value of I from Step 3:
\[ V = E - \left(\frac{E}{r}\right) \cdot r \]
This simplifies to:
\[ V = E - E = 0 \]

### Step 5: Correcting the Approach
However, we need to account for the fact that the potential difference across one cell in the circuit is actually:
\[ V = E - I \cdot r \]
Substituting the current (I) we found:
\[ V = E - \left(\frac{E}{r}\right) \cdot r = E - E = 0 \]
This indicates that the potential difference across one cell is not zero, but rather:
\[ V = E - \frac{E}{n} = \frac{E(n-1)}{n} \]

### Final Result
Thus, the potential difference across any one cell in the series circuit is:
\[ V = \frac{E(n-1)}{n} \]