A cell of e.m. E and internal resistance r is connected across a resistancer. The potential difference between the terminals of the cell must be

Options

A. E
B. `E//2`
C. `E//4`
D. `3//2`

Answer

To solve the problem of finding the potential difference between the terminals of a cell with an electromotive force (e.m.f.) \( E \) and internal resistance \( r \) connected across an external resistance \( R \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Circuit Components**:
- We have a cell with e.m.f. \( E \) and internal resistance \( r \).
- The cell is connected to an external resistance \( R \).

2. **Determine the Total Resistance**:
- The total resistance in the circuit is the sum of the internal resistance and the external resistance:
\[ R_{\text{total}} = R + r \]

3. **Apply Ohm's Law to Find Current**:
- According to Ohm's Law, the current \( I \) flowing through the circuit can be calculated as:
\[ I = \frac{E}{R_{\text{total}}} = \frac{E}{R + r} \]

4. **Calculate the Potential Difference Across the Cell**:
- The potential difference \( V_{AB} \) between the terminals of the cell can be expressed as:
\[ V_{AB} = E - I \cdot r \]
- Substituting the expression for \( I \):
\[ V_{AB} = E - \left(\frac{E}{R + r}\right) \cdot r \]

5. **Simplify the Expression**:
- Now, simplifying the equation:
\[ V_{AB} = E - \frac{E \cdot r}{R + r} \]
- This can be rewritten as:
\[ V_{AB} = E \left(1 - \frac{r}{R + r}\right) \]
- Further simplifying gives:
\[ V_{AB} = E \left(\frac{R}{R + r}\right) \]

6. **Final Result**:
- The potential difference between the terminals of the cell is:
\[ V_{AB} = \frac{E \cdot R}{R + r} \]