Two sources of equal emf are connected to an external resistance R. The internal resistance of the two sources are `R_1 and R_2(R_1gtR_1).` if the potential difference across the source having internal resistance `R_2` is zero, then

Options

A. `R = R_2-R_1`
B. `R = R_2 xx (R_1+R_2)//(R_2-R_1)`
C. `R= R_1R_2//(R_2-R_1)`
D. `R = R_1R_2//(R_1-R_2)`

Explanation

To solve the problem step by step, we need to analyze the circuit with two sources of equal EMF connected to an external resistance \( R \) and internal resistances \( R_1 \) and \( R_2 \). Given that the potential difference across the source with internal resistance \( R_2 \) is zero, we can derive the required relationship.

### Step-by-Step Solution:

1. **Understanding the Circuit Configuration**:
- We have two sources of equal EMF \( V \).
- The internal resistances are \( R_1 \) and \( R_2 \) respectively, with the condition \( R_1 > R_2 \).
- The external resistance is \( R \).

2. **Setting Up the Circuit**:
- Let’s denote the current flowing through the circuit as \( I \).
- The total EMF in the circuit is \( 2V \) (since both sources contribute equally).
- The total internal resistance in the circuit is \( R_1 + R_2 \).

3. **Calculating the Current**:
- The equivalent resistance \( R_{eq} \) of the circuit is given by:
\[ R_{eq} = R + R_1 + R_2 \]
- The current \( I \) flowing through the circuit can be expressed as:
\[ I = \frac{2V}{R + R_1 + R_2} \]

4. **Analyzing the Potential Difference Across \( R_2 \)**:
- The potential difference across the source with internal resistance \( R_2 \) is given as zero. This means:
\[ V_A - V_B = 0 \]
- Where \( V_A \) is the potential at the positive terminal of the source with \( R_2 \) and \( V_B \) is the potential at the negative terminal.
- Using Ohm's law, we can express this as:
\[ V_A - V + I \cdot R_2 = V_B \]
- Since \( V_A - V_B = 0 \), we have:
\[ V - I \cdot R_2 = 0 \quad \Rightarrow \quad V = I \cdot R_2 \]

5. **Substituting for Current \( I \)**:
- Substitute the expression for \( I \):
\[ V = \left(\frac{2V}{R + R_1 + R_2}\right) R_2 \]

6. **Simplifying the Equation**:
- Cancel \( V \) from both sides (assuming \( V \neq 0 \)):
\[ 1 = \frac{2R_2}{R + R_1 + R_2} \]
- Cross-multiplying gives:
\[ R + R_1 + R_2 = 2R_2 \]
- Rearranging this equation leads to:
\[ R + R_1 = R_2 \]

7. **Final Relationship**:
- Thus, we find:
\[ R = R_2 - R_1 \]

### Conclusion:
The relationship derived from the conditions given in the problem is:
\[ R = R_2 - R_1 \]