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The given problem involves two cells with electromotive forces \(E_1\) and \(E_2\), connected in series with internal resistances \(r_1\) and \(r_2\), to a resistor \(R\). The voltmeter shows 0V, meaning no potential difference between the two points across the voltmeter. We need to find the value of the resistor \(R\).

### Step-by-step Solution:

1. **Understanding the zero reading in the voltmeter:**
   - A reading of 0V across the voltmeter means that the potential drop across the external resistor \(R\) balances the potential difference created by the two cells and their internal resistances.

2. **Circuit setup and Kirchhoff's Voltage Law (KVL):**
   - The total EMF of the cells is \(E_1 - E_2\) since they are opposing.
   - The internal resistances are \(r_1\) and \(r_2\), which will also drop some voltage when current \(I\) flows through the circuit.

3. **Apply KVL in the loop:**
   - The total voltage in the circuit is \(E_1 - E_2\).
   - The total resistance in the circuit is the sum of the internal resistances plus \(R\), i.e., \(r_1 + R + r_2\).

   Using Ohm's law for the total circuit:
   \[
   I = \frac{E_1 - E_2}{r_1 + R + r_2}
   \]
   
4. **Condition for zero potential across the voltmeter:**
   - For the voltmeter to read 0V, the current must flow such that the voltage drops across each element balance each other.
   - The voltage drop across the internal resistances must cancel out the EMF difference between the two cells.

5. **Determine \(R\):**
   - To satisfy the zero potential condition, the external resistor \(R\) must equal the difference between the internal resistances:
   \[
   R = r_1 - r_2
   \]

Thus, the resistance \(R\) is:

\[
R = r_1 - r_2
\]

### Answer:
The correct answer is \( \boxed{B} \).

Would you like further clarification on any steps?

Here are 5 related questions you might consider:
1. How does Kirchhoff's Voltage Law apply to this circuit?
2. What is the significance of the voltmeter reading 0V in a circuit?
3. How do internal resistances affect the total resistance in a circuit?
4. Can we apply this method to circuits with more than two cells?
5. How does reversing the orientation of one of the cells affect the circuit?

**Tip:** In circuits involving multiple EMFs, always analyze how the internal resistances and external loads contribute to the potential differences across components.

Two cells, each with electromotive force E1 and E2 but different internal resistances r1 and r2, are connected in series to a resistor R. The reading on the voltmeter is 0V. What is the resistance of R?

In this problem, two cells with electromotive forces and internal resistances are connected in series to a resistor. The voltmeter reads 0V, and we determine the resistance R using Kirchhoff's Voltage Law and Ohm's Law.

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