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In this problem, we have two cells, each with electromotive forces \( E \) but with different internal resistances \( r_1 \) and \( r_2 \). These cells are connected in series with a resistor \( R \), and the voltmeter reads **0V**. We are asked to find the value of the resistance \( R \).

### Step-by-Step Solution:

#### 1. **Understanding the Zero Voltmeter Reading:**
- A voltmeter reading of 0V means that there is **no potential difference** across the voltmeter, which implies that the voltage drops across the components on either side of the voltmeter must cancel each other out.

#### 2. **Circuit Setup and Applying Kirchhoff's Voltage Law (KVL):**
- The two EMFs \( E \) are connected in opposition, which means that one is working against the other.
- The internal resistances of the two cells are \( r_1 \) and \( r_2 \).
  
For the total EMF in the circuit, we have:
\[
\text{Total EMF} = E - E = 0
\]
Thus, the current in the circuit is determined by the internal resistances and the resistor \( R \).

#### 3. **Current in the Circuit:**
Let’s apply Kirchhoff's Voltage Law (KVL). According to KVL:
- The sum of the voltage drops around any closed loop must equal zero.
- The voltage drops across the resistors are caused by the current flowing through them.

The current in the circuit \( I \) is given by:
\[
I = \frac{E}{r_1 + R + r_2}
\]

#### 4. **Voltage Across the Resistor \( R \):**
Since the voltmeter reads 0V, this implies that the potential difference across \( R \) is exactly balanced by the difference in potential drops across \( r_1 \) and \( r_2 \).

Thus, the voltage drop across \( R \) must equal the difference between the voltage drops across \( r_1 \) and \( r_2 \). Using Ohm’s law:
\[
I \cdot R = I \cdot (r_1 - r_2)
\]

Canceling out the current \( I \) from both sides (since \( I \neq 0 \)):
\[
R = r_1 - r_2
\]

### Conclusion:

The resistance \( R \) is equal to the difference between the internal resistances \( r_1 \) and \( r_2 \). Thus, the answer is:

\[
\boxed{B \: (r_1 - r_2)}
\]

Would you like further explanation on any step?

Here are 5 related questions you might consider:
1. How does Kirchhoff’s Voltage Law apply in this situation with opposing EMFs?
2. What happens if \( r_1 = r_2 \)? How does that affect the circuit and the voltmeter reading?
3. Could this problem be solved if the EMFs were not opposing each other?
4. How would the solution change if the voltmeter showed a small voltage instead of 0V?
5. What would happen to the current in the circuit if \( R = r_1 + r_2 \)?

**Tip:** In circuits with opposing EMFs, the current and potential differences are influenced by the difference in the EMFs and internal resistances. Always consider these opposing effects carefully when analyzing the circuit.

Two cells, each with electromotive force E, 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?

Learn how to calculate the resistance R in a series circuit with two cells of electromotive force E, different internal resistances r1 and r2, and a zero voltmeter reading. This problem uses Kirchhoff's Voltage Law and Ohm's Law for the solution. Suitable for Grades 11-12.

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