A potentiometer wire has length `4 m` and resistance `8 Omega`. The resistance that must be connected in series with the wire and an accumulator of e.m.f. `2 V`, so as the get a potential gradient `1 mV` per cm` on the wire is

Options

A. `32Omega`
B. `40Omega`
C. `44Omega`
D. `48Omega`

Answer

To solve the problem, we need to determine the resistance that must be connected in series with the potentiometer wire and the accumulator to achieve a potential gradient of 1 mV/cm.

### Step-by-step Solution:

1. **Identify Given Values:**
- Length of potentiometer wire, \( L = 4 \, \text{m} \)
- Resistance of potentiometer wire, \( R_w = 8 \, \Omega \)
- EMF of the accumulator, \( E = 2 \, \text{V} \)
- Desired potential gradient, \( V_g = 1 \, \text{mV/cm} = 1 \times 10^{-3} \, \text{V/cm} = 0.1 \, \text{V/m} \)

2. **Calculate Total Resistance:**
- Let \( R \) be the resistance that needs to be connected in series.
- The total resistance in the circuit will be:
\[ R_{\text{total}} = R_w + R = 8 \, \Omega + R \]

3. **Calculate Current in the Circuit:**
- Using Ohm's law, the current \( I \) in the circuit can be expressed as:
\[ I = \frac{E}{R_{\text{total}}} = \frac{2 \, \text{V}}{8 \, \Omega + R} \]

4. **Calculate Potential Drop Across the Potentiometer Wire:**
- The potential drop \( V_p \) across the potentiometer wire is given by:
\[ V_p = I \times R_w = \left(\frac{2}{8 + R}\right) \times 8 \]

5. **Calculate Potential Gradient:**
- The potential gradient \( V_g \) along the length of the wire is given by:
\[ V_g = \frac{V_p}{L} = \frac{I \times R_w}{L} \]
- Substituting for \( V_p \):
\[ V_g = \frac{8 \times \frac{2}{8 + R}}{4} = \frac{16}{4(8 + R)} = \frac{4}{8 + R} \]

6. **Set the Potential Gradient Equal to Desired Value:**
- We want \( V_g = 0.1 \, \text{V/m} \):
\[ \frac{4}{8 + R} = 0.1 \]

7. **Solve for R:**
- Cross-multiplying gives:
\[ 4 = 0.1(8 + R) \]
- Simplifying:
\[ 4 = 0.8 + 0.1R \]
\[ 4 - 0.8 = 0.1R \]
\[ 3.2 = 0.1R \]
\[ R = \frac{3.2}{0.1} = 32 \, \Omega \]

### Final Answer:
The resistance that must be connected in series with the wire is \( R = 32 \, \Omega \).