Copper and carbon wires are connected in series and the combined resistor is kept at `0^(@)C`. Assuming the combined resistance does not vary with temperature the ratio of the resistances of carbon and copper wires at `0^(@)C` is (Temperature coefficient of resistivity of copper and carbon respectively are `4xx(10^(-3))/(``^(@)C)` and `-0.5xx(10^(-3))/(``^(@)C)`

Options

A. 2
B. 4
C. 8
D. 6

Explanation

To solve the problem, we need to determine the ratio of the resistances of carbon (R_C) and copper (R_u) wires when they are connected in series at a temperature of 0°C.

### Step-by-Step Solution:

1. **Understanding the Temperature Coefficient of Resistivity**:
- The temperature coefficient of resistivity (α) indicates how much the resistance of a material changes with temperature.
- For copper, α_Cu = 4 × 10^(-3) /°C.
- For carbon, α_C = -0.5 × 10^(-3) /°C.

2. **Resistance Change with Temperature**:
- The resistance of a material at a temperature T can be expressed as:
\[ R(T) = R_0(1 + α(T - T_0)) \]
- Here, R_0 is the resistance at the reference temperature T_0 (which is 0°C in this case).

3. **Setting Up the Equation**:
- Since the combined resistance does not vary with temperature, we can set up the equation based on the temperature coefficients:
\[ α_Cu \cdot R_C + α_C \cdot R_u = 0 \]
- This implies that the total change in resistance due to temperature for both materials must equal zero.

4. **Substituting the Values**:
- Substitute the values of α for copper and carbon:
\[ (4 × 10^{-3})R_C + (-0.5 × 10^{-3})R_u = 0 \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ 4R_C = 0.5R_u \]
- This can be rewritten as:
\[ \frac{R_C}{R_u} = \frac{0.5}{4} = \frac{1}{8} \]

6. **Final Ratio**:
- Therefore, the ratio of the resistances of carbon to copper is:
\[ \frac{R_C}{R_u} = \frac{1}{8} \]

### Conclusion:
The ratio of the resistances of carbon and copper wires at 0°C is \( \frac{1}{8} \).