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. 4
B. 8
C. 6
D. 2

Answer

To find the ratio of the resistances of carbon and copper wires at \(0^\circ C\), we will use the temperature dependence of resistance. The formula for resistance at a temperature \(T\) is given by:

\[ R = R_0 (1 + \alpha \Delta T) \]

where:
- \(R_0\) is the resistance at \(0^\circ C\),
- \(\alpha\) is the temperature coefficient of resistivity,
- \(\Delta T\) is the change in temperature.

### Step 1: Define the resistances
Let:
- \(R_1\) be the resistance of the copper wire at \(0^\circ C\),
- \(R_2\) be the resistance of the carbon wire at \(0^\circ C\).

### Step 2: Write the equations for the resistances
The resistance of copper at a temperature \(T\) is:
\[ R_1(T) = R_1(0) \left(1 + \alpha_1 T\right) \]
The resistance of carbon at a temperature \(T\) is:
\[ R_2(T) = R_2(0) \left(1 + \alpha_2 T\right) \]

### Step 3: Set up the equation for series resistances
Since the combined resistance does not vary with temperature, we can write:
\[ R_1 + R_2 = R_1(1 + \alpha_1 T) + R_2(1 + \alpha_2 T) \]

### Step 4: Simplify the equation
Expanding the right-hand side:
\[ R_1 + R_2 = R_1 + \alpha_1 R_1 T + R_2 + \alpha_2 R_2 T \]
Cancelling \(R_1\) and \(R_2\) from both sides, we get:
\[ 0 = \alpha_1 R_1 T + \alpha_2 R_2 T \]

### Step 5: Factor out \(T\)
Since \(T\) is not zero (we assume a small change in temperature), we can divide by \(T\):
\[ 0 = \alpha_1 R_1 + \alpha_2 R_2 \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ \alpha_1 R_1 = -\alpha_2 R_2 \]
Thus, we can express the ratio of the resistances:
\[ \frac{R_2}{R_1} = -\frac{\alpha_1}{\alpha_2} \]

### Step 7: Substitute the values of \(\alpha_1\) and \(\alpha_2\)
Given:
- \(\alpha_1 = 4 \times 10^{-3} \, ^\circ C^{-1}\)
- \(\alpha_2 = -0.5 \times 10^{-3} \, ^\circ C^{-1}\)

Substituting these values:
\[ \frac{R_2}{R_1} = -\frac{4 \times 10^{-3}}{-0.5 \times 10^{-3}} = \frac{4}{0.5} = 8 \]

### Step 8: Conclusion
Thus, the ratio of the resistances of carbon to copper at \(0^\circ C\) is:
\[ \frac{R_{\text{carbon}}}{R_{\text{copper}}} = 8 \]