An aluminium` ( alpha_(Al) = 4 xx 10^(-3) //""^(0) C)` wire resistance `R_1`' and carbon wire `( alpha_c = -0.5 xx 10^(-3) //""^(@) C)` resistance `R_2`' are connected in series to have a resultant resistance of 18 ohm at all temperatures. The values of `R_1` and `R_2` in ohms

Options

A. `2,16`
B. `12,6`
C. `13,5`
D. `14,4`

Answer

To solve the problem, we need to find the individual resistances \( R_1 \) and \( R_2 \) of an aluminum wire and a carbon wire connected in series, given their temperature coefficients of resistance and the condition that their total resistance remains constant at 18 ohms at all temperatures.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two resistances \( R_1 \) (aluminum) and \( R_2 \) (carbon) connected in series.
- The total resistance \( R_{eq} = R_1 + R_2 = 18 \, \Omega \).
- The temperature coefficient of resistance for aluminum \( \alpha_{Al} = 4 \times 10^{-3} \, \text{°C}^{-1} \).
- The temperature coefficient of resistance for carbon \( \alpha_{C} = -0.5 \times 10^{-3} \, \text{°C}^{-1} \).

2. **Setting Up the Equation for Constant Resistance**:
- For the total resistance to remain constant with temperature changes, the overall change in resistance must be zero:
\[ \alpha_{Al} R_1 + \alpha_C R_2 = 0 \]
- Substituting the values of \( \alpha_{Al} \) and \( \alpha_C \):
\[ 4 \times 10^{-3} R_1 - 0.5 \times 10^{-3} R_2 = 0 \]
- Rearranging gives:
\[ 4 R_1 = 0.5 R_2 \quad \Rightarrow \quad R_2 = \frac{4}{0.5} R_1 = 8 R_1 \]

3. **Substituting into the Total Resistance Equation**:
- Now substitute \( R_2 = 8 R_1 \) into the total resistance equation:
\[ R_1 + R_2 = 18 \quad \Rightarrow \quad R_1 + 8 R_1 = 18 \]
- This simplifies to:
\[ 9 R_1 = 18 \]

4. **Solving for \( R_1 \)**:
- Dividing both sides by 9:
\[ R_1 = \frac{18}{9} = 2 \, \Omega \]

5. **Finding \( R_2 \)**:
- Now use \( R_1 \) to find \( R_2 \):
\[ R_2 = 8 R_1 = 8 \times 2 = 16 \, \Omega \]

### Final Answer:
- The values of \( R_1 \) and \( R_2 \) are:
- \( R_1 = 2 \, \Omega \)
- \( R_2 = 16 \, \Omega \)