The temperature coefficient of resistance of tungsten is `4.5 xx 10^(-3) ""^(@)C^(-1)` and that of germanium is ` - 5 xx 10^(-2)""^(@)C^(-1)` . A tungsten wire of resistance ` 100 Omega` is connected in series with a germanium wire of resistance R. The value of R for which the resistance of combination does not change with temperature is .

Options

A. `9 Omega`
B. `11 .11 Omega`
C. `0.9 Omega`
D. `111.1 Omega`

Answer

To solve the problem, we need to find the value of resistance \( R \) for the germanium wire such that the total resistance of the series combination of tungsten and germanium does not change with temperature.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Temperature coefficient of resistance for tungsten (\( \alpha_1 \)): \( 4.5 \times 10^{-3} \, ^\circ C^{-1} \)
- Temperature coefficient of resistance for germanium (\( \alpha_2 \)): \( -5 \times 10^{-2} \, ^\circ C^{-1} \)
- Resistance of tungsten wire (\( R_1 \)): \( 100 \, \Omega \)
- Resistance of germanium wire (\( R_2 \)): \( R \)

2. **Understand the Condition for No Change in Resistance:**
- The effective temperature coefficient of resistance for the series combination should be zero for the total resistance to remain unchanged with temperature. The formula for the effective temperature coefficient (\( \alpha \)) in series is:
\[ \alpha = \frac{\alpha_1 R_1 + \alpha_2 R_2}{R_1 + R_2} \]
- Setting \( \alpha = 0 \) gives:
\[ \alpha_1 R_1 + \alpha_2 R_2 = 0 \]

3. **Substitute the Known Values:**
- Substitute \( \alpha_1 \), \( R_1 \), \( \alpha_2 \), and \( R_2 \) into the equation:
\[ 4.5 \times 10^{-3} \times 100 + (-5 \times 10^{-2}) \times R = 0 \]

4. **Simplify the Equation:**
- Calculate \( 4.5 \times 10^{-3} \times 100 \):
\[ 4.5 \times 10^{-1} = 0.45 \]
- Now the equation becomes:
\[ 0.45 - 5 \times 10^{-2} R = 0 \]

5. **Rearranging the Equation:**
- Rearranging gives:
\[ 5 \times 10^{-2} R = 0.45 \]

6. **Solve for \( R \):**
- Divide both sides by \( 5 \times 10^{-2} \):
\[ R = \frac{0.45}{5 \times 10^{-2}} = \frac{0.45}{0.05} = 9 \]

7. **Final Answer:**
- The value of \( R \) for which the resistance of the combination does not change with temperature is:
\[ R = 9 \, \Omega \]