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. A. `9 Omega`
B. B. `11 .11 Omega`
C. C. `0.9 Omega`
D. 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 combination of tungsten and germanium does not change with temperature.

### Step-by-Step Solution:

1. **Identify Given Data**:
- Temperature coefficient of resistance of tungsten, \( \alpha_T = 4.5 \times 10^{-3} \, ^\circ C^{-1} \)
- Temperature coefficient of resistance of germanium, \( \alpha_G = -5 \times 10^{-2} \, ^\circ C^{-1} \)
- Resistance of tungsten wire, \( R_T = 100 \, \Omega \)
- Resistance of germanium wire, \( R_G = R \, \Omega \) (unknown)

2. **Write the Formula for Resistance Change**:
The resistance of a conductor at a temperature change \( \Delta T \) can be expressed as:
\[ R = R_0 (1 + \alpha \Delta T) \]
where \( R_0 \) is the initial resistance and \( \alpha \) is the temperature coefficient.

3. **Calculate the Resistance of Tungsten Wire**:
The resistance of the tungsten wire at a temperature change \( \Delta T \) is:
\[ R_T' = R_T (1 + \alpha_T \Delta T) = 100 (1 + 4.5 \times 10^{-3} \Delta T) \]

4. **Calculate the Resistance of Germanium Wire**:
The resistance of the germanium wire at the same temperature change is:
\[ R_G' = R (1 + \alpha_G \Delta T) = R (1 - 5 \times 10^{-2} \Delta T) \]

5. **Total Resistance of the Combination**:
The total resistance \( R_{total} \) of the series combination is:
\[ R_{total} = R_T' + R_G' = 100 (1 + 4.5 \times 10^{-3} \Delta T) + R (1 - 5 \times 10^{-2} \Delta T) \]

6. **Set Condition for No Change in Resistance**:
For the total resistance to remain constant with temperature, the coefficient of \( \Delta T \) in the total resistance equation must equal zero:
\[ 100 \cdot 4.5 \times 10^{-3} + R \cdot (-5 \times 10^{-2}) = 0 \]

7. **Solve for \( R \)**:
Rearranging the equation gives:
\[ 100 \cdot 4.5 \times 10^{-3} = R \cdot 5 \times 10^{-2} \]
\[ R = \frac{100 \cdot 4.5 \times 10^{-3}}{5 \times 10^{-2}} \]
\[ R = \frac{0.45}{0.05} = 9 \, \Omega \]

### Conclusion:
The value of \( R \) for which the resistance of the combination does not change with temperature is \( 9 \, \Omega \).