Resistance of the wires is measured as `2Omega and 3Omega` at `10^(@)C` and `30^(@)C` respectively. Temperature co-efficient of resistance of the material of the wire is :

Options

A. `0.033"^(@)C^(-1)`
B. `-0.033"^(@)C^(-1)`
C. `0.011"^(@)C^(-1)`
D. `0.055"^(@)C^(-1)`

Explanation

To find the temperature coefficient of resistance (α) of the material of the wire, we can use the formula for resistance change with temperature:

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

Where:
- \( R \) is the resistance at temperature \( T \),
- \( R_0 \) is the resistance at the reference temperature (usually taken as 0°C),
- \( \alpha \) is the temperature coefficient of resistance,
- \( \Delta T \) is the change in temperature.

### Step-by-step Solution:

1. **Identify Given Values:**
- Resistance at \( 10°C \) (let's call it \( R_1 \)): \( 2 \, \Omega \)
- Resistance at \( 30°C \) (let's call it \( R_2 \)): \( 3 \, \Omega \)
- Initial temperature for \( R_1 \): \( T_1 = 10°C \)
- Initial temperature for \( R_2 \): \( T_2 = 30°C \)

2. **Calculate Change in Temperature:**
- For \( R_1 \): \( \Delta T_1 = T_1 - 0 = 10°C \)
- For \( R_2 \): \( \Delta T_2 = T_2 - 0 = 30°C \)

3. **Set Up Equations Using the Resistance Formula:**
- For \( R_1 \):
\[ R_1 = R_0 (1 + \alpha \Delta T_1) \implies 2 = R_0 (1 + 10\alpha) \quad \text{(Equation 1)} \]
- For \( R_2 \):
\[ R_2 = R_0 (1 + \alpha \Delta T_2) \implies 3 = R_0 (1 + 30\alpha) \quad \text{(Equation 2)} \]

4. **Divide Equation 2 by Equation 1:**
\[ \frac{3}{2} = \frac{R_0 (1 + 30\alpha)}{R_0 (1 + 10\alpha)} \]
Simplifying gives:
\[ \frac{3}{2} = \frac{1 + 30\alpha}{1 + 10\alpha} \]

5. **Cross Multiply to Eliminate the Fraction:**
\[ 3(1 + 10\alpha) = 2(1 + 30\alpha) \]
Expanding both sides:
\[ 3 + 30\alpha = 2 + 60\alpha \]

6. **Rearranging the Equation:**
\[ 3 - 2 = 60\alpha - 30\alpha \]
\[ 1 = 30\alpha \]

7. **Solve for α:**
\[ \alpha = \frac{1}{30} \]

8. **Convert to Decimal:**
\[ \alpha \approx 0.0333 \, \text{°C}^{-1} \]

### Final Answer:
The temperature coefficient of resistance \( \alpha \) is approximately \( 0.0333 \, \text{°C}^{-1} \).