The resistance of a conductor at `15^(@)C` is `16 Omega` and at `100^(@)C` is `20 Omega`. What will be the temperature coefficient of resistance of the conductor ?

Options

A. `0.010^(@)C^(-1)`
B. `0.033^(@)C^(-1)`
C. `0.003^(@)C^(-1)`
D. `0.042^(@)C^(-1)`

Answer

To find the temperature coefficient of resistance (α) of the conductor, we can use the formula for resistance at two different temperatures:

1. **Given Information:**
- Resistance at \( t_1 = 15^\circ C \) is \( R_1 = 16 \, \Omega \)
- Resistance at \( t_2 = 100^\circ C \) is \( R_2 = 20 \, \Omega \)

2. **Formula for Resistance:**
The resistance of a conductor at a temperature \( t \) can be expressed as:
\[ R_t = R_0 (1 + \alpha t) \]
where \( R_0 \) is the resistance at \( 0^\circ C \), and \( \alpha \) is the temperature coefficient of resistance.

3. **Setting Up the Equations:**
For the two temperatures, we can write:
\[ R_1 = R_0 (1 + \alpha t_1) \quad \text{(1)} \]
\[ R_2 = R_0 (1 + \alpha t_2) \quad \text{(2)} \]

4. **Dividing the Equations:**
Dividing equation (2) by equation (1):
\[ \frac{R_2}{R_1} = \frac{R_0 (1 + \alpha t_2)}{R_0 (1 + \alpha t_1)} \]
This simplifies to:
\[ \frac{R_2}{R_1} = \frac{1 + \alpha t_2}{1 + \alpha t_1} \]

5. **Substituting Known Values:**
Substitute \( R_1 = 16 \, \Omega \), \( R_2 = 20 \, \Omega \), \( t_1 = 15 \), and \( t_2 = 100 \):
\[ \frac{20}{16} = \frac{1 + \alpha \cdot 100}{1 + \alpha \cdot 15} \]
Simplifying the left side:
\[ 1.25 = \frac{1 + 100\alpha}{1 + 15\alpha} \]

6. **Cross Multiplying:**
Cross multiplying gives:
\[ 1.25(1 + 15\alpha) = 1 + 100\alpha \]
Expanding the left side:
\[ 1.25 + 18.75\alpha = 1 + 100\alpha \]

7. **Rearranging the Equation:**
Rearranging gives:
\[ 1.25 - 1 = 100\alpha - 18.75\alpha \]
\[ 0.25 = 81.25\alpha \]

8. **Solving for α:**
Dividing both sides by \( 81.25 \):
\[ \alpha = \frac{0.25}{81.25} \approx 0.00307 \, \text{per} \, ^\circ C \]

9. **Final Answer:**
The temperature coefficient of resistance \( \alpha \) is approximately:
\[ \alpha \approx 0.003 \, \text{per} \, ^\circ C \]