At what temperature will the resistannce of a copper wire become three times its value at `0^(@)C` (Temperature coefficient of resistance for copper `=4xx10^(-3)"per".^(@)C`:-

Options

A. `400^(@)C`
B. `450^(@)C`
C. `500^(@)C`
D. `500^(@)C`

Answer

To solve the problem of determining the temperature at which the resistance of a copper wire becomes three times its value at 0°C, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Resistance Change**:
The resistance \( R \) at a temperature \( T \) can be expressed in terms of its resistance at 0°C \( R_0 \) using the formula:
\[ R_T = R_0 (1 + \alpha \Delta T) \]
where:
- \( R_T \) is the resistance at temperature \( T \),
- \( R_0 \) is the resistance at 0°C,
- \( \alpha \) is the temperature coefficient of resistance,
- \( \Delta T = T - 0 = T \).

2. **Set Up the Equation**:
According to the problem, we want the resistance at temperature \( T \) to be three times the resistance at 0°C:
\[ R_T = 3R_0 \]
Substituting this into the resistance formula gives:
\[ 3R_0 = R_0 (1 + \alpha T) \]

3. **Cancel \( R_0 \)**:
Since \( R_0 \) is common on both sides and is not zero, we can divide both sides by \( R_0 \):
\[ 3 = 1 + \alpha T \]

4. **Rearrange the Equation**:
Rearranging the equation to isolate \( \alpha T \):
\[ \alpha T = 3 - 1 \]
\[ \alpha T = 2 \]

5. **Substitute the Value of \( \alpha \)**:
We know that the temperature coefficient of resistance for copper is given as \( \alpha = 4 \times 10^{-3} \, \text{per} \, °C \). Substituting this value into the equation:
\[ 4 \times 10^{-3} \, T = 2 \]

6. **Solve for \( T \)**:
Now, solve for \( T \):
\[ T = \frac{2}{4 \times 10^{-3}} = \frac{2}{0.004} = 500 \, °C \]

### Conclusion:
The temperature at which the resistance of the copper wire becomes three times its value at 0°C is \( 500 \, °C \).