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. `J lt 12`
B. `J gt 12`
C. `J lt 10`
D. `J gt 10`

Explanation

To find the temperature at which the resistance of a copper wire becomes three times its value at 0°C, we can use the formula for resistance change with temperature. The formula is given by:

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

Where:
- \( R_T \) is the resistance at temperature \( T \).
- \( R_0 \) is the resistance at 0°C.
- \( \alpha \) is the temperature coefficient of resistance.
- \( \Delta T \) is the change in temperature, which can be expressed as \( T - 0 = T \).

Given:
- \( R_T = 3 R_0 \)
- \( \alpha = 4 \times 10^{-3} \, \text{per} \, °C \)

### Step-by-step Solution:

1. **Set up the equation using the given information:**
\[ R_T = 3 R_0 \]
Substituting this into the resistance change formula:
\[ 3 R_0 = R_0 (1 + \alpha T) \]

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

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

4. **Substitute the value of \( \alpha \):**
\[ 2 = (4 \times 10^{-3}) T \]

5. **Solve for \( T \):**
\[ T = \frac{2}{4 \times 10^{-3}} = \frac{2}{0.004} = 500 \]

6. **Final Result:**
The temperature at which the resistance of the copper wire becomes three times its value at 0°C is:
\[ T = 500 \, °C \]