The resistance of a silver wire at `0^(@)C " is " 1.25 Omega`. Upto what temperature it must be heated so that its resistance is doubled ? The temperature coefficient of resistance of silver is `0.00375^(@)C^(-1)`. Will the temperature be same for all silver conductors of all shapes ?

Answer

To solve the problem, we need to find the temperature at which the resistance of a silver wire doubles from its initial value. The initial resistance \( R_0 \) is given as \( 1.25 \, \Omega \), and we want to find the temperature \( T \) at which the resistance becomes \( R = 2.5 \, \Omega \).

### Step-by-Step Solution:

1. **Identify the initial resistance and the final resistance:**
\[ R_0 = 1.25 \, \Omega \quad \text{(initial resistance)} \]
\[ R = 2 \times R_0 = 2 \times 1.25 = 2.5 \, \Omega \quad \text{(final resistance)} \]

2. **Use the formula for resistance change with temperature:**
The resistance of a conductor changes with temperature according to the formula:
\[ R = R_0 (1 + \alpha \Delta T) \]
where:
- \( R \) is the final resistance,
- \( R_0 \) is the initial resistance,
- \( \alpha \) is the temperature coefficient of resistance,
- \( \Delta T \) is the change in temperature.

3. **Substitute the known values into the formula:**
We know:
- \( R = 2.5 \, \Omega \)
- \( R_0 = 1.25 \, \Omega \)
- \( \alpha = 0.00375 \, ^\circ C^{-1} \)

Substituting these values into the formula gives:
\[ 2.5 = 1.25 (1 + 0.00375 \Delta T) \]

4. **Solve for \( \Delta T \):**
First, divide both sides by \( 1.25 \):
\[ 2 = 1 + 0.00375 \Delta T \]
Now, subtract 1 from both sides:
\[ 1 = 0.00375 \Delta T \]
Finally, solve for \( \Delta T \):
\[ \Delta T = \frac{1}{0.00375} = 266.67 \, ^\circ C \]

5. **Calculate the final temperature:**
Since the initial temperature \( T_0 \) is \( 0 \, ^\circ C \), the final temperature \( T \) is:
\[ T = T_0 + \Delta T = 0 + 266.67 = 266.67 \, ^\circ C \]

### Conclusion:
The silver wire must be heated to approximately \( 266.67 \, ^\circ C \) for its resistance to double.

### Additional Note:
The temperature will not be the same for all silver conductors of all shapes because the resistance also depends on the physical dimensions (length and cross-sectional area) of the conductor, in addition to the material properties.