To find the rise in temperature of the wire when a condenser is discharged through it, we can follow these steps:
### Step 1: Calculate the energy stored in the capacitor
The energy (E) stored in a capacitor is given by the formula:
\[
E = \frac{1}{2} C V^2
\]
Where:
- \( C = 21 \, \mu F = 21 \times 10^{-6} \, F \)
- \( V = 100 \, V \)
Substituting the values:
\[
E = \frac{1}{2} \times 21 \times 10^{-6} \times (100)^2
\]
\[
E = \frac{1}{2} \times 21 \times 10^{-6} \times 10000
\]
\[
E = \frac{21 \times 10^{-2}}{2} \, J
\]
\[
E = 10.5 \times 10^{-2} \, J
\]
### Step 2: Convert the energy to calories
Since the specific heat is given in calories, we need to convert joules to calories. The conversion factor is:
\[
1 \, \text{cal} = 4.2 \, \text{J}
\]
Thus, the energy in calories is:
\[
E = \frac{10.5 \times 10^{-2} \, J}{4.2 \, J/cal} = \frac{10.5 \times 10^{-2}}{4.2} \, cal
\]
Calculating this gives:
\[
E \approx 0.025 \, cal
\]
### Step 3: Use the specific heat formula to find the rise in temperature
The formula relating heat energy, mass, specific heat, and temperature change is:
\[
Q = m \cdot c \cdot \Delta T
\]
Where:
- \( Q \) = energy in calories
- \( m = 0.25 \, g \)
- \( c = 0.1 \, \frac{cal}{g \cdot °C} \)
- \( \Delta T \) = rise in temperature in °C
Rearranging the formula to find \( \Delta T \):
\[
\Delta T = \frac{Q}{m \cdot c}
\]
Substituting the known values:
\[
\Delta T = \frac{0.025 \, cal}{0.25 \, g \cdot 0.1 \, \frac{cal}{g \cdot °C}}
\]
Calculating this gives:
\[
\Delta T = \frac{0.025}{0.025} = 1 \, °C
\]
### Final Answer
The rise in temperature of the wire is \( 1 \, °C \).
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