The resistance of wire in a heater at room temperature is `65Oemga`. When the heater is connected to a `220V` supply the current settles after a few seconds to 2.8A. What is the steady temperature of the wire. (Temperature coefficient of resistance `alpha=1.70xx10^(-4) ^(@).C^(-)`)

Options

A. `955^(@)C`
B. `1055^(@)C`
C. `1155^(@)C`
D. `1258^(@)C`

Explanation

To find the steady temperature of the wire in the heater, we can follow these steps:

### Step 1: Identify Given Values
- Resistance at room temperature (R1) = 65 Ω
- Room temperature (T1) = 27 °C
- Voltage (V) = 220 V
- Current (I) = 2.8 A
- Temperature coefficient of resistance (α) = 1.7 × 10^(-4) °C^(-1)

### Step 2: Calculate the Resistance at Steady State (R2)
Using Ohm's Law, we can calculate the resistance when the heater is connected to the supply:
\[ R2 = \frac{V}{I} \]
Substituting the values:
\[ R2 = \frac{220 \, \text{V}}{2.8 \, \text{A}} \]
\[ R2 = 78.57 \, \Omega \] (approximately 78.6 Ω)

### Step 3: Use the Resistance-Temperature Relationship
The relationship between resistance and temperature is given by:
\[ R2 = R1 \left(1 + \alpha (T2 - T1)\right) \]
Substituting the known values into the equation:
\[ 78.6 = 65 \left(1 + 1.7 \times 10^{-4} (T2 - 27)\right) \]

### Step 4: Solve for T2
First, divide both sides by 65:
\[ \frac{78.6}{65} = 1 + 1.7 \times 10^{-4} (T2 - 27) \]
Calculating the left side:
\[ 1.2092 = 1 + 1.7 \times 10^{-4} (T2 - 27) \]

Now, subtract 1 from both sides:
\[ 0.2092 = 1.7 \times 10^{-4} (T2 - 27) \]

Next, divide by \(1.7 \times 10^{-4}\):
\[ T2 - 27 = \frac{0.2092}{1.7 \times 10^{-4}} \]
Calculating the right side:
\[ T2 - 27 \approx 1231.76 \]

Finally, add 27 to both sides to find T2:
\[ T2 \approx 1231.76 + 27 \]
\[ T2 \approx 1258.76 \, °C \]

### Step 5: Round the Result
Thus, the steady temperature of the wire is approximately:
\[ T2 \approx 1259 \, °C \]

### Summary of Steps:
1. Identify given values.
2. Calculate resistance at steady state using Ohm's Law.
3. Use the resistance-temperature relationship to set up the equation.
4. Solve for the steady temperature T2.
5. Round the final result.