A wire of length 1 m and radius `10^-3`m is carrying a heavy current and is assumed to radiate as a black body. At equilibrium, its temperature is 900 K while that of surrounding is 300 K. The resistivity of the material of the wire at 300 K is `pi^@xx10^-8` ohm m and its temperature coefficient of resistance is `7.8xx10^(-3)//C` (stefan's constant`sigma=5.68xx10^-8W//m^(2) K^(4)`).
Q. The resistivity of wire at 900 K is nearly

Options

A. `2.4xx10^7ohmm`
B. `2.4xx10^-7ohmm`
C. `1.2xx10^-7ohmm`
D. `1.2xx10^-7ohmm`

Explanation

To find the resistivity of the wire at 900 K, we can use the formula that relates the resistivity at different temperatures:

\[ \rho(T_f) = \rho(T_i) \left(1 + \alpha (T_f - T_i)\right) \]

Where:
- \(\rho(T_f)\) = resistivity at final temperature \(T_f\)
- \(\rho(T_i)\) = resistivity at initial temperature \(T_i\)
- \(\alpha\) = temperature coefficient of resistivity
- \(T_f\) = final temperature (900 K)
- \(T_i\) = initial temperature (300 K)

### Step 1: Identify the given values
- \(\rho(T_i) = \pi \times 10^{-8} \, \text{ohm m}\)
- \(\alpha = 7.8 \times 10^{-3} \, \text{per degree Celsius}\)
- \(T_f = 900 \, \text{K}\)
- \(T_i = 300 \, \text{K}\)

### Step 2: Calculate the temperature difference
Convert the temperatures to Celsius:
- \(T_i = 300 \, \text{K} - 273.15 = 26.85 \, \text{°C}\)
- \(T_f = 900 \, \text{K} - 273.15 = 626.85 \, \text{°C}\)

Now calculate the difference:
\[ T_f - T_i = 626.85 - 26.85 = 600 \, \text{°C} \]

### Step 3: Substitute the values into the formula
Now substitute the values into the resistivity formula:
\[ \rho(900) = \pi \times 10^{-8} \left(1 + 7.8 \times 10^{-3} \times 600\right) \]

### Step 4: Calculate the factor
Calculate the factor:
\[ 1 + 7.8 \times 10^{-3} \times 600 = 1 + 4.68 = 5.68 \]

### Step 5: Calculate the resistivity at 900 K
Now substitute this back into the resistivity equation:
\[ \rho(900) = \pi \times 10^{-8} \times 5.68 \]

### Step 6: Compute the final value
Using the approximate value of \(\pi \approx 3.14\):
\[ \rho(900) \approx 3.14 \times 10^{-8} \times 5.68 \approx 17.83 \times 10^{-8} \, \text{ohm m} \]

### Step 7: Convert to scientific notation
Convert this to scientific notation:
\[ \rho(900) \approx 1.78 \times 10^{-7} \, \text{ohm m} \]

### Final Answer
The resistivity of the wire at 900 K is approximately:
\[ \rho(900) \approx 1.78 \times 10^{-7} \, \text{ohm m} \]