When the centre of earth is at a distance of `1.5 xx 10^(11)m` from the centre of sun, the intensity of solar radiation reaching at the earth's surface is `1.26kW//m^(2)`. There is a spherical cloud of cosmic dust, containing iron particles. The melting point for iron particles in the cloud is 2000 K. Find the distance of iron particles from the centre of sun at which the iron particle starts melting. (Assume sun and cloud as a black body, `sigma = 5.8 xx 10^(-8) W//m^(2)K^(4))`:

Options

A. `2.81 xx 10^(5)m`
B. `2.81 xx 10^(10)m`
C. `2.81 xx 10^(9)m`
D. `1.40 xx 10^(10)m`

Explanation

To find the distance of iron particles from the center of the Sun at which they start melting, we can use the Stefan-Boltzmann Law and the given information. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We know that the intensity of solar radiation at the Earth's surface is given as \( I = 1.26 \, \text{kW/m}^2 = 1.26 \times 10^3 \, \text{W/m}^2 \). The melting point of iron is \( T = 2000 \, \text{K} \). We need to find the distance \( d_1 \) from the center of the Sun where the temperature of the iron particles reaches 2000 K.

### Step 2: Use the Stefan-Boltzmann Law
The power radiated per unit area of a black body is given by the Stefan-Boltzmann Law:
\[ P = \sigma T^4 \]
Where \( \sigma = 5.67 \times 10^{-8} \, \text{W/m}^2 \text{K}^4 \) is the Stefan-Boltzmann constant.

### Step 3: Calculate the Power Radiated by the Sun
The total power \( P_{\text{sun}} \) emitted by the Sun can be calculated using the intensity at the Earth's distance \( d_2 \):
\[ P_{\text{sun}} = I \times A \]
Where \( A \) is the surface area of a sphere with radius \( d_2 \):
\[ A = 4 \pi d_2^2 \]
Given \( d_2 = 1.5 \times 10^{11} \, \text{m} \):
\[ A = 4 \pi (1.5 \times 10^{11})^2 \]
Calculating \( A \):
\[ A = 4 \pi (2.25 \times 10^{22}) \approx 2.83 \times 10^{23} \, \text{m}^2 \]
Now, substituting back to find \( P_{\text{sun}} \):
\[ P_{\text{sun}} = 1.26 \times 10^3 \times 2.83 \times 10^{23} \approx 3.57 \times 10^{26} \, \text{W} \]

### Step 4: Relate the Power to the Distance \( d_1 \)
At distance \( d_1 \), the intensity \( I_1 \) can be expressed as:
\[ I_1 = \frac{P_{\text{sun}}}{4 \pi d_1^2} \]
Setting \( I_1 \) equal to the intensity needed to melt iron:
\[ \sigma T^4 = \frac{P_{\text{sun}}}{4 \pi d_1^2} \]
Substituting \( T = 2000 \, \text{K} \):
\[ \sigma (2000)^4 = \frac{3.57 \times 10^{26}}{4 \pi d_1^2} \]

### Step 5: Calculate \( \sigma (2000)^4 \)
Calculating \( \sigma (2000)^4 \):
\[ \sigma (2000)^4 = 5.67 \times 10^{-8} \times (16 \times 10^{12}) = 9.072 \times 10^5 \, \text{W/m}^2 \]

### Step 6: Solve for \( d_1 \)
Now we can rearrange to solve for \( d_1 \):
\[ d_1^2 = \frac{3.57 \times 10^{26}}{4 \pi (9.072 \times 10^5)} \]
Calculating the denominator:
\[ 4 \pi (9.072 \times 10^5) \approx 1.141 \times 10^7 \]
Now substituting back:
\[ d_1^2 = \frac{3.57 \times 10^{26}}{1.141 \times 10^7} \approx 3.13 \times 10^{19} \]
Taking the square root:
\[ d_1 \approx \sqrt{3.13 \times 10^{19}} \approx 5.6 \times 10^{9} \, \text{m} \]

### Final Answer
The distance of the iron particles from the center of the Sun at which they start melting is approximately \( 5.6 \times 10^{9} \, \text{m} \).

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