The radiation power from a source at temperature T and2m away is `2W//m^(2)`. If the temperature of the source increases by 100% the radiation power at a distance 4m from the source will increase by:

Options

A. 1
B. 2
C. 6
D. None of the above

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We have the following information:
- Initial radiation power at a distance of \(2 \, \text{m}\) from the source is \(2 \, \text{W/m}^2\).
- The temperature of the source is increased by \(100\%\), which means it is doubled (from \(T\) to \(2T\)).
- We need to find the radiation power at a distance of \(4 \, \text{m}\) from the source after the temperature increase.

### Step 2: Use the Formula for Radiation Power
According to Stefan-Boltzmann Law, the power radiated per unit area (\(P\)) is given by:
\[ P \propto \frac{T^4}{r^2} \]
where \(T\) is the temperature and \(r\) is the distance from the source.

### Step 3: Calculate Initial Power
Let’s denote the initial power at \(2 \, \text{m}\) as \(P_1\):
\[ P_1 = k \frac{T^4}{(2)^2} = k \frac{T^4}{4} \]
Given that \(P_1 = 2 \, \text{W/m}^2\), we can write:
\[ 2 = k \frac{T^4}{4} \]
From this, we can find \(k \frac{T^4}{4} = 2\).

### Step 4: Calculate New Power after Temperature Increase
Now, we need to find the new power \(P_2\) at a distance of \(4 \, \text{m}\) when the temperature is \(2T\):
\[ P_2 = k \frac{(2T)^4}{(4)^2} \]
Calculating this gives:
\[ P_2 = k \frac{16T^4}{16} = k T^4 \]

### Step 5: Relate \(P_2\) to \(P_1\)
From our previous calculation, we know:
\[ k \frac{T^4}{4} = 2 \implies k T^4 = 8 \]
Thus, substituting back, we find:
\[ P_2 = k T^4 = 8 \, \text{W/m}^2 \]

### Step 6: Conclusion
The radiation power at a distance of \(4 \, \text{m}\) from the source after the temperature increase is \(8 \, \text{W/m}^2\).

### Final Answer
The radiation power at a distance of \(4 \, \text{m}\) from the source will increase to \(8 \, \text{W/m}^2\).

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