A body of area `0.8xx10^(-2) m^(2)` and mass `5xx10^(-4) kg` directly faces the sun on a clear day. The body has an emissivity of 0.8 and specific heat of `0.8 ca//kg` K. The surroundings are at `27^@C`. (solar constant`=1.4 kW//m^(2)`).
The maximum attainable temperature of the body is

Options

A. `396K`
B. `396^@C`
C. `85^@C`
D. `85K`

Explanation

To find the maximum attainable temperature of the body facing the sun, we can follow these steps:

### Step 1: Understand the Heat Transfer Mechanisms
The body absorbs solar radiation and emits thermal radiation. The rate of heat absorbed (Q_abs) and the rate of heat emitted (Q_emit) can be expressed as:
- **Heat Absorbed (Q_abs)**: \( Q_{abs} = S \cdot A \cdot e \)
- **Heat Emitted (Q_emit)**: \( Q_{emit} = \sigma \cdot e \cdot A \cdot T^4 \)

Where:
- \( S \) = Solar constant = \( 1.4 \, \text{kW/m}^2 = 1.4 \times 10^3 \, \text{W/m}^2 \)
- \( A \) = Area = \( 0.8 \times 10^{-2} \, \text{m}^2 \)
- \( e \) = Emissivity = \( 0.8 \)
- \( \sigma \) = Stefan-Boltzmann constant = \( 5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4 \)
- \( T \) = Temperature in Kelvin

### Step 2: Set Up the Equation for Maximum Temperature
At maximum temperature, the rate of heat absorbed equals the rate of heat emitted:
\[ Q_{abs} = Q_{emit} \]

This gives us:
\[ S \cdot A \cdot e = \sigma \cdot e \cdot A \cdot T^4 \]

### Step 3: Simplify the Equation
Since \( e \) and \( A \) are present on both sides, we can cancel them out:
\[ S = \sigma \cdot T^4 \]

### Step 4: Solve for Temperature (T)
Rearranging the equation gives:
\[ T^4 = \frac{S}{\sigma} \]
\[ T = \left( \frac{S}{\sigma} \right)^{1/4} \]

### Step 5: Substitute the Values
Now we can substitute the known values into the equation:
\[ T = \left( \frac{1.4 \times 10^3 \, \text{W/m}^2}{5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4} \right)^{1/4} \]

### Step 6: Calculate the Value
Calculating the fraction:
\[ \frac{1.4 \times 10^3}{5.67 \times 10^{-8}} = 2.47 \times 10^{10} \]

Now take the fourth root:
\[ T = (2.47 \times 10^{10})^{1/4} \]

Calculating this gives:
\[ T \approx 396 \, \text{K} \]

### Conclusion
The maximum attainable temperature of the body is approximately **396 K**.

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