A body at `300^(@)C` radiates `10 J cm^(-2) s^(-1)`. If Sun radiates `10^(5) Jcm^(-2) s^(-1)`,then its temperature is:

Options

A. `3000^(@)C`
B. `5457^(@)C`
C. `300 xx 10^(4)``""^(@)C`
D. `5730^(@)C`

Explanation

To solve the problem, we will use the Stefan-Boltzmann law, which states that the intensity (I) of radiation emitted by a black body is directly proportional to the fourth power of its absolute temperature (T). The relationship can be expressed as:

\[ I \propto T^4 \]

Given:
- Intensity of the body (I1) = \(10 \, \text{J cm}^{-2} \text{s}^{-1}\)
- Intensity of the Sun (I2) = \(10^5 \, \text{J cm}^{-2} \text{s}^{-1}\)
- Temperature of the body (T1) = \(300 \, \text{°C}\)

First, we need to convert the temperature from Celsius to Kelvin:

\[ T1 = 300 + 273 = 573 \, \text{K} \]

Now, we can set up the ratio of the intensities and temperatures:

\[ \frac{I_1}{I_2} = \frac{T_1^4}{T_2^4} \]

Substituting the known values:

\[ \frac{10}{10^5} = \frac{(573)^4}{T_2^4} \]

This simplifies to:

\[ \frac{1}{10^4} = \frac{(573)^4}{T_2^4} \]

Cross-multiplying gives:

\[ T_2^4 = 10^4 \cdot (573)^4 \]

Taking the fourth root of both sides to solve for \(T_2\):

\[ T_2 = 573 \cdot 10 \]

Calculating \(T_2\):

\[ T_2 = 5730 \, \text{K} \]

Now, we need to convert this temperature back to Celsius:

\[ T_2 (\text{in °C}) = 5730 - 273 = 5457 \, \text{°C} \]

Thus, the temperature of the Sun is:

\[ \boxed{5457 \, \text{°C}} \]