If the temperature of the sun were to increase form T to 2T and its radius from R to 2R, then the ratio of the radiant energy received on earth to what it was previously will be

Options

A. `32`
B. `16`
C. `4`
D. `64`

Explanation

To find the ratio of the radiant energy received on Earth after the changes in the Sun's temperature and radius, we can follow these steps:

### Step 1: Understand the Stefan-Boltzmann Law
The Stefan-Boltzmann Law states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature (T). The formula is given by:
\[ E = \sigma T^4 \]
where \( E \) is the energy radiated per unit area, \( \sigma \) is the Stefan-Boltzmann constant, and \( T \) is the temperature.

### Step 2: Calculate the Initial Energy Radiated
Initially, the Sun has a temperature \( T \) and radius \( R \). The energy radiated per unit area from the Sun is:
\[ E_{\text{initial}} = \sigma T^4 \]

### Step 3: Calculate the Final Energy Radiated
After the changes, the temperature of the Sun increases to \( 2T \) and the radius increases to \( 2R \). The energy radiated per unit area from the Sun now becomes:
\[ E_{\text{final}} = \sigma (2T)^4 = \sigma \cdot 16T^4 \]

### Step 4: Calculate the Area of the Sun
The area \( A \) of the Sun is given by the formula for the surface area of a sphere:
\[ A = 4\pi R^2 \]
After the radius changes to \( 2R \), the new area becomes:
\[ A_{\text{final}} = 4\pi (2R)^2 = 16\pi R^2 \]

### Step 5: Calculate the Total Energy Radiated
The total energy radiated by the Sun initially is:
\[ E_{\text{total initial}} = E_{\text{initial}} \cdot A = \sigma T^4 \cdot 4\pi R^2 \]

The total energy radiated by the Sun after the changes is:
\[ E_{\text{total final}} = E_{\text{final}} \cdot A_{\text{final}} = \sigma \cdot 16T^4 \cdot 16\pi R^2 \]

### Step 6: Find the Ratio of Energies
To find the ratio of the total energy received on Earth after the changes to what it was previously, we can set up the ratio:
\[ \text{Ratio} = \frac{E_{\text{total final}}}{E_{\text{total initial}}} = \frac{\sigma \cdot 16T^4 \cdot 16\pi R^2}{\sigma T^4 \cdot 4\pi R^2} \]

### Step 7: Simplify the Ratio
Canceling out common terms:
\[ \text{Ratio} = \frac{16 \cdot 16}{4} = \frac{256}{4} = 64 \]

### Final Answer
Thus, the ratio of the radiant energy received on Earth after the changes to what it was previously is:
\[ \text{Ratio} = 64 \]

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