If the temperature of the sun (black body) is doubled, the rate of energy received on earth will be increase by a factor of

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the problem of how the rate of energy received on Earth changes when the temperature of the Sun (considered as a black body) is doubled, we can use Stefan-Boltzmann's law. Here’s a step-by-step solution:

### Step 1: Understand Stefan-Boltzmann Law
According to Stefan-Boltzmann Law, the energy radiated per unit area of a black body is directly proportional to the fourth power of its absolute temperature (T). This can be expressed mathematically as:
\[ E \propto T^4 \]
where \( E \) is the energy emitted per unit area, and \( T \) is the temperature in Kelvin.

### Step 2: Set Up the Initial and Final Conditions
Let the initial temperature of the Sun be \( T_1 \) and the initial energy received on Earth be \( E_1 \). If the temperature of the Sun is doubled, the new temperature \( T_2 \) will be:
\[ T_2 = 2T_1 \]

### Step 3: Apply the Stefan-Boltzmann Law
Using the Stefan-Boltzmann Law, we can write the energy received at the two temperatures:
\[ E_1 \propto T_1^4 \]
\[ E_2 \propto T_2^4 = (2T_1)^4 \]

### Step 4: Calculate \( E_2 \)
Now, substituting \( T_2 \) into the equation:
\[ E_2 \propto (2T_1)^4 = 16T_1^4 \]

### Step 5: Find the Ratio of Energies
Now, we can find the ratio of the energy received before and after the temperature change:
\[ \frac{E_2}{E_1} = \frac{16T_1^4}{T_1^4} = 16 \]

### Step 6: Conclusion
Thus, when the temperature of the Sun is doubled, the rate of energy received on Earth increases by a factor of 16.

### Final Answer:
The rate of energy received on Earth will increase by a factor of **16**.

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