Suppose the sun expands so that its radius becomes `100` times its present radius and its surface temperature becomes half of its present value. The total energy emited by it then will increase by a factor of :

Options

A. `10^(4)`
B. 625
C. 256
D. 16

Explanation

To solve the problem, we will use the Stefan-Boltzmann law, which states that the total energy emitted per unit surface area of a black body is proportional to the fourth power of its absolute temperature. The formula can be expressed as:

\[ Q = \sigma A T^4 \]

where:
- \( Q \) is the total energy emitted,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area,
- \( T \) is the absolute temperature.

### Step 1: Calculate the initial energy emitted by the Sun

The initial radius of the Sun is \( R \) and the initial temperature is \( T \). The surface area \( A \) of the Sun is given by:

\[ A = 4\pi R^2 \]

Thus, the initial energy emitted \( Q \) can be expressed as:

\[ Q = \sigma A T^4 = \sigma (4\pi R^2) T^4 \]

### Step 2: Calculate the new energy emitted after the changes

According to the problem, the radius of the Sun expands to \( 100R \) and the surface temperature becomes \( \frac{T}{2} \).

The new surface area \( A' \) is:

\[ A' = 4\pi (100R)^2 = 4\pi (10000R^2) = 10000 \times 4\pi R^2 \]

The new temperature \( T' \) is:

\[ T' = \frac{T}{2} \]

Now, we can calculate the new energy emitted \( Q' \):

\[ Q' = \sigma A' (T')^4 = \sigma (10000 \times 4\pi R^2) \left(\frac{T}{2}\right)^4 \]

Calculating \( \left(\frac{T}{2}\right)^4 \):

\[ \left(\frac{T}{2}\right)^4 = \frac{T^4}{16} \]

Substituting this back into the equation for \( Q' \):

\[ Q' = \sigma (10000 \times 4\pi R^2) \left(\frac{T^4}{16}\right) \]

### Step 3: Simplify the expression for \( Q' \)

Now we can simplify \( Q' \):

\[ Q' = \sigma (4\pi R^2) T^4 \times \frac{10000}{16} \]

Since \( Q = \sigma (4\pi R^2) T^4 \), we can substitute \( Q \) into the equation:

\[ Q' = Q \times \frac{10000}{16} \]

Calculating \( \frac{10000}{16} \):

\[ \frac{10000}{16} = 625 \]

Thus, we have:

\[ Q' = 625 Q \]

### Conclusion

The total energy emitted by the Sun after the changes will increase by a factor of **625**.