AIIMS 2004 Physics Stefan-Boltzmann Law MCQ Question

To solve the question, we need to apply the Stefan-Boltzmann Law, which states that the total power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature. The formula can be expressed as:

$$P = \sigma A T^4$$

where:

• $P$ is the total power (energy emitted per second),
• $\sigma$ is the Stefan-Boltzmann constant,
• $A$ is the surface area of the radiating body, and
• $T$ is the absolute temperature.

Step 1: Calculate Initial Conditions

1. Initial Radius: Let the initial radius of the sun be $R$.
2. Initial Temperature: Let the initial temperature of the sun be $T$.

Using the formula for surface area $A$ of a sphere:

$$A = 4 \pi R^2$$

The initial power emitted by the sun can be expressed as:

$$P_{\text{initial}} = \sigma (4 \pi R^2) T^4$$

Step 2: Calculate New Conditions

Now, according to the problem:

• The new radius becomes $R' = 100R$.
• The new temperature becomes $T' = \frac{1}{2} T$.

The new surface area will be:

$$A' = 4 \pi (R')^2 = 4 \pi (100R)^2 = 4 \pi (10000 R^2) = 10000 (4 \pi R^2)$$

The new power emitted will be:

$$P_{\text{new}} = \sigma (A') (T')^4 = \sigma (10000 (4 \pi R^2)) \left( \frac{1}{2} T \right)^4$$

Calculating $\left( \frac{1}{2} T \right)^4$:

$$\left( \frac{1}{2} T \right)^4 = \frac{1}{16} T^4$$

Substituting this into the equation for $P_{\text{new}}$:

$$P_{\text{new}} = \sigma (10000 (4 \pi R^2)) \left( \frac{1}{16} T^4 \right)$$

Now simplifying:

$$P_{\text{new}} = \sigma (4 \pi R^2) T^4 \cdot 10000 \cdot \frac{1}{16} = P_{\text{initial}} \cdot 10000 \cdot \frac{1}{16}$$

Step 3: Calculate the Increase Factor

Now, calculating the factor by which the total energy emitted increases:

$$\text{Factor} = \frac{P_{\text{new}}}{P_{\text{initial}}} = 10000 \cdot \frac{1}{16} = 625$$

Conclusion

Thus, the total energy emitted by the sun when it expands to 100 times its radius and its temperature decreases to half is increased by a factor of 625.

Correct Answer: B) 625

Clarification of Incorrect Options

• A) 10⁴: This would imply that the energy emitted increases by a factor of $10^4$ without accounting for the temperature reduction. This option does not consider the $\frac{1}{16}$ factor from the temperature change.

• C) 256: This factor might arise from a misunderstanding of $\left( \frac{1}{2} \right)^4 = \frac{1}{16}$, but does not correctly apply the area increase.

• D) 16: This option considers only the decrease in temperature without factoring in the increase in area.

In summary, the correct answer is derived from applying the Stefan-Boltzmann Law accurately, considering both changes in radius and temperature.