If the temperature of the sun were to increase from $$T$$ 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

Solution & Explanation

1. Apply Stefan-Boltzmann Law for Total Radiated Power

According to the Stefan-Boltzmann Law, the total radiant energy emitted per second (thermal power, $$E$$) by a perfectly blackbody spherical star like the Sun is directly proportional to its surface area and the fourth power of its absolute temperature:

$$E = \sigma \cdot A \cdot T^4$$

Since the surface area of a spherical star with radius $$R$$ is $$A = 4\pi \cdot R^2$$, we can substitute this geometric value into the equation:

$$E = \sigma \cdot (4\pi \cdot R^2) \cdot T^4$$

Since the distance between the Earth and the Sun remains unchanged, the radiant energy received on Earth per unit area per second (intensity) is directly proportional to the total power emitted by the Sun:

$$E \propto R^2 \cdot T^4$$

2. Establish the Proportionality Ratio

Let us write down the ratio comparing the final radiant energy state ($$E_{\text{final}}$$) to the initial radiant energy state ($$E_{\text{initial}}$$):

$$\frac{E_{\text{final}}}{E_{\text{initial}}} = \left( \frac{R_{\text{final}}}{R_{\text{initial}}} \right)^2 \cdot \left( \frac{T_{\text{final}}}{T_{\text{initial}}} \right)^4$$

3. Substitute the Given Values

From the problem statement, we have the following scaling parameters for the Sun's physical properties:

• Final Radius: $$R_{\text{final}} = 2R$$
• Final Temperature: $$T_{\text{final}} = 2T$$

Plugging these values back into our ratio equation:

$$\frac{E_{\text{final}}}{E_{\text{initial}}} = \left( \frac{2R}{R} \right)^2 \cdot \left( \frac{2T}{T} \right)^4$$

$$\frac{E_{\text{final}}}{E_{\text{initial}}} = (2)^2 \cdot (2)^4$$

$$\frac{E_{\text{final}}}{E_{\text{initial}}} = 4 \cdot 16 = 64$$

Therefore, the ratio of the newly received radiant energy to the previous value is $$64 : 1$$.

Concept Check: Doubling the radius increases the emitting surface area by a factor of 4 ($$2^2$$). Concurrently, doubling the absolute temperature increases the radiation output per unit area by a massive factor of 16 ($$2^4$$) due to its highly sensitive fourth-power dependence. Compounding both physical growth factors yields a total energy increase of exactly 64 times.

Correct Option Key: Option D ($64$)