Question

If the temperature of the sun is doubled, the rate of energy received by the earth will be increased by a factor

• A. $8$
• B. $2$
• C. $4$
• D. $16$

Hint:

Always remember that radiation power depends on the fourth power of absolute temperature ($T^4$). A simple mental shortcut for these types of questions is to take the scaling multiplier (2) and raise it to the fourth power: $2^4 = 16$. If the temperature had tripled, the factor would be $3^4 = 81$!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem asks for the scaling factor by which the thermal radiation power received on Earth will increase if the absolute thermodynamic temperature of the Sun is exactly doubled.

Step 2: Key Formula or Approach:
According to the Stefan-Boltzmann Law, the total radiant energy emitted per unit surface area of a blackbody per unit time ($E$) is directly proportional to the fourth power of its absolute temperature ($T$): $$E = \sigma T^4 \implies E \propto T^4$$ Since the geometric distance between the Earth and the Sun remains constant, the rate of energy received by the Earth scales identically with the total power emitted by the Sun.

Step 3: Detailed Explanation:
Let the initial absolute temperature of the sun be $T_1 = T$, and the initial rate of energy received be $E_1$. The temperature is doubled, so the final absolute temperature is $T_2 = 2T$. Let's set up the ratio for the final energy emission rate $E_2$: $$\frac{E_2}{E_1} = \left(\frac{T_2}{T_1}\right)^4$$ Substitute our temperature values into the ratio equation: $$\frac{E_2}{E_1} = \left(\frac{2T}{T}\right)^4 = (2)^4$$ Evaluating $2$ raised to the power of $4$: $$2^4 = 2 \times 2 \times 2 \times 2 = 16$$ $$\frac{E_2}{E_1} = 16 \implies E_2 = 16 E_1$$ Therefore, the total rate of energy received increases by a factor of 16.

Step 4: Final Answer:
The rate of energy received increases by a factor of $16$, corresponding to option (D).