Question

A black sphere has radius R whose rate of radiation is E at temperature T. If radius is made $R/2$ and temperature $3T$, the rate of radiation will be

• A. $3E/2$
• B. $27E/8$
• C. $81E/4$
• D. $9E/4$

Hint:

Radiation power is highly sensitive to temperature changes ($\text{Power} \propto T^4$).

Answer 1

The Correct Option is C

Step 1: Formula
According to Stefan-Boltzmann Law, Rate of radiation $E = \sigma A T^4$. For a sphere, $E = \sigma (4\pi R^2) T^4$.

Step 2: Analysis
$E \propto R^2 T^4$.
Let $E'$ be the new rate: $E' \propto (R/2)^2 (3T)^4$.

Step 3: Calculation
$E' \propto (\frac{R^2}{4}) (81 T^4) = \frac{81}{4} (R^2 T^4)$.
Since $E = R^2 T^4$, then $E' = \frac{81}{4}E$.

Step 4: Conclusion
Hence, the new rate of radiation is $81E/4$. Final Answer: (C)