Question

A black sphere has radius \(R\) whose rate of radiation is \(E\) at temperature \(T\). If the radius is made \( \dfrac{R}{3} \) and the temperature is made \(3T\), the new rate of radiation will be

• A. \(3E\)
• B. \(E\)
• C. \(9E\)
• D. \(6E\)

Hint:

Radiation power depends very strongly on temperature (\( \propto T^4 \)).

Answer 1

The Correct Option is C

Step 1: Stefan–Boltzmann law.
Rate of radiation:
\[ E \propto A T^4 \]
Step 2: Surface area dependence.
\[ A \propto R^2 \]
Step 3: Comparing old and new conditions.
\[ \frac{E_2}{E_1} = \left(\frac{R/3}{R}\right)^2 \left(\frac{3T}{T}\right)^4 \]
Step 4: Simplifying.
\[ \frac{E_2}{E_1} = \frac{1}{9} \times 81 = 9 \]
Step 5: Conclusion.
\[ E_2 = 9E \]