Question

A black body at a temperature of 125$^\circ$C emits heat at the rate of 32 W. The rate of heat emitted by the body when the temperature of the body is increased by 398 K is:

• A. 64
• B. 128
• C. 512
• D. 256

Hint:

The power radiated by a black body increases with the fourth power of its temperature. When the temperature increases, the rate of heat emission increases significantly.

Answer 1

The Correct Option is C

Step 1:
The rate of heat emission from a black body is proportional to the fourth power of its temperature: \[ P \propto T^4 \]
Step 2:
Let the initial temperature be \( T_1 = 125 ^\circ C = 125 + 273 = 398 \, K \), and the final temperature be \( T_2 = 398 + 398 = 796 \, K \).
Step 3:
Using the Stefan-Boltzmann law: \[ \frac{P_2}{P_1} = \left( \frac{T_2}{T_1} \right)^4 \] \[ \frac{P_2}{32} = \left( \frac{796}{398} \right)^4 \] \[ \frac{P_2}{32} = 2^4 = 16 \] \[ P_2 = 32 \times 16 = 512 \, \text{W m$^-2$} \]