Question

A sphere having temperature 600 K is losing heat due to radiation. At this temperature its rate of cooling is \(R\). The rate of cooling of this sphere at 400 K is \(\frac{x}{243}\). The value of \(x\) is (Temperature of surrounding is 300 K)

• A. \(50R\)
• B. \(45R\)
• C. \(40R\)
• D. \(35R\)

Hint:

Rate of cooling is proportional to T to the fourth minus T0 to the fourth for radiation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
By Stefan's law, the net radiation loss is proportional to \(T^4-T_0^4\), with surroundings at \(T_0=300\) K. The rate of cooling is proportional to this loss.

Step 2: At 600 K
\[ 600^4-300^4=(6^4-3^4)\times10^8=(1296-81)\times10^8=1215\times10^8 \]

Step 3: At 400 K
\[ 400^4-300^4=(4^4-3^4)\times10^8=(256-81)\times10^8=175\times10^8 \]

Step 4: Ratio
\[ \frac{R'}{R}=\frac{175}{1215}=\frac{35}{243} \]
So \(R'=\dfrac{35R}{243}\).

Step 5: Compare with the form given
The rate is written as \(\dfrac{x}{243}\), so \(x=35R\), option (D).

Final Answer:
The ratio of the radiation terms is 175 to 1215, which is 35/243, so x = 35R, option (D). \[ \boxed{35R} \]