Question

Two spherical black bodies have radii \(R_{1}\) and \(R_{2}\). Their surface temperatures are \(T_{1}\) K and \(T_{2}\) K respectively. If they radiate the same power, the ratio \(\frac{R_{1}}{R_{2}}\) is

• A. \(\left(\frac{T_{1}}{T_{2}}\right)^{4}\)
• B. \(\left(\frac{T_{1}}{T_{2}}\right)^{2}\)
• C. \(\left(\frac{T_{2}}{T_{1}}\right)^{4}\)
• D. \(\left(\frac{T_{2}}{T_{1}}\right)^{2}\)

Hint:

Use the Stefan law P = sigma A T to the fourth power and equate the powers.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A black sphere radiates power \(P=\sigma(4\pi R^2)T^4\) according to Stefan's law.

Step 2: Equate the powers
\(R_1^2T_1^4=R_2^2T_2^4\)

Step 3: Rearrange
\(\dfrac{R_1^2}{R_2^2}=\dfrac{T_2^4}{T_1^4}\Rightarrow\dfrac{R_1}{R_2}=\left(\dfrac{T_2}{T_1}\right)^2\)

Final Answer:
The ratio \(\dfrac{R_1}{R_2}\) is \(\left(\dfrac{T_2}{T_1}\right)^2\). \[ \boxed{\text{(D) }\left(\dfrac{T_{2}}{T_{1}}\right)^{2}} \]