Question

Two spherical black bodies of radii \(R_1\) and \(R_2\) having surface temperature \(T_1\) and \(T_2\) respectively, radiate the same power. The ratio of \(R_1\) to \(R_2\) will be

• A. \((\frac{T_1}{T_2})^2\)
• B. \((\frac{T_1}{T_2})^4\)
• C. \((\frac{T_2}{T_1})^2\)
• D. \((\frac{T_2}{T_1})^4\)

Hint:

Use Stefan law: power is sigma times area times T to the fourth.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For a black body, the power radiated is \(P = \sigma A T^4\), with \(A = 4\pi R^2\) for a sphere.

Step 2: Key Formula or Approach:
Equal powers: \(4\pi R_1^2\sigma T_1^4 = 4\pi R_2^2\sigma T_2^4\).

Step 3: Detailed Explanation:
\(R_1^2 T_1^4 = R_2^2 T_2^4\).
\(\frac{R_1^2}{R_2^2} = \frac{T_2^4}{T_1^4}\).
\[ \frac{R_1}{R_2} = \left(\frac{T_2}{T_1}\right)^2 \]
The hotter body must be smaller to radiate the same power. Options A and B have the temperatures in the wrong order, and option D has the wrong power.

Final Answer:
\(\frac{R_1}{R_2} = \left(\frac{T_2}{T_1}\right)^2\), option (C). \[ \boxed{\left(\frac{T_2}{T_1}\right)^2} \]