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: Cancel common factors:
Both bodies have the same $4\pi\sigma$, so $R^2T^4$ is the same for both.

Step 2: Take the square root:
$R_1T_1^2 = R_2T_2^2$, so $\frac{R_1}{R_2} = \frac{T_2^2}{T_1^2}$.

Final Answer:
The ratio is $\left(\frac{T_2}{T_1}\right)^2$, option (C). \[ \boxed{\left(\frac{T_2}{T_1}\right)^2} \]