Question

Two spherical black bodies have radii $R_{1}$ and $R_{2}$ and temperatures $T_{1}$ and $T_{2}$. If they radiate same power, then $R_2/R_1$ is}

• A. $\frac{T_{2}}{T_{1}}$
• B. $\frac{T_{1}}{T_{2}}$
• C. $(T_{1}/T_{2})^{2}$
• D. $(T_{2}/T_{1})^{2}$

Hint:

For equal power, radius squared is inversely proportional to temperature to the fourth power ($R^2 \propto 1/T^4$).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We use Stefan-Boltzmann law to equate the power radiated by two spherical black bodies.

Step 2: Key Formula or Approach:
Power radiated \( P = \sigma A T^4 \).
For a sphere, surface area \( A = 4\pi R^2 \).

Step 3: Detailed Explanation:
Given \( P_1 = P_2 \).
\[ \sigma (4\pi R_1^2) T_1^4 = \sigma (4\pi R_2^2) T_2^4 \]
Cancelling common constants \( \sigma \) and \( 4\pi \):
\[ R_1^2 T_1^4 = R_2^2 T_2^4 \]
Rearranging for the ratio \( \frac{R_2}{R_1} \):
\[ \frac{R_2^2}{R_1^2} = \frac{T_1^4}{T_2^4} \]
Taking square root on both sides:
\[ \frac{R_2}{R_1} = \frac{T_1^2}{T_2^2} = \left( \frac{T_1}{T_2} \right)^2 \]

Step 4: Final Answer:
The ratio \( \frac{R_2}{R_1} \) is \( \left( \frac{T_1}{T_2} \right)^2 \).