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: Formula
By Stefan-Boltzmann Law, Power $P = \sigma A T^{4}$. For a sphere, $A = 4\pi R^{2}$.

Step 2: Analysis
$P = \sigma (4\pi R^{2}) T^{4}$. Since $P_1 = P_2$:
$\sigma 4\pi R_{1}^{2} T_{1}^{4} = \sigma 4\pi R_{2}^{2} T_{2}^{4}$.

Step 3: Calculation
$R_{1}^{2} T_{1}^{4} = R_{2}^{2} T_{2}^{4} \implies \frac{R_{2}^{2}}{R_{1}^{2}} = \frac{T_{1}^{4}}{T_{2}^{4}}$.
Taking the square root: $\frac{R_2}{R_1} = \frac{T_{1}^{2}}{T_{2}^{2}} = (T_{1}/T_{2})^{2}$.

Step 4: Conclusion
Hence, the ratio of radii is $(T_{1}/T_{2})^{2}$. Final Answer: (C)