Question

Consider two spherical perfect blackbodies with radii \(R_1\) and \(R_2\) at temperatures \(T_1 = 1000\, \text{K}\) and \(T_2 = 2000\, \text{K}\), respectively. They both emit radiation of power 1 kW. The ratio of their radii, \(R_1/R_2\), is given by ...............

Hint:

The total power radiated by a blackbody varies as \(A T^4\). For equal power output, the radius scales inversely with the square of temperature.

Answer 1

Correct Answer: 4

Step 1: Apply Stefan–Boltzmann law.
For a blackbody, \[ P = \sigma A T^4 = \sigma (4\pi R^2) T^4 \] Given \(P_1 = P_2 = 1\, \text{kW}\), \[ R_1^2 T_1^4 = R_2^2 T_2^4 \] \[ \frac{R_1}{R_2} = \left(\frac{T_2}{T_1}\right)^{-2} = \left(\frac{2000}{1000}\right)^{-2} = \frac{1}{4}^2 = 0.0625 \]

Step 2: Conclusion.
Hence, the ratio of radii \(R_1 / R_2 = 0.0625\).