Final answer:
The ratio of energy radiated by two spheres at different temperatures is determined using the Stefan-Boltzmann law, accounting for surface area differences and the fourth power of each sphere's absolute temperature in kelvins.
Step-by-step explanation:
The question concerns the calculation of the ratio of energy radiated by two spheres of different temperatures but the same color, using the Stefan-Boltzmann law of radiation. To calculate the energy radiated, we use the formula P = oAeT^4, where o is the Stefan-Boltzmann constant (5.67 × 10^-8 J/s · m^2 · K^4), A is the surface area, e is the emissivity (assumed to be the same for both), and T is the absolute temperature in kelvins.
First, we convert the temperatures from degrees Celsius to kelvins (by adding 273 to each temperature), getting 400 K for sphere P and 800 K for sphere Q. The surface area A for a sphere is 4πr^2; thus, the surface areas for P and Q are 4π(8 cm)^2 and 4π(2 cm)^2, respectively.
The power radiated by each sphere is obtained by plugging the above values into the Stefan-Boltzmann formula. After calculating the power radiated by both spheres, we take the ratio of P's power to Q's power to determine the final answer. Since the spheres are identical in material and presumably in emissivity, e cancels out, and we are left with the ratio depending primarily on the surface area and the fourth power of their respective temperatures.