Introduction
The cooling of a solid sphere by radiation can be described using the principles of thermodynamics and heat transfer. In this scenario, we assume that the density (ρ) of the sphere is constant.

Explanation
The rate of cooling of a solid sphere can be expressed as the rate at which heat is lost from the surface of the sphere. This heat loss occurs through radiation, which is the transfer of thermal energy in the form of electromagnetic waves.

Stefan-Boltzmann Law
The rate of heat transfer by radiation is governed by the Stefan-Boltzmann law, which states that the rate of radiation heat transfer is proportional to the fourth power of the absolute temperature (T) of the object. Mathematically, it can be expressed as:

Q = εσAT^4

Where:
Q is the rate of heat transfer by radiation
ε is the emissivity of the sphere (a measure of how well it radiates compared to a perfect black body, typically between 0 and 1)
σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4)
A is the surface area of the sphere

Proportional Cooling
Since the rate of heat transfer by radiation is proportional to the fourth power of the temperature, we can conclude that the rate of cooling of the sphere is also proportional to the fourth power of the temperature. This means that as the temperature decreases, the rate of cooling also decreases.

Temperature Dependence
The temperature dependence of the cooling rate can be explained by considering the energy balance of the sphere. As the sphere cools down, the temperature difference between the sphere and its surroundings decreases. This results in a decrease in the rate of heat transfer by radiation, leading to a slower cooling rate.

Conclusion
In summary, the rate of cooling of a solid sphere by radiation is proportional to the fourth power of the temperature. This relationship is governed by the Stefan-Boltzmann law, which describes the rate of radiation heat transfer. As the temperature decreases, the rate of cooling also decreases, leading to a slower cooling process.

Rc is directly proportional to T^4