Derivation of Newton's Law of Cooling from Stefan's Law of Radiation:

Newton's Law of Cooling describes how the temperature of an object changes over time when it is in contact with a cooler surrounding medium. This law states that the rate of heat loss of an object is proportional to the temperature difference between the object and its surroundings.

Stefan's Law of Radiation:
Stefan's Law of Radiation states that the rate at which an object radiates energy is directly proportional to the fourth power of its temperature. Mathematically, it can be expressed as:
\[ P = \sigma \cdot A \cdot T^4 \]

Where:
- \( P \) is the power radiated
- \( \sigma \) is the Stefan-Boltzmann constant
- \( A \) is the surface area of the object
- \( T \) is the temperature of the object

Deriving Newton's Law of Cooling:
1. Consider a body at a temperature \( T_1 \) in a surroundings at a temperature \( T_0 \).
2. According to Stefan's Law, the rate of radiated energy by the body is \( \sigma \cdot A \cdot T_1^4 \).
3. The rate at which the body loses heat to its surroundings is given by Newton's Law as \( k \cdot (T_1 - T_0) \), where \( k \) is the cooling constant.
4. Equating the two rates of energy loss:
\[ \sigma \cdot A \cdot T_1^4 = k \cdot (T_1 - T_0) \]
5. Rearranging the equation yields:
\[ \frac{dT_1}{dt} = - k \cdot (T_1 - T_0) \]
6. This is the differential equation for Newton's Law of Cooling, where \( \frac{dT_1}{dt} \) represents the rate of change of temperature of the body with respect to time.
7. By solving this equation, we can obtain the temperature of the body at any given time, following the principles of Newton's Law of Cooling.