Consider a thermometer with thermal resistance and thermal capacitance having a temperature . The thermometer is immediately and totally submerged in a liquid bath of temperature :
1. Differential Equation: Formulate the differential equation governing the temperature of the thermometer in response to the sudden submersion in the liquid bath.
2. Initial Condition: Specify any initial condition for the differential equation, considering the state of the thermometer before submersion.
3. Solving Process: Outline the steps involved in solving the differential equation to determine the temperature as a function of time.
4. Physical Interpretation: Explain the physical significance of the solution, addressing how the temperature of the thermometer evolves over time in the new environment.
Provide a comprehensive overview of the scenario, covering the differential equation, initial condition, solving process, and the physical interpretation of the solution.