Two solid objects, \(A\) and \(B\), are in contact. In which case will thermal energy transfer from \(\mathrm{A}\) to \(\mathrm{B} ?\) a) \(\mathrm{A}\) is at \(20{ }^{\circ} \mathrm{C},\) and \(\mathrm{B}\) is at \(27{ }^{\circ} \mathrm{C}\) b) \(A\) is at \(15^{\circ} \mathrm{C},\) and \(\mathrm{B}\) is at \(15^{\circ} \mathrm{C}\). c) \(\mathrm{A}\) is at \(0{ }^{\circ} \mathrm{C},\) and \(\mathrm{B}\) is at \(-10{ }^{\circ} \mathrm{C}\).

Heat conduction is a mode of transfer of thermal energy from a region of higher temperature to a region of lower temperature within an object or between objects in direct contact. The fundamental principle guiding heat conduction is that molecules at a higher temperature have more kinetic energy. When these molecules come in contact with cooler molecules, they transfer some of their energy to them, which results in the overall spread of heat. The transfer continues until thermal equilibrium is reached, meaning the objects involved achieve the same temperature.

In the context of the exercise, when solid object A is at a higher temperature than solid object B (case c), the kinetic energy of the molecules in object A is higher. As they are in contact, this energy is transferred to the cooler molecules of object B through the process of heat conduction.

The temperature difference between two objects is the driving force for heat conduction. When two objects are at different temperatures and come into contact, the temperature difference creates a 'thermal gradient.' This gradient is what causes thermal energy to move from the hotter object to the cooler one. The greater the temperature difference, the faster the rate of thermal energy transfer.

As afore mentioned in the exercise, case b demonstrates where both objects A and B have no temperature difference. Without a temperature gradient, there is no driving force for the heat to flow, which is why no heat conduction takes place in this scenario. It's critical to remember that thermal energy naturally flows from higher to lower temperatures until it is balanced or equalized across the objects involved.

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. There are four laws of thermodynamics, but the concept of energy transfer is rooted in the first and second laws. The first law, also known as the law of energy conservation, states that energy can neither be created nor destroyed, only transformed from one form to another. This concept is demonstrated with the transfer of thermal energy in conduction.

The second law of thermodynamics states that total entropy, or chaos, of an isolated system can never decrease over time. It suggests that thermal energy transfer from a hot object to a cold one (such as object A to object B when A is warmer), without the input of work, is a spontaneous process that increases the overall entropy of the system. These fundamental principles of thermodynamics are essential to understanding the flow of heat and energy distribution in any system, including the exercise presented.