Understanding the Problem:
The problem states that a fin will be necessary and effective only when the thermal conductivity of the fin material (k) is large and the convective heat transfer coefficient between the fin surface and the environment temperature (h) is small. We need to determine why this combination of parameters is necessary for the fin to be effective.

Explanation:

Role of Thermal Conductivity (k):
The thermal conductivity (k) of a material determines how well it can conduct heat. A high thermal conductivity means that the material can transfer heat efficiently. In the context of a fin, a large value of k implies that the fin can conduct heat from its base to its tip effectively.

Role of Convective Heat Transfer Coefficient (h):
The convective heat transfer coefficient (h) represents the effectiveness of heat transfer between the fin surface and the surrounding environment. It depends on various factors such as the nature of the fluid, flow conditions, and geometry of the fin. A small value of h indicates poor heat transfer between the fin surface and the environment.

Reasoning:
To understand why a large k and a small h are necessary for an effective fin, let's consider the heat transfer process in a fin.

Heat is transferred from the base of the fin to the tip through conduction. The fin's purpose is to increase the surface area available for heat transfer and enhance the heat dissipation from the system. The effectiveness of the fin is determined by how efficiently it can transfer heat from the base to the tip and then dissipate it into the surrounding environment.

If the thermal conductivity (k) of the fin material is small, it means that the fin cannot conduct heat effectively from the base to the tip. This would result in poor heat transfer along the length of the fin, making it less efficient in dissipating heat.

On the other hand, if the convective heat transfer coefficient (h) between the fin surface and the environment is large, it means that there is efficient heat transfer between the fin and the surrounding fluid. In this case, the convection process becomes dominant, and the fin's ability to conduct heat internally becomes less important.

Conclusion:
Based on the above reasoning, we can conclude that a fin will be necessary and effective when the thermal conductivity (k) is large and the convective heat transfer coefficient (h) is small. In this scenario, the fin can effectively conduct heat from the base to the tip and efficiently transfer it to the surrounding environment through convection.