# Potential energy of external force

The potential energy is a property of a system, being defined only for internal conservative forces. From the work energy theorem, we have $$W_{ext} + W_{\text{internal cons. forces}} = \Delta K$$ We define the (change in) potential energy as $$\Delta U = -W_{icf}$$ giving us $$W_{ext} = \Delta K + \Delta U$$ My question is, why leave out the external force? If the external forces are also conservative, then they can also store some potential energy. We can take that term and define a potential energy change for the external force too can't we? For example, if we had a ball and the earth, we could take the ball as the system, and due to the external force of gravity $$mg$$ it would have a potential energy $$mgh$$ with it.

You can think of a change in the potential energy of a system as being work done by external forces on the system. However, some work done by external forces may also go into changing the kinetic energy of the system - this is why $$W_{ext} = \Delta K + \Delta U$$. So we want to restrict the change in potential energy to being work done by external forces that stores energy internally within the system. But a less ambiguous way to express this is to say that the change in potential energy is the work done by internal forces in opposition to external forces (as a result of Newton's Third Law).

In your ball example, it is the ball and the Earth that are the system, and the potential energy is the energy stored in the gravitational attraction between them, which is an internal force. The work done by external forces on the system may be greater than $$mgh$$ if it increases the kinetic energy of the system - or it may be less than $$mgh$$ if the kinetic energy of the system decreases.

We define the (change in) potential energy as $$\Delta U = -W_{icf}$$

If $$W_{icf}$$ stands for internal conservative force only, then the definition is incorrect. $$W$$ can be the work done by internal or external forces, as long as the forces are conservative. So it should be written

$$\Delta U=-W_c$$

Whether the conservative forces are internal or external to a system depends on how the system is defined. But regardless of whether the force is internal or external, the change in potential energy depends only on the work done by conservative forces.

Hope this helps.