Is potential energy calculated between a system? More specifically, if we say 'potential energy of a ball with respect to earth', does it mean that the Earth + the ball is a system?
Answer
Potential energy is the energy due to configuration of the system.
If you keep three charges very, very far from each other, then the potential energy of the system is very effectively zero.
But when you bring them close together to a specified coordinate, then the potential energy of the system increases from $0$ to a positive value given by $$U= \frac{1}{2} \sum_{j=1}^3\sum_{k\ne j} \frac{1}{4\pi\epsilon_0} \cdot \frac{q_j q_k}{r_{jk}}\; .$$
You can't say some of it belongs to charge no.1 & some belongs to the second charge or so.
Potential energy is the property of the system rather than of either particles or entities that constitute the system.
So, potential energy gained by lifting the ball against gravity is not the potential energy of the ball but solely belongs to the ball-Earth system. But since, the Earth is mammoth compared to the petty ball, you can say most of the potential energy of this system gets converted to kinetic energy of the ball.
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