When can the potential be in a negative value especially when it come to atoms and bind formation?
Potential energy can have negative value whenever you like, since the formula includes the integration constant that can be set to an arbitrary value. That's why we usually only concern ourselves with the difference between potential energies at some two points in the field(ΔU).
If you want to know when is the difference in PE negative, then it depends on the direction of the force and the direction of displacement.
When the force is attractive, like with gravity, strong nuclear force, or Coulomb force for odd charges, AND the displacement is towards the centre of the field, then the ΔU is negative. Meaning, there is a release of energy as the objects comprising the system get closer.
If the displacement is away from the centre of the field, the ΔU is positive - it requires input of energy to move the objects apart.
In a repulsive force field(e.g., like charges repelling) it is the other way around.
Yeah yeah I got your point that potential energy is an arbitrary value, but but if we consider a metal is out of a Magnet's magnetic field it's potential should be zero but saying the potential is zero is a little bit misleading as it could be in the field and it's potential is still zero as it's an arbitrary value, how can I precisely say that the body is out of the field in terms of potential energy ???
Just set the potential energy to be zero at infinity.
What is meant by zero at infinity ?
Rigorously, this means that one sets the integral used to define potential energy equal to zero at the limit as distance from some chosen origin goes to infinity. I think the simplest example is gravitational potential energy. Have you taken calculus? You don't really need calculus to get this idea, but the argument is a bit more "hand-wavey". Using gravity, you can find a value for ##r## that will make the potential energy arbitrarily close to 0. You just need to choose larger and larger values of ##r##. In the limit, one says that the potential energy is "zero at infinity" to mean that one can make the potential arbitrarily close to zero by taking a larger ##r##.
You can apply the same idea to magnetic potential energy, but it is a bit trickier because the angle of the dipole also matters.
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