The reason why it is not possible to define potential energy for a magnetic field in closed paths is rooted in the fundamental nature of magnetic fields and their interaction with charged particles.
In electrostatics, electric potential energy can be defined for a conservative electric field. When a positive charge is moved in an electric field, the work done on the charge is path-independent, meaning it depends only on the initial and final positions of the charge. This work done per unit charge is defined as the change in electric potential, or voltage.
However, magnetic fields are different from electric fields in a crucial aspect: they do no work on charged particles in a vacuum. The force experienced by a charged particle moving through a magnetic field is always perpendicular to its velocity, resulting in no net work being done on the particle. As a consequence, there is no concept of potential energy associated with a magnetic field.
This behavior is encapsulated in one of Maxwell's equations, specifically Ampere's law with no electric current:
∇ × B = μ₀ε₀∂E/∂t
The right-hand side of this equation involves the time rate of change of the electric field, implying that magnetic fields are intimately connected to changing electric fields. In other words, magnetic fields are typically generated by moving charges or changing electric fields.
However, it is important to note that although there is no potential energy associated with a magnetic field itself, magnetic fields can still store energy indirectly. For example, in an inductor (a coil of wire), when a current flows through it, a magnetic field is generated. The energy stored in the magnetic field is actually stored in the form of electromagnetic potential energy within the electric field generated by the changing current.
In summary, the absence of potential energy associated with a magnetic field in closed paths is a consequence of the perpendicular force between magnetic fields and charged particles, resulting in no net work being done on the particles by the magnetic field. However, magnetic fields can store energy indirectly within the associated electric fields or through their interactions with currents.