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Ampere's law produces inconsistent magnetic field values when applied to different surfaces of an Amperian loop around a charging capacitor because current does not pass through all surfaces equally. The law assumes steady current, but a capacitor's charging creates a time-varying electric field that Ampere's law in its original form cannot account for, revealing a fundamental limitation in the classical formulation.

Displacement current is a fictitious current introduced by Maxwell to account for changing electric fields. It is defined as the product of the permittivity of free space and the time rate of change of electric flux through a surface. Unlike real conduction current, displacement current produces a magnetic field even when no actual charge movement occurs, enabling Ampere's law to work in all situations.

As a capacitor charges, the electric field between its plates increases, causing the electric flux through the bulging surface of an Amperian loop to increase with time. The value of electric flux can be expressed in terms of the charge accumulating on the capacitor plates, directly linking the time-varying charge to the changing electromagnetic field.

The Ampere-Maxwell law, also called the generalized Ampere's law, modifies the original Ampere's law by adding a displacement current term to the real conduction current. This modification ensures the law works independently of the surface chosen for measurement and accounts for magnetic fields produced by changing electric fields, making it valid in all electromagnetic situations.

Maxwell introduced displacement current to resolve the inconsistency in Ampere's law when applied to time-varying electric fields. He predicted that a changing electric field produces a non-zero magnetic field between capacitor plates through this fictitious current, providing a unified framework that explains how magnetic fields arise from both real currents and changing electric fields.

Real conduction current involves actual charge movement through a conductor, while displacement current arises from a time-varying electric field with no charge motion. Displacement current is analogous to real current in Ampere's law and produces magnetic fields identically, but it can generate magnetic fields even when no real current is present, making it essential for understanding electromagnetic phenomena.

By including the displacement current term, the modified Ampere's law accounts for all sources of magnetic fields—both real currents and changing electric fields. This ensures that regardless of which surface bounds an Amperian loop, the total contribution from conduction and displacement currents remains constant, making the law surface-independent and universally applicable.