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The Ampère-Maxwell Law is an extended version of Ampère's Law, accommodating time-varying electric fields. This law is pivotal in the case of charging and discharging capacitors because it includes the concept of displacement current. The displacement current is not a real current like the flow of electrons, but it allows us to apply Ampère's Law to capacitors where no actual charge is moving through the dielectric. The law can be mathematically expressed as: \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} \]This equation accounts for both current through a conductor and time-varying electric fields through the term \( \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} \), which represents the displacement current \( I_d \). This term is crucial for understanding how electric and magnetic fields can exist and propagate in empty space, providing a cornerstone for electromagnetic wave theory.

A parallel-plate capacitor consists of two large plates separated by a certain distance. In this setup, the plates store electric charges, and an electric field is formed between them. When discussing charging, it's important to understand that while electrons accumulate on the plates, within the space between them exists an electric field. This results in a concept called 'electric flux', which measures how much of the electric field passes through a given area. The capacity of a capacitor to store charge is quantified by its capacitance, which depends on the area of the plates and the distance between them. A key point here is that displacement current plays a role between the plates during the charging phase, keeping the relationship with magnetic fields intact. This non-movement of charges answers how a magnetic field can be present even when no actual current flows through the dielectric medium of the capacitor.

The magnetic field, symbolized by \( B \), is a vector field that represents the influence magnetic forces will experience in a region. In the context of a charging capacitor, the magnetic field is crucial because it accounts for the effects of both conduction and displacement currents. Using the Ampère-Maxwell Law, we see how a changing electric field between capacitor plates induces a magnetic field along loops encircling the plates, even when the current isn't carried by moving charges. In simple terms, the magnetic field arises around the loop due to the equivalence of the changing electrical environment between the plates. These concepts help us comprehend much more complex phenomena in nature, such as how electromagnetic waves, like light, are capable of energy transfer through the vacuum of space.