Final answer:
Displacement current is necessary to make Ampère's law consistent for a charging capacitor, as it accounts for the magnetic field generated by changing electric fields. Maxwell extended Ampère's law by including a term for displacement current, leading to a consistent electromagnetic theory and the understanding of electromagnetic waves.
Step-by-step explanation:
To show that displacement current is necessary to make Ampère's law consistent for a charging capacitor, we need to understand how Ampère's law relates to magnetic fields and electric currents. Originally, Ampère's law stated that the magnetic field in a loop of wire is proportional to the electric current that passes through the loop. However, this did not account for the effects seen when electric fields change over time, such as in the case of a charging capacitor where there is no actual movement of charges in the space between the capacitor plates.
Maxwell extended Ampère's law by adding a term for displacement current, which considers the changing electric field as a form of current even where no actual (real) current exists. This amendment is necessary for the consistency of electromagnetic theory, specifically when dealing with scenarios where electric fields vary. It led to the symmetrical formulation in what is known as the Ampère-Maxwell law, which is part of Maxwell's equations. These equations describe all electromagnetic phenomena and are the basis for understanding electromagnetic waves.
Displacement current is thus as crucial in producing a magnetic field as is the real electrical current. The importance of displacement current extends beyond just the charging capacitor; it is fundamental to the symmetry of Maxwell's equations and underlies the theory of electromagnetic wave propagation, where changing electric fields produce magnetic fields and vice versa.