A uniform electric field is increasing at \(1.5(\mathrm{V} / \mathrm{m}) / \mu\) s. Find the displacement current through a \(1-\mathrm{cm}^{2}\) area perpendicular to the field.

James Clerk Maxwell, in the 19th century, contributed significantly to the field of electromagnetism by formulating a set of equations known as Maxwell's equations. These equations describe how electric and magnetic fields are generated by charges, currents, and changes of these fields. There are four equations in total, but in the context of our exercise, we're particularly interested in the one that relates the displacement current to the electric field.

The concept of the displacement current is crucial in making Maxwell's equations consistent. It is defined as a form of fictitious current that is thought to exist in a space where the electric field and the electric charge are changing with time. Displacement current allows for the continuity of current in a circuit, even through a capacitor where there is no actual movement of charge from one plate to another. Maxwell's inclusion of the displacement current term was critical in realizing that light is an electromagnetic wave—understanding which paved the way to modern electromagnetic theory.

An electric field represents a region around a charged particle, or within a changing magnetic field, where a force would be exerted on other charges. This field can be visualized as a pattern of imaginary lines emanating from a positive charge and terminating on a negative charge. The strength of an electric field is described by the amount of force per unit charge it would exert on a positive test charge placed within the field.

In our exercise, the electric field is not static; rather, it's changing over time. This changing field is the source of what we call a displacement current. To understand how a changing electric field contributes to the creation of a displacement current, it helps to think of the electric field lines as being elastic. As these lines stretch with time due to the increasing field strength, the associated displacement current also grows in magnitude, simulating the effects of a real current in the associated space, despite the absence of actual charge carriers.

Permittivity of the vacuum, often denoted by the symbol \( \epsilon_{0} \) is a physical constant that describes how an electric field affects, and is affected by, a vacuum. Essentially, it's a measure of the ability of a vacuum to permit electric field lines to permeate through it. Its exact value is approximately \( 8.85 \times 10^{-12} \text{ F/m} \), where F/m stands for farads per meter.

The permittivity of vacuum plays a vital role when considering the interaction between charges or in the propagation of electromagnetic waves through space. In the context of our exercise, it is a proportionality constant that relates the electric field to the displacement current. The higher the permittivity of vacuum, the more the space 'allows' for the electric field lines to permeate through it and, therefore, the greater the effect an electric field will have—in this case, resulting in a higher displacement current for a given rate of change of the electric field.