An imperfect insulator of conductivity \(\sigma\) and relative permittivity \(\varepsilon_{r}\) fills the space between the plates of a parallel plate capacitor. An alternating voltage \(V_{0} \sin \omega t\) is applied across the plates. Compare the magnitudes of the displacement and conduction current densities. Show that for many actual materials the displacement current is negligible for frequencies in the radio wave region.

Electrical conductivity, often denoted with the symbol \(\sigma\), is essentially a measure of a material's ability to conduct electric current. When an electric field is applied to a material, it causes the charge carriers, which could be electrons or ions, to move, thus creating a current. The larger the conductivity, the easier it is for the current to flow through the material.

Let's relate this to the exercise at hand. In the context of the parallel plate capacitor filled with an imperfect insulator, conductivity plays a critical role in determining the conduction current density \(\vec{J}_{cond}\). According to Ohm's law, the conduction current density is directly proportional to the electric field \(\vec{E}\) and the material’s conductivity. The formula is expressed as \(\vec{J}_{cond} = \sigma \vec{E}\). The higher the conductivity, the greater the conduction current density for a given electric field.

Understanding the concept of electrical conductivity helps in realizing why materials with a higher \(\sigma\) are typically better conductors of electricity and why those with lower \(\sigma\) are considered insulators or semiconductors. It's also significant in the solution to our exercise as it helps determine the dominance of conduction current in certain materials, particularly at lower frequency ranges, compared to displacement current.

Relative permittivity, indicated by \(\varepsilon_{r}\), is a dimensionless parameter that signifies how much the electric field \(\vec{E}\) is reduced inside a material compared with a vacuum. It's often called the 'dielectric constant,' and it's a metric of how easily a material can be polarized by the imposition of an external electric field. For a vacuum, this value is 1, which forms the baseline against which other materials are assessed. Materials with high relative permittivity can store more electrical energy at a given electric field strength than those with lower values.

In our parallel plate capacitor scenario, \(\varepsilon_{r}\) is vital for determining the displacement field \(\vec{D}\), which is related to the electric field via the equation \(\vec{D} = \varepsilon_{0} \varepsilon_{r} \vec{E}\), where \(\varepsilon_{0}\) represents vacuum permittivity. The exercise leads us to understand how the displacement current density \(\vec{J}_{disp}\) stems from the rate of change of the electric displacement field and how it's influenced by \(\varepsilon_{r}\). The greater the relative permittivity, the larger the displacement current density for a given rate of change in the electric field.

With the concept of relative permittivity, we can begin to understand why different materials may have different responses to the same electromagnetic field and why this affects the displacement current in a parallel plate capacitor.

A parallel plate capacitor is a simple yet fundamental electrical component that consists of two conductive plates separated by a distance \(d\), with an insulating material (dielectric) sandwiched between them. When a voltage is applied across these plates, an electric field is established, and the capacitor stores energy in the form of an electrostatic field.

The strength of the electric field \(\vec{E}\) in a parallel plate capacitor with a voltage \(V\) applied is given by \(\vec{E} = \frac{V}{d}\). In our exercise, the voltage is alternating, taking the form \(V_{0} \sin \omega t\), which will cause both the electric field and displacement field to vary with time. The displacement current density \(\vec{J}_{disp}\) in the capacitor arises from the change in the displacement field over time, and this is particularly essential when dealing with alternating currents or electromagnetic waves, as it aids in understanding the complete electrical behavior of capacitive circuits.

The parallel plate capacitor concept is the cornerstone in the step-by-step solution for comparing displacement and conduction current densities. With this, students can appreciate the roles of these different currents and the conditions under which one may be more significant than the other, thereby deepening their understanding of the electrodynamic processes in AC circuits and capacitive components.