Two large parallel plates with a narrow separation carry...

Two large parallel plates with a narrow separation carry fixed, uniform surface charge densities $\sigma$ and $-\sigma$ respectively. Two large slabs with identical surface areas as the plates and permittivities $\varepsilon_1$ and $\varepsilon_2$ are then slotted between the plates such that the gap within the plates is filled completely. If the slab with permittivity $\varepsilon_1$ is closer to the plate with surface charge density $\sigma$, determine the electric displacements, electric fields and polarizations in the two slabs. Finally, determine the bound charges everywhere.

Key Concepts

Electric Displacement Field

This concept describes a field (denoted as D) that directly relates to the free charges present in a system. It is central to Gauss’s law in dielectrics where the total flux of D through a closed surface equals the enclosed free charge, regardless of any bound charges residing within the material. It is used to simplify problems involving dielectric materials by focusing on free charges, with the relation D = ?E bridging between the displacement and the electric field.

Electric Field

The electric field (E) represents the force per unit charge exerted on a test charge in space. In dielectric materials, the relationship between the electric field and the displacement field is modified by the medium’s permittivity via the relation D = ?E. Dielectric materials often reduce the effective electric field due to polarization effects, altering the field distribution compared to vacuum.

Polarization

Polarization (P) refers to the dipole moment per unit volume induced in a dielectric when an electric field is applied. It quantifies the displacement of bound charges within the material, and is related to the electric susceptibility and permittivity of the material. Polarization is fundamental in the formation of bound charges at interfaces and within the volume of dielectric materials.

Bound Charges

Bound charges are charges that arise in dielectrics as a consequence of polarization. Unlike free charges that are externally imposed, bound charges result from the internal separation of charges in response to an applied field. They can appear as volumetric bound charge density (given by -?·P) or as surface bound charge density (given by P·n? at interfaces), and play a critical role in determining the overall electric field in and around dielectric materials.

Interface Conditions in Dielectrics

At the boundary between different dielectric materials or between a dielectric and a conductor, specific continuity conditions apply. The normal component of the electric displacement field (D) is continuous across an interface in the absence of free surface charge, while the tangential component of the electric field (E) must be continuous. These conditions are essential for determining field distributions in systems with multiple dielectric media.

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