The polarization \(P\) of a dielectric material positioned within a parallel- plate capacitor is to be \(4.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2}\) (a) What must be the dielectric constant if an electric field of \(10^{5} \mathrm{V} / \mathrm{m}\) is applied? (b) What will be the dielectric displacement \(D ?\)

Polarization in a material occurs when an electric field causes a shift in the position of charges, leading to a separation of positive and negative charges. This creates an induced electric dipole moment in the material. In the context of a dielectric material, which is an insulator, placed in an electric field, the field tends to align the positive and negatively charged particles in opposite directions. This alignment results in an internal electric field that opposes the applied field.
Imagine tiny magnets inside the dielectric that all line up when an external magnetic field is applied; similarly, electric dipoles in a dielectric line up in response to an electric field. The degree of polarization, represented by the symbol \(P\), is defined as the electric dipole moment per unit volume. In the given problem, the polarization is provided as \(4.0 \times 10^{-6}\) Coulombs per square meter (\(\mathrm{C/m^2}\)).
However, the polarization does not just depend on the material's inherent properties, but also on the strength of the external electric field. Enhanced understanding comes from visualizing this process on the atomic level--picture each molecule with a slightly stretched distribution of charge, aligning within the field to produce the overall polarized effect. This microscopic behavior of polarized charges is what gives rise to the macroscopic properties of the dielectric material.

The electric field, commonly denoted as \(E\), is a vector field surrounding any electrically charged object. It represents the force per unit charge that would be exerted on a positive test charge placed in the field. The strength of the electric field is measured in volts per meter (\(\mathrm{V/m}\)).

In our exercise, the electric field of \(10^5 \mathrm{V/m}\) is applied across the dielectric material. The electric field exerts forces on the charges within the dielectric, trying to move the positive charges in the direction of the field and the negative charges in the opposite direction. This establishes the basis for the polarization of the dielectric. Understanding the electric field's role is crucial in grasping how the dielectric constant and dielectric displacement are influenced.

Visualizing the Electric Field

Imagine a grid with arrows representing the direction and strength of the field; the longer the arrow, the stronger the field. This visualization helps in understanding how the force would act on charges in different regions of the field. The field itself is an essential concept in electromagnetism and plays a pivotal role in various phenomena such as electrostatics, current flow, and even in the functioning of capacitors.

Dielectric displacement, denoted as \(D\), reflects the total electric charge displacement per unit area in a dielectric material when it is subjected to an electric field. It encompasses the contributions from the free charges supplied by the external field and the bound charges associated with the material's polarization.

Mathematically, dielectric displacement is given by adding the product of the permittivity of free space (\(\epsilon_0\)) and the electric field (\(E\)) to the polarization (\(P\)). When we calculate the dielectric displacement for the given exercise, we're essentially quantifying the combined effect of the field's forces and the dielectric's response to those forces.

Significance of Dielectric Displacement

In the context of capacitors, dielectric displacement helps determine how much electric field 'passes through' the dielectric. It's a key parameter for understanding how dielectrics enhance the capacitor's storage capacity. It also provides insight into the overall behavior of dielectric materials under the influence of electric fields, which is crucial for designing materials with specific electrical properties. The calculation from our exercise indicates that the dielectric material's displacement is approximately \(4.8854 \times 10^{-6} \mathrm{C/m^2}\), which is a direct measure of the material's ability to store electrical energy.