91Ó°ÊÓ

A dielectric constant, also known as relative permittivity, is a dimensionless measure that describes how easily a material can become polarized by an electric field. In other words, it shows how well a material can store electrical energy when placed in an electric field. The dielectric constant is typically denoted by the symbol \( K \) or \( \varepsilon_r \).

One important formula used in problems involving dielectrics is \( E = \frac{E_0}{K} \), where \( E_0 \) is the electric field without the dielectric, and \( E \) is the electric field with the dielectric present. By rearranging this formula, you can find the dielectric constant \( K \) by calculating:

• \( K = \frac{E_0}{E} \)

In our example exercise, the electric field without and with the dielectric is given, allowing us to compute the dielectric constant as \( K = \frac{3.20 \times 10^5}{2.50 \times 10^5} = 1.28 \).

This value indicates how much the dielectric material reduces the electric field compared to a vacuum.

Charge density represents how the electric charge is distributed over a surface, a volume, or along a line. When a dielectric is present, the charge density can include contributions from both the free charges and those induced by polarization.

For dielectric materials, the total surface charge density \( \sigma \) is often considered to be the sum of the free surface charge density \( \sigma_f \) and the polarization surface charge density \( \sigma_p \).

In the provided exercise, to find the free surface charge density, the equation \( \sigma_f = \epsilon_0 E_0 \) is used, where \( \epsilon_0 \) is the permittivity of free space (approximately \( 8.85 \times 10^{-12} \text{ F/m} \)). Calculating, we find that \( \sigma_f = 2.832 \times 10^{-6} \text{ C/m}^2 \).

However, because the total surface charge density calculated through the electric displacement field \( D \) matches \( \sigma_f \), the polarization charge density \( \sigma_p \) is zero.

The electric displacement field, denoted as \( D \), is a vector quantity that accounts for both free and bound charges within a dielectric. It's used in Maxwell's equations and is crucial when analyzing electric fields in materials.

The displacement field \( D \) is defined as:

• \( D = \epsilon E \)

where \( \epsilon = \epsilon_0 K \) is the permittivity of the dielectric, \( E \) is the electric field with dielectric, and \( \epsilon_0 \) is the permittivity of free space. For free surface charges, \( D \) equates to the surface charge density \( \sigma_f \).

In the scenario provided, we computed \( D = \epsilon E = \epsilon_0 K E \) and obtained \( D = 2.832 \times 10^{-6} \text{ C/m}^2 \), reinforcing the result that the non-polarized charge density is entirely free surface charge, ensuring \( \sigma_p = 0 \text{ C/m}^2 \).

Understanding \( D \) helps clarify how electric fields interact with dielectrics and exhibits the behavior of materials under electric influence.