Question

What is the relationship between electric field and electric displacement?

Answer 1

The electric field \(\mathbf{E}\) and the electric displacement field \(\mathbf{D}\) are related concepts in electromagnetism, particularly in the context of materials. Here’s how they are related:

1. **Electric Field (\(\mathbf{E}\))**: This is a vector field that represents the force experienced by a unit positive charge placed in a given region of space. It is defined as:
   \[
   \mathbf{E} = \frac{\mathbf{F}}{q}
   \]
   where \(\mathbf{F}\) is the force on a test charge \(q\).

2. **Electric Displacement Field (\(\mathbf{D}\))**: This field accounts for the effect of free and bound charge in a material. It is defined in terms of the electric field and the polarization of the material. In a linear, isotropic dielectric material, it is given by:
   \[
   \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}
   \]
   where \(\varepsilon_0\) is the permittivity of free space and \(\mathbf{P}\) is the polarization vector of the material. In the case of linear dielectrics, this can be simplified to:
   \[
   \mathbf{D} = \varepsilon \mathbf{E}
   \]
   where \(\varepsilon\) is the permittivity of the material.

The relationship between \(\mathbf{E}\) and \(\mathbf{D}\) is crucial for understanding how electric fields interact with materials. In free space or in materials without polarization, \(\mathbf{D}\) and \(\mathbf{E}\) are directly proportional, with the proportionality constant being the permittivity of the material. In materials with significant polarization, \(\mathbf{D}\) includes contributions from both the free charges and the bound charges due to polarization.

Answer 2

The relationship between the electric field (\( \mathbf{E} \)) and electric displacement field (\( \mathbf{D} \)) is fundamental in electromagnetism. They are related through the permittivity of the medium. Specifically:

\[ \mathbf{D} = \epsilon \mathbf{E} \]

Here:
- \( \mathbf{D} \) is the electric displacement field.
- \( \mathbf{E} \) is the electric field.
- \( \epsilon \) is the permittivity of the medium, which can be expressed as \( \epsilon = \epsilon_0 \epsilon_r \), where \( \epsilon_0 \) is the permittivity of free space and \( \epsilon_r \) is the relative permittivity (dielectric constant) of the material.

In a dielectric medium, \( \mathbf{D} \) accounts for both the free charges and the bound charges due to polarization. The electric displacement field is particularly useful in situations where the material's polarization affects the overall electric field.