What is the electric field inside a dielectric?
2 Answers
The electric field inside a dielectric material can be understood by considering how the dielectric responds to an external electric field. Here’s a detailed explanation:
### 1. **Understanding Dielectrics:**
A dielectric is an insulating material that, when exposed to an electric field, becomes polarized. This means that the positive and negative charges within the dielectric material are displaced in response to the external electric field, but the material does not conduct electricity.
### 2. **Electric Field Without Dielectric:**
In a vacuum or air (which can be approximated as a vacuum for many practical purposes), the electric field \(\mathbf{E}_0\) due to a charge distribution or between charged plates can be described by Coulomb's law or Gauss's law.
### 3. **Introduction of a Dielectric:**
When a dielectric material is introduced into the space between the plates of a capacitor or in any region of an electric field, it modifies the electric field in the following way:
- **Polarization:** The dielectric material becomes polarized, meaning dipole moments are induced in the material. The polarization \(\mathbf{P}\) is the vector field representing the dipole moment per unit volume.
- **Electric Displacement Field \(\mathbf{D}\):** The electric displacement field \(\mathbf{D}\) is related to the external electric field \(\mathbf{E}_0\) and the polarization \(\mathbf{P}\). It is given by:
\[
\mathbf{D} = \epsilon_0 \mathbf{E}_0 + \mathbf{P}
\]
where \(\epsilon_0\) is the permittivity of free space.
- **Relationship between \(\mathbf{D}\) and \(\mathbf{E}\):** The relationship between the electric field \(\mathbf{E}\) inside the dielectric and \(\mathbf{D}\) is:
\[
\mathbf{D} = \epsilon \mathbf{E}
\]
where \(\epsilon\) is the permittivity of the dielectric material, which is greater than \(\epsilon_0\) and can be expressed as:
\[
\epsilon = \epsilon_r \epsilon_0
\]
with \(\epsilon_r\) being the relative permittivity or dielectric constant of the material.
### 4. **Electric Field Inside the Dielectric:**
Combining the equations above, we can derive the electric field \(\mathbf{E}\) inside the dielectric:
\[
\mathbf{E} = \frac{\mathbf{D}}{\epsilon}
\]
Substituting \(\mathbf{D} = \epsilon_0 \mathbf{E}_0 + \mathbf{P}\) and considering \(\mathbf{P} = \epsilon_0 (\epsilon_r - 1) \mathbf{E}\), we get:
\[
\mathbf{E} = \frac{\mathbf{E}_0}{\epsilon_r}
\]
### 5. **Summary:**
In summary, the electric field inside a dielectric material is reduced compared to the electric field in a vacuum or air. The reduction factor is the relative permittivity \(\epsilon_r\) of the dielectric. If \(\mathbf{E}_0\) is the electric field in a vacuum or air, the electric field \(\mathbf{E}\) inside the dielectric is given by:
\[
\mathbf{E} = \frac{\mathbf{E}_0}{\epsilon_r}
\]
This means that the presence of a dielectric material decreases the electric field within it relative to what it would be without the dielectric.
### 1. **Understanding Dielectrics:**
A dielectric is an insulating material that, when exposed to an electric field, becomes polarized. This means that the positive and negative charges within the dielectric material are displaced in response to the external electric field, but the material does not conduct electricity.
### 2. **Electric Field Without Dielectric:**
In a vacuum or air (which can be approximated as a vacuum for many practical purposes), the electric field \(\mathbf{E}_0\) due to a charge distribution or between charged plates can be described by Coulomb's law or Gauss's law.
### 3. **Introduction of a Dielectric:**
When a dielectric material is introduced into the space between the plates of a capacitor or in any region of an electric field, it modifies the electric field in the following way:
- **Polarization:** The dielectric material becomes polarized, meaning dipole moments are induced in the material. The polarization \(\mathbf{P}\) is the vector field representing the dipole moment per unit volume.
- **Electric Displacement Field \(\mathbf{D}\):** The electric displacement field \(\mathbf{D}\) is related to the external electric field \(\mathbf{E}_0\) and the polarization \(\mathbf{P}\). It is given by:
\[
\mathbf{D} = \epsilon_0 \mathbf{E}_0 + \mathbf{P}
\]
where \(\epsilon_0\) is the permittivity of free space.
- **Relationship between \(\mathbf{D}\) and \(\mathbf{E}\):** The relationship between the electric field \(\mathbf{E}\) inside the dielectric and \(\mathbf{D}\) is:
\[
\mathbf{D} = \epsilon \mathbf{E}
\]
where \(\epsilon\) is the permittivity of the dielectric material, which is greater than \(\epsilon_0\) and can be expressed as:
\[
\epsilon = \epsilon_r \epsilon_0
\]
with \(\epsilon_r\) being the relative permittivity or dielectric constant of the material.
### 4. **Electric Field Inside the Dielectric:**
Combining the equations above, we can derive the electric field \(\mathbf{E}\) inside the dielectric:
\[
\mathbf{E} = \frac{\mathbf{D}}{\epsilon}
\]
Substituting \(\mathbf{D} = \epsilon_0 \mathbf{E}_0 + \mathbf{P}\) and considering \(\mathbf{P} = \epsilon_0 (\epsilon_r - 1) \mathbf{E}\), we get:
\[
\mathbf{E} = \frac{\mathbf{E}_0}{\epsilon_r}
\]
### 5. **Summary:**
In summary, the electric field inside a dielectric material is reduced compared to the electric field in a vacuum or air. The reduction factor is the relative permittivity \(\epsilon_r\) of the dielectric. If \(\mathbf{E}_0\) is the electric field in a vacuum or air, the electric field \(\mathbf{E}\) inside the dielectric is given by:
\[
\mathbf{E} = \frac{\mathbf{E}_0}{\epsilon_r}
\]
This means that the presence of a dielectric material decreases the electric field within it relative to what it would be without the dielectric.