What is the relation between displacement vector and polarization vector?
2 Answers
The **displacement vector** \(\mathbf{D}\) and the **polarization vector** \(\mathbf{P}\) are two key concepts in electromagnetism that describe how electric fields interact with materials. Understanding their relationship involves recognizing the role they play in dielectric (non-conducting) materials when exposed to an electric field.
### 1. **Polarization Vector \(\mathbf{P}\)**
The **polarization vector** \(\mathbf{P}\) describes the degree to which a dielectric material becomes polarized when placed in an electric field. In simple terms, polarization occurs when the positive and negative charges within the material slightly separate, creating tiny electric dipoles throughout the material. These dipoles align with the external electric field, and the **polarization vector** represents the dipole moment per unit volume.
- **Formula:**
\[
\mathbf{P} = \frac{\text{Total dipole moment}}{\text{Volume of the dielectric material}}
\]
\(\mathbf{P}\) points in the direction of the average dipole moment inside the material, and its magnitude tells us how strongly the material is polarized.
### 2. **Displacement Vector \(\mathbf{D}\)**
The **electric displacement vector** \(\mathbf{D}\) takes into account both the free charges and the polarization effects within a dielectric material. It is used to describe the electric field in the presence of dielectrics and simplifies Maxwell's equations when dealing with materials that can become polarized.
The electric displacement vector \(\mathbf{D}\) is related to the **electric field vector** \(\mathbf{E}\) and the polarization vector \(\mathbf{P}\). The relation between them is:
\[
\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}
\]
Where:
- \(\varepsilon_0\) is the permittivity of free space (a constant).
- \(\mathbf{E}\) is the electric field strength.
- \(\mathbf{P}\) is the polarization vector.
### 3. **Relation Between Displacement Vector \(\mathbf{D}\) and Polarization Vector \(\mathbf{P}\)**
The **displacement vector** \(\mathbf{D}\) can be thought of as the total electric field within a material, which includes two components:
1. The contribution from the external electric field (\(\varepsilon_0 \mathbf{E}\)).
2. The contribution from the material's internal polarization (\(\mathbf{P}\)).
- **In a vacuum**, where there is no polarization, \(\mathbf{P} = 0\), so the displacement vector reduces to:
\[
\mathbf{D} = \varepsilon_0 \mathbf{E}
\]
- **In a dielectric material**, the polarization vector \(\mathbf{P}\) represents the dipoles induced within the material. The displacement vector \(\mathbf{D}\) then becomes a measure of the total electric effect (both the external field and the field due to the material's polarization).
The key relationship is that **\(\mathbf{D}\) includes the effects of both the free charges in the system (via the external field) and the bound charges in the dielectric (via the polarization)**. This makes \(\mathbf{D}\) particularly useful for analyzing systems with complex material properties.
### 4. **Summary of the Relationship**
- The **polarization vector** \(\mathbf{P}\) represents the response of a material to an applied electric field, describing the internal dipoles formed within the material.
- The **displacement vector** \(\mathbf{D}\) accounts for both the external electric field and the material's response to it (polarization).
- The **relationship** between them is given by:
\[
\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}
\]
In this equation:
- \(\varepsilon_0 \mathbf{E}\) represents the contribution from the external electric field.
- \(\mathbf{P}\) represents the material's polarization (internal dipole moment per unit volume).
Thus, the **displacement vector** \(\mathbf{D}\) is influenced by both the applied electric field and the material's polarization response.
### 1. **Polarization Vector \(\mathbf{P}\)**
The **polarization vector** \(\mathbf{P}\) describes the degree to which a dielectric material becomes polarized when placed in an electric field. In simple terms, polarization occurs when the positive and negative charges within the material slightly separate, creating tiny electric dipoles throughout the material. These dipoles align with the external electric field, and the **polarization vector** represents the dipole moment per unit volume.
- **Formula:**
\[
\mathbf{P} = \frac{\text{Total dipole moment}}{\text{Volume of the dielectric material}}
\]
\(\mathbf{P}\) points in the direction of the average dipole moment inside the material, and its magnitude tells us how strongly the material is polarized.
### 2. **Displacement Vector \(\mathbf{D}\)**
The **electric displacement vector** \(\mathbf{D}\) takes into account both the free charges and the polarization effects within a dielectric material. It is used to describe the electric field in the presence of dielectrics and simplifies Maxwell's equations when dealing with materials that can become polarized.
The electric displacement vector \(\mathbf{D}\) is related to the **electric field vector** \(\mathbf{E}\) and the polarization vector \(\mathbf{P}\). The relation between them is:
\[
\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}
\]
Where:
- \(\varepsilon_0\) is the permittivity of free space (a constant).
- \(\mathbf{E}\) is the electric field strength.
- \(\mathbf{P}\) is the polarization vector.
### 3. **Relation Between Displacement Vector \(\mathbf{D}\) and Polarization Vector \(\mathbf{P}\)**
The **displacement vector** \(\mathbf{D}\) can be thought of as the total electric field within a material, which includes two components:
1. The contribution from the external electric field (\(\varepsilon_0 \mathbf{E}\)).
2. The contribution from the material's internal polarization (\(\mathbf{P}\)).
- **In a vacuum**, where there is no polarization, \(\mathbf{P} = 0\), so the displacement vector reduces to:
\[
\mathbf{D} = \varepsilon_0 \mathbf{E}
\]
- **In a dielectric material**, the polarization vector \(\mathbf{P}\) represents the dipoles induced within the material. The displacement vector \(\mathbf{D}\) then becomes a measure of the total electric effect (both the external field and the field due to the material's polarization).
The key relationship is that **\(\mathbf{D}\) includes the effects of both the free charges in the system (via the external field) and the bound charges in the dielectric (via the polarization)**. This makes \(\mathbf{D}\) particularly useful for analyzing systems with complex material properties.
### 4. **Summary of the Relationship**
- The **polarization vector** \(\mathbf{P}\) represents the response of a material to an applied electric field, describing the internal dipoles formed within the material.
- The **displacement vector** \(\mathbf{D}\) accounts for both the external electric field and the material's response to it (polarization).
- The **relationship** between them is given by:
\[
\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}
\]
In this equation:
- \(\varepsilon_0 \mathbf{E}\) represents the contribution from the external electric field.
- \(\mathbf{P}\) represents the material's polarization (internal dipole moment per unit volume).
Thus, the **displacement vector** \(\mathbf{D}\) is influenced by both the applied electric field and the material's polarization response.
The displacement vector **\(\mathbf{D}\)** and the polarization vector **\(\mathbf{P}\)** are related concepts in electromagnetism, particularly in the context of dielectric materials.
1. **Displacement Vector (\(\mathbf{D}\))**: This vector represents the total electric field within a material, accounting for both the external electric field and the effects of polarization. It is given by:
\[
\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}
\]
where \(\varepsilon_0\) is the permittivity of free space, \(\mathbf{E}\) is the electric field, and \(\mathbf{P}\) is the polarization vector.
2. **Polarization Vector (\(\mathbf{P}\))**: This vector represents the dipole moment per unit volume of the material. It describes how the material responds to an electric field by aligning its dipoles. The polarization vector can be related to the electric field through the material's electric susceptibility \(\chi_e\):
\[
\mathbf{P} = \varepsilon_0 \chi_e \mathbf{E}
\]
In summary, the displacement vector **\(\mathbf{D}\)** includes the contribution from the external electric field **\(\mathbf{E}\)** and the polarization **\(\mathbf{P}\)** of the material. The polarization vector **\(\mathbf{P}\)** reflects how the material's dipole moments are oriented due to the applied electric field.
1. **Displacement Vector (\(\mathbf{D}\))**: This vector represents the total electric field within a material, accounting for both the external electric field and the effects of polarization. It is given by:
\[
\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}
\]
where \(\varepsilon_0\) is the permittivity of free space, \(\mathbf{E}\) is the electric field, and \(\mathbf{P}\) is the polarization vector.
2. **Polarization Vector (\(\mathbf{P}\))**: This vector represents the dipole moment per unit volume of the material. It describes how the material responds to an electric field by aligning its dipoles. The polarization vector can be related to the electric field through the material's electric susceptibility \(\chi_e\):
\[
\mathbf{P} = \varepsilon_0 \chi_e \mathbf{E}
\]
In summary, the displacement vector **\(\mathbf{D}\)** includes the contribution from the external electric field **\(\mathbf{E}\)** and the polarization **\(\mathbf{P}\)** of the material. The polarization vector **\(\mathbf{P}\)** reflects how the material's dipole moments are oriented due to the applied electric field.