What is the equation for the relation between electric displacement vector D and polarization P?
2 Answers
The relationship between the electric displacement vector \(\mathbf{D}\) and the polarization \(\mathbf{P}\) in a dielectric material is given by the equation:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P},
\]
where:
- \(\mathbf{D}\) is the electric displacement vector (measured in coulombs per square meter, \(C/m^2\)),
- \(\epsilon_0\) is the permittivity of free space (approximately \(8.85 \times 10^{-12} \, \text{F/m}\)),
- \(\mathbf{E}\) is the electric field vector (measured in volts per meter, \(V/m\)),
- \(\mathbf{P}\) is the polarization vector (measured in coulombs per square meter, \(C/m^2\)).
### Explanation of the Equation
1. **Electric Displacement Vector (\(\mathbf{D}\))**: This vector accounts for the effects of free charge and bound charge within a material. It is used to simplify the description of electric fields in materials, particularly in the presence of dielectrics.
2. **Electric Field (\(\mathbf{E}\))**: This is the physical electric field that exists in space due to free charges. It is the force per unit charge that a test charge would experience at any point in space.
3. **Polarization (\(\mathbf{P}\))**: Polarization represents the dipole moment per unit volume of the dielectric material. It arises due to the alignment of microscopic dipoles (either induced or permanent) within the material when it is subjected to an electric field.
### Derivation and Physical Interpretation
When an electric field \(\mathbf{E}\) is applied to a dielectric material, the material's molecules become polarized. This polarization creates bound charges, which modify the total electric field within the material.
The total electric displacement \(\mathbf{D}\) is therefore a combination of the electric field \(\mathbf{E}\) due to free charges and the additional field contribution from the bound charges represented by the polarization \(\mathbf{P}\). In free space (vacuum), where there is no polarization, the relationship simplifies to:
\[
\mathbf{D} = \epsilon_0 \mathbf{E}.
\]
However, in a material medium, the polarization \(\mathbf{P}\) modifies this relationship, leading to the more general form:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}.
\]
This equation is fundamental in understanding the behavior of dielectrics in the presence of electric fields, and it plays a crucial role in various applications, including capacitors, insulators, and other electronic components.
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P},
\]
where:
- \(\mathbf{D}\) is the electric displacement vector (measured in coulombs per square meter, \(C/m^2\)),
- \(\epsilon_0\) is the permittivity of free space (approximately \(8.85 \times 10^{-12} \, \text{F/m}\)),
- \(\mathbf{E}\) is the electric field vector (measured in volts per meter, \(V/m\)),
- \(\mathbf{P}\) is the polarization vector (measured in coulombs per square meter, \(C/m^2\)).
### Explanation of the Equation
1. **Electric Displacement Vector (\(\mathbf{D}\))**: This vector accounts for the effects of free charge and bound charge within a material. It is used to simplify the description of electric fields in materials, particularly in the presence of dielectrics.
2. **Electric Field (\(\mathbf{E}\))**: This is the physical electric field that exists in space due to free charges. It is the force per unit charge that a test charge would experience at any point in space.
3. **Polarization (\(\mathbf{P}\))**: Polarization represents the dipole moment per unit volume of the dielectric material. It arises due to the alignment of microscopic dipoles (either induced or permanent) within the material when it is subjected to an electric field.
### Derivation and Physical Interpretation
When an electric field \(\mathbf{E}\) is applied to a dielectric material, the material's molecules become polarized. This polarization creates bound charges, which modify the total electric field within the material.
The total electric displacement \(\mathbf{D}\) is therefore a combination of the electric field \(\mathbf{E}\) due to free charges and the additional field contribution from the bound charges represented by the polarization \(\mathbf{P}\). In free space (vacuum), where there is no polarization, the relationship simplifies to:
\[
\mathbf{D} = \epsilon_0 \mathbf{E}.
\]
However, in a material medium, the polarization \(\mathbf{P}\) modifies this relationship, leading to the more general form:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}.
\]
This equation is fundamental in understanding the behavior of dielectrics in the presence of electric fields, and it plays a crucial role in various applications, including capacitors, insulators, and other electronic components.
In electromagnetism, the electric displacement vector \(\mathbf{D}\) and the polarization \(\mathbf{P}\) are related through the following equation:
\[ \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} \]
Here's a detailed explanation of each term in this equation:
1. **Electric Displacement Vector (\(\mathbf{D}\))**:
This vector field is used to describe how electric fields interact with materials, particularly how they are influenced by the presence of dielectric materials. It combines the effects of the electric field \(\mathbf{E}\) and the material's response to that field.
2. **Electric Field (\(\mathbf{E}\))**:
This is the vector field that represents the force per unit charge exerted on a charged particle at a given point in space. It is a fundamental quantity in electromagnetism.
3. **Vacuum Permittivity (\(\varepsilon_0\))**:
Also known as the permittivity of free space, this is a constant that measures the ability of a vacuum to permit electric field lines. Its approximate value is \(8.854 \times 10^{-12} \, \text{F/m} \) (farads per meter).
4. **Polarization (\(\mathbf{P}\))**:
Polarization describes how a material responds to an electric field by developing dipole moments within it. The polarization vector \(\mathbf{P}\) represents the dipole moment per unit volume of the material.
To break it down further:
- **In a dielectric material**, when an electric field \(\mathbf{E}\) is applied, it induces a polarization \(\mathbf{P}\) in the material. This polarization arises because the electric field causes positive and negative charges within the material to separate slightly, creating an internal electric field.
- **The electric displacement field \(\mathbf{D}\)** takes into account both the applied electric field and the effect of the polarization. The term \(\varepsilon_0 \mathbf{E}\) represents the electric displacement in the absence of polarization (like in a vacuum), while \(\mathbf{P}\) represents the additional displacement due to the polarization of the material.
In summary, the equation \(\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}\) shows that the electric displacement vector is the sum of the electric field contribution (scaled by the permittivity of free space) and the polarization contribution of the material.
\[ \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} \]
Here's a detailed explanation of each term in this equation:
1. **Electric Displacement Vector (\(\mathbf{D}\))**:
This vector field is used to describe how electric fields interact with materials, particularly how they are influenced by the presence of dielectric materials. It combines the effects of the electric field \(\mathbf{E}\) and the material's response to that field.
2. **Electric Field (\(\mathbf{E}\))**:
This is the vector field that represents the force per unit charge exerted on a charged particle at a given point in space. It is a fundamental quantity in electromagnetism.
3. **Vacuum Permittivity (\(\varepsilon_0\))**:
Also known as the permittivity of free space, this is a constant that measures the ability of a vacuum to permit electric field lines. Its approximate value is \(8.854 \times 10^{-12} \, \text{F/m} \) (farads per meter).
4. **Polarization (\(\mathbf{P}\))**:
Polarization describes how a material responds to an electric field by developing dipole moments within it. The polarization vector \(\mathbf{P}\) represents the dipole moment per unit volume of the material.
To break it down further:
- **In a dielectric material**, when an electric field \(\mathbf{E}\) is applied, it induces a polarization \(\mathbf{P}\) in the material. This polarization arises because the electric field causes positive and negative charges within the material to separate slightly, creating an internal electric field.
- **The electric displacement field \(\mathbf{D}\)** takes into account both the applied electric field and the effect of the polarization. The term \(\varepsilon_0 \mathbf{E}\) represents the electric displacement in the absence of polarization (like in a vacuum), while \(\mathbf{P}\) represents the additional displacement due to the polarization of the material.
In summary, the equation \(\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}\) shows that the electric displacement vector is the sum of the electric field contribution (scaled by the permittivity of free space) and the polarization contribution of the material.