What is the relationship between electric displacement susceptibility and permittivity?
2 Answers
To understand the relationship between electric displacement susceptibility (\( \chi_D \)) and permittivity (\( \epsilon \)), we need to delve into a few fundamental concepts in electromagnetism.
### Key Concepts
1. **Electric Field (\( \mathbf{E} \))**: This is a vector field that represents the force experienced by a unit positive charge placed in the field.
2. **Electric Displacement Field (\( \mathbf{D} \))**: This field accounts for free and bound charge in materials. It is defined as:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}
\]
where:
- \( \epsilon_0 \) is the permittivity of free space (vacuum),
- \( \mathbf{P} \) is the polarization vector, representing the dipole moment per unit volume of the material.
3. **Polarization (\( \mathbf{P} \))**: When an electric field is applied to a dielectric material, the molecules within the material can become polarized, leading to the formation of bound charges. The degree of this polarization is expressed through the polarization vector.
4. **Permittivity (\( \epsilon \))**: This is a measure of how much electric field ( \( \mathbf{E} \) ) is reduced within a medium compared to vacuum. It is defined as:
\[
\epsilon = \epsilon_0 (1 + \chi_e)
\]
where \( \chi_e \) is the electric susceptibility of the material, indicating how easily a material can be polarized by an electric field.
### Relationship Between \( \chi_D \) and \( \epsilon \)
Electric displacement susceptibility (\( \chi_D \)) is a measure that relates the electric displacement field \( \mathbf{D} \) to the electric field \( \mathbf{E} \). The relationship is given by:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} = \epsilon_0 \mathbf{E} + \epsilon_0 \chi_e \mathbf{E}
\]
This can be simplified to:
\[
\mathbf{D} = \epsilon_0 (1 + \chi_e) \mathbf{E} = \epsilon \mathbf{E}
\]
Now, from the definition of the electric displacement susceptibility:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \epsilon_0 \chi_D \mathbf{E}
\]
This can also be rewritten as:
\[
\mathbf{D} = \epsilon_0 (1 + \chi_D) \mathbf{E}
\]
### Connecting \( \chi_D \) to \( \chi_e \)
From the two equations we have derived, we can equate them:
\[
\epsilon \mathbf{E} = \epsilon_0 (1 + \chi_D) \mathbf{E}
\]
This leads to the relationship:
\[
\epsilon = \epsilon_0 (1 + \chi_D)
\]
By substituting \( \epsilon = \epsilon_0 (1 + \chi_e) \) into this equation, we get:
\[
\epsilon_0 (1 + \chi_e) = \epsilon_0 (1 + \chi_D)
\]
Thus, we arrive at:
\[
\chi_D = \chi_e
\]
### Summary
In summary, electric displacement susceptibility (\( \chi_D \)) and electric susceptibility (\( \chi_e \)) are related through the permittivity of a material. They essentially describe the same phenomenon in different contexts. Both parameters illustrate how materials respond to electric fields, but \( \chi_D \) specifically relates to the electric displacement field \( \mathbf{D} \), while \( \chi_e \) relates to the electric field \( \mathbf{E} \).
In practical terms, the understanding of these relationships is crucial for designing materials used in capacitors, insulators, and various electronic components, as it helps predict how materials will behave in electric fields.
### Key Concepts
1. **Electric Field (\( \mathbf{E} \))**: This is a vector field that represents the force experienced by a unit positive charge placed in the field.
2. **Electric Displacement Field (\( \mathbf{D} \))**: This field accounts for free and bound charge in materials. It is defined as:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}
\]
where:
- \( \epsilon_0 \) is the permittivity of free space (vacuum),
- \( \mathbf{P} \) is the polarization vector, representing the dipole moment per unit volume of the material.
3. **Polarization (\( \mathbf{P} \))**: When an electric field is applied to a dielectric material, the molecules within the material can become polarized, leading to the formation of bound charges. The degree of this polarization is expressed through the polarization vector.
4. **Permittivity (\( \epsilon \))**: This is a measure of how much electric field ( \( \mathbf{E} \) ) is reduced within a medium compared to vacuum. It is defined as:
\[
\epsilon = \epsilon_0 (1 + \chi_e)
\]
where \( \chi_e \) is the electric susceptibility of the material, indicating how easily a material can be polarized by an electric field.
### Relationship Between \( \chi_D \) and \( \epsilon \)
Electric displacement susceptibility (\( \chi_D \)) is a measure that relates the electric displacement field \( \mathbf{D} \) to the electric field \( \mathbf{E} \). The relationship is given by:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} = \epsilon_0 \mathbf{E} + \epsilon_0 \chi_e \mathbf{E}
\]
This can be simplified to:
\[
\mathbf{D} = \epsilon_0 (1 + \chi_e) \mathbf{E} = \epsilon \mathbf{E}
\]
Now, from the definition of the electric displacement susceptibility:
\[
\mathbf{D} = \epsilon_0 \mathbf{E} + \epsilon_0 \chi_D \mathbf{E}
\]
This can also be rewritten as:
\[
\mathbf{D} = \epsilon_0 (1 + \chi_D) \mathbf{E}
\]
### Connecting \( \chi_D \) to \( \chi_e \)
From the two equations we have derived, we can equate them:
\[
\epsilon \mathbf{E} = \epsilon_0 (1 + \chi_D) \mathbf{E}
\]
This leads to the relationship:
\[
\epsilon = \epsilon_0 (1 + \chi_D)
\]
By substituting \( \epsilon = \epsilon_0 (1 + \chi_e) \) into this equation, we get:
\[
\epsilon_0 (1 + \chi_e) = \epsilon_0 (1 + \chi_D)
\]
Thus, we arrive at:
\[
\chi_D = \chi_e
\]
### Summary
In summary, electric displacement susceptibility (\( \chi_D \)) and electric susceptibility (\( \chi_e \)) are related through the permittivity of a material. They essentially describe the same phenomenon in different contexts. Both parameters illustrate how materials respond to electric fields, but \( \chi_D \) specifically relates to the electric displacement field \( \mathbf{D} \), while \( \chi_e \) relates to the electric field \( \mathbf{E} \).
In practical terms, the understanding of these relationships is crucial for designing materials used in capacitors, insulators, and various electronic components, as it helps predict how materials will behave in electric fields.
Electric displacement susceptibility and permittivity are closely related concepts in electromagnetism, and understanding their relationship requires a bit of background on each.
### Permittivity (\(\epsilon\))
Permittivity is a measure of how much electric field (E-field) is reduced within a material compared to a vacuum. It's a fundamental property of materials that quantifies their ability to allow electric field lines to pass through them. In general, permittivity (\(\epsilon\)) can be expressed as:
\[ \epsilon = \epsilon_0 \epsilon_r \]
where:
- \(\epsilon_0\) is the permittivity of free space (vacuum), approximately \(8.854 \times 10^{-12} \, \text{F/m}\).
- \(\epsilon_r\) is the relative permittivity (or dielectric constant) of the material, a dimensionless number that represents how much the material can increase the capacitance of a capacitor compared to a vacuum.
### Electric Displacement Field (\(\mathbf{D}\)) and Susceptibility (\(\chi_e\))
The electric displacement field \(\mathbf{D}\) accounts for both the electric field \(\mathbf{E}\) and the effects of polarization in a material. It is defined as:
\[ \mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} \]
where \(\mathbf{P}\) is the polarization of the material, which results from the alignment of electric dipoles in response to an electric field.
Electric displacement susceptibility (\(\chi_e\)) is related to how the polarization \(\mathbf{P}\) responds to the electric field \(\mathbf{E}\). It is defined as:
\[ \mathbf{P} = \epsilon_0 \chi_e \mathbf{E} \]
where \(\chi_e\) is the electric susceptibility, a dimensionless quantity that measures how easily a material becomes polarized in response to an electric field.
### Relationship Between Electric Displacement Susceptibility and Permittivity
To see how these quantities relate, substitute the polarization \(\mathbf{P}\) in the displacement field equation:
\[ \mathbf{D} = \epsilon_0 \mathbf{E} + \epsilon_0 \chi_e \mathbf{E} \]
Simplify to:
\[ \mathbf{D} = \epsilon_0 (1 + \chi_e) \mathbf{E} \]
Since \(\epsilon = \epsilon_0 (1 + \chi_e)\), we can rewrite this as:
\[ \mathbf{D} = \epsilon \mathbf{E} \]
Thus, the permittivity \(\epsilon\) and the electric displacement susceptibility \(\chi_e\) are related through:
\[ \epsilon = \epsilon_0 (1 + \chi_e) \]
In summary:
- **Permittivity (\(\epsilon\))** measures the material's overall ability to allow electric field lines to pass through, including the effects of polarization.
- **Electric Displacement Susceptibility (\(\chi_e\))** measures the material's tendency to become polarized in response to an electric field.
The relationship between them shows that permittivity is directly related to susceptibility, with permittivity incorporating both the vacuum permittivity and the material's susceptibility.
### Permittivity (\(\epsilon\))
Permittivity is a measure of how much electric field (E-field) is reduced within a material compared to a vacuum. It's a fundamental property of materials that quantifies their ability to allow electric field lines to pass through them. In general, permittivity (\(\epsilon\)) can be expressed as:
\[ \epsilon = \epsilon_0 \epsilon_r \]
where:
- \(\epsilon_0\) is the permittivity of free space (vacuum), approximately \(8.854 \times 10^{-12} \, \text{F/m}\).
- \(\epsilon_r\) is the relative permittivity (or dielectric constant) of the material, a dimensionless number that represents how much the material can increase the capacitance of a capacitor compared to a vacuum.
### Electric Displacement Field (\(\mathbf{D}\)) and Susceptibility (\(\chi_e\))
The electric displacement field \(\mathbf{D}\) accounts for both the electric field \(\mathbf{E}\) and the effects of polarization in a material. It is defined as:
\[ \mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} \]
where \(\mathbf{P}\) is the polarization of the material, which results from the alignment of electric dipoles in response to an electric field.
Electric displacement susceptibility (\(\chi_e\)) is related to how the polarization \(\mathbf{P}\) responds to the electric field \(\mathbf{E}\). It is defined as:
\[ \mathbf{P} = \epsilon_0 \chi_e \mathbf{E} \]
where \(\chi_e\) is the electric susceptibility, a dimensionless quantity that measures how easily a material becomes polarized in response to an electric field.
### Relationship Between Electric Displacement Susceptibility and Permittivity
To see how these quantities relate, substitute the polarization \(\mathbf{P}\) in the displacement field equation:
\[ \mathbf{D} = \epsilon_0 \mathbf{E} + \epsilon_0 \chi_e \mathbf{E} \]
Simplify to:
\[ \mathbf{D} = \epsilon_0 (1 + \chi_e) \mathbf{E} \]
Since \(\epsilon = \epsilon_0 (1 + \chi_e)\), we can rewrite this as:
\[ \mathbf{D} = \epsilon \mathbf{E} \]
Thus, the permittivity \(\epsilon\) and the electric displacement susceptibility \(\chi_e\) are related through:
\[ \epsilon = \epsilon_0 (1 + \chi_e) \]
In summary:
- **Permittivity (\(\epsilon\))** measures the material's overall ability to allow electric field lines to pass through, including the effects of polarization.
- **Electric Displacement Susceptibility (\(\chi_e\))** measures the material's tendency to become polarized in response to an electric field.
The relationship between them shows that permittivity is directly related to susceptibility, with permittivity incorporating both the vacuum permittivity and the material's susceptibility.