Question

Which of the following relations is(are) valid for linear dielectrics?
\(E\) = Electric field, \(P\) = Polarization, \(D\) = Electric displacement, \(\epsilon_0\) = Permittivity of free space, \(\epsilon\) = Dielectric permittivity, \(\chi_e\) = Electric susceptibility, \(\rho_f\) = Free charge density, \(\rho_b\) = Bound charge density

• A. \(\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}\)
• B. \(\epsilon = \epsilon_0 (1 + \chi_e)\)
• C. \(\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}\)
• D. \(\nabla \cdot \mathbf{D} = \rho_f + \rho_b\)

Hint:

Remember the three fundamental vectors in electrostatics of materials: \(\mathbf{E}\) (the total field), \(\mathbf{P}\) (the material's response), and \(\mathbf{D}\) (an auxiliary field related to free charges). The definition \(\mathbf{D} = \epsilon_0\mathbf{E} + \mathbf{P}\) is always true. The relation \(\mathbf{P} = \epsilon_0\chi_e\mathbf{E}\) is true specifically for linear dielectrics. The great advantage of \(\mathbf{D}\) is that its divergence is equal to the free charge density, \(\nabla \cdot \mathbf{D} = \rho_f\), which simplifies problems with dielectrics.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks to identify the fundamental relations that define and describe the behavior of linear dielectric materials in the presence of an electric field. Linear dielectrics are materials where the induced polarization is directly proportional to the applied electric field.
Step 2: Key Formula or Approach:
We need to examine each given equation and determine its validity based on the standard definitions and relationships in the theory of dielectrics.
Step 3: Detailed Explanation:
(A) \(\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}\)
This is the definition of a linear dielectric. The polarization \(\mathbf{P}\) (dipole moment per unit volume) is proportional to the total electric field \(\mathbf{E}\) inside the material. The constant of proportionality is \(\epsilon_0 \chi_e\), where \(\chi_e\) is the electric susceptibility, a dimensionless measure of how easily the material polarizes. So, this relation is valid for linear dielectrics by definition. Statement (A) is correct.
(B) \(\epsilon = \epsilon_0 (1 + \chi_e)\)
The electric displacement \(\mathbf{D}\) is defined as \(\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}\). For a linear dielectric, we can substitute \(\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}\) into this definition: \[ \mathbf{D} = \epsilon_0 \mathbf{E} + \epsilon_0 \chi_e \mathbf{E} = \epsilon_0 (1 + \chi_e) \mathbf{E} \] We also define the relationship between \(\mathbf{D}\) and \(\mathbf{E}\) in a linear material as \(\mathbf{D} = \epsilon \mathbf{E}\), where \(\epsilon\) is the permittivity of the material. Comparing the two expressions for \(\mathbf{D}\), we get: \[ \epsilon \mathbf{E} = \epsilon_0 (1 + \chi_e) \mathbf{E} \] \[ \epsilon = \epsilon_0 (1 + \chi_e) \] This is a standard and valid relation for linear dielectrics. Statement (B) is correct.
(C) \(\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}\)
This is the general definition of the electric displacement vector \(\mathbf{D}\). It is a fundamental equation in electromagnetism that is valid for all materials, including linear dielectrics, non-linear dielectrics, and even vacuum (where \(\mathbf{P}=0\)). Since it is universally valid, it is certainly valid for the specific case of linear dielectrics. Statement (C) is correct.
(D) \(\nabla \cdot \mathbf{D} = \rho_f + \rho_b\)
One of Maxwell's equations (Gauss's law in differential form) states that \(\nabla \cdot \mathbf{E} = \frac{\rho_{total}}{\epsilon_0} = \frac{\rho_f + \rho_b}{\epsilon_0}\). The divergence of the polarization is related to the bound charge density by \(\nabla \cdot \mathbf{P} = -\rho_b\). Let's take the divergence of the definition of \(\mathbf{D}\) from (C): \[ \nabla \cdot \mathbf{D} = \nabla \cdot (\epsilon_0 \mathbf{E} + \mathbf{P}) = \epsilon_0 (\nabla \cdot \mathbf{E}) + (\nabla \cdot \mathbf{P}) \] Substitute the expressions for \(\nabla \cdot \mathbf{E}\) and \(\nabla \cdot \mathbf{P}\): \[ \nabla \cdot \mathbf{D} = \epsilon_0 \left(\frac{\rho_f + \rho_b}{\epsilon_0}\right) + (-\rho_b) = (\rho_f + \rho_b) - \rho_b = \rho_f \] So, the correct relation is \(\nabla \cdot \mathbf{D} = \rho_f\). This is Gauss's law for dielectrics, and its main utility is that \(\mathbf{D}\) is related only to the free charges. The statement \(\nabla \cdot \mathbf{D} = \rho_f + \rho_b\) is incorrect. Statement (D) is incorrect.
Step 4: Final Answer:
The valid relations for linear dielectrics are (A), (B), and (C).