Derive Equation 26.33 for the induced surface charge density on the dielectric in a capacitor.

Verified step by step guidance

1. Start by understanding the physical setup: A parallel-plate capacitor is partially filled with a dielectric material of permittivity \( \varepsilon \). The goal is to derive the induced surface charge density \( \sigma_{\text{ind}} \) on the dielectric's surface due to polarization.
2. Recall that the electric displacement field \( \mathbf{D} \) is related to the free surface charge density \( \sigma_f \) on the capacitor plates by \( \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} \), where \( \mathbf{P} \) is the polarization vector. For a parallel-plate capacitor, \( \mathbf{D} = \sigma_f \).
3. The polarization \( \mathbf{P} \) in the dielectric is related to the electric field \( \mathbf{E} \) by \( \mathbf{P} = \chi_e \varepsilon_0 \mathbf{E} \), where \( \chi_e \) is the electric susceptibility of the dielectric. Substituting this into the expression for \( \mathbf{D} \), we get \( \mathbf{D} = \varepsilon_0 \mathbf{E} + \chi_e \varepsilon_0 \mathbf{E} = \varepsilon \mathbf{E} \), where \( \varepsilon = \varepsilon_0 (1 + \chi_e) \).
4. The induced surface charge density \( \sigma_{\text{ind}} \) is related to the polarization \( \mathbf{P} \) by \( \sigma_{\text{ind}} = \mathbf{P} \cdot \hat{n} \), where \( \hat{n} \) is the unit normal to the surface. Substituting \( \mathbf{P} = \chi_e \varepsilon_0 \mathbf{E} \), we find \( \sigma_{\text{ind}} = \chi_e \varepsilon_0 \mathbf{E} \cdot \hat{n} \).
5. Finally, express \( \mathbf{E} \) in terms of the free surface charge density \( \sigma_f \) using \( \mathbf{D} = \sigma_f \) and \( \mathbf{D} = \varepsilon \mathbf{E} \). This gives \( \mathbf{E} = \frac{\sigma_f}{\varepsilon} \). Substituting this into the expression for \( \sigma_{\text{ind}} \), we get \( \sigma_{\text{ind}} = \chi_e \varepsilon_0 \frac{\sigma_f}{\varepsilon} \). Simplify further using \( \varepsilon = \varepsilon_0 (1 + \chi_e) \) to obtain the final expression for \( \sigma_{\text{ind}} \).

Key Concepts

Dielectric Materials

Dielectric materials are insulating substances that can be polarized by an electric field. When placed in a capacitor, they increase the capacitor's ability to store charge by reducing the electric field within the material. This polarization leads to the formation of bound charges on the surface of the dielectric, which is crucial for understanding induced surface charge density.

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. It is defined as C = Q/V, where C is capacitance, Q is the charge stored, and V is the voltage across the capacitor. The presence of a dielectric increases the capacitance of a capacitor by a factor known as the dielectric constant, which is essential for deriving the induced surface charge density.

Induced Surface Charge Density

Induced surface charge density refers to the distribution of charge that appears on the surface of a dielectric material when it is subjected to an external electric field. This phenomenon occurs due to the alignment of dipoles within the dielectric, leading to a net charge on the surface. Understanding this concept is key to deriving the equation for the induced surface charge density in a capacitor.