6: Dielectrics

Section 1: Electrostatics

6: Dielectrics

Dielectrics are insulating materials (e.g., glass, plastic) that exhibit no free charges but polarize in an external electric field. Unlike conductors, they lack mobile charges, but their bound charges (electrons bound to atoms/molecules) redistribute under an applied field, reducing the net field within the material.

Polarization
When an external field $\mathbf{E}_0$ acts on a dielectric, it induces polarization:

• Nonpolar molecules (e.g., $\text{CH}_4$) develop induced dipole moments as electron clouds shift relative to nuclei.
• Polar molecules (e.g., $\text{H}_2\text{O}$) experience alignment of permanent dipoles with the field.
  The net effect is a volume distribution of bound charges. Polarization $\mathbf{P}$ (dipole moment per unit volume) is defined as:

$$\mathbf{P} = \chi_e \varepsilon_0 \mathbf{E}$$

where $\chi_e$ is the electric susceptibility, and $\mathbf{E}$ is the net field inside the dielectric.

Dielectric Constant & Capacitance
The relative permittivity (dielectric constant) $\kappa \geq 1$ quantifies a dielectric’s ability to reduce $\mathbf{E}$:

$$\kappa = \frac{\varepsilon}{\varepsilon_0} = 1 + \chi_e$$

Here, $\varepsilon$ is the permittivity of the material, and $\mathbf{E} = \frac{\mathbf{E}_0}{\kappa}$.
In capacitors, inserting a dielectric between plates:

• Increases capacitance: $C = \kappa C_0$ (where $C_0$ is vacuum capacitance).
• Reduces potential difference for fixed charge: $V = \frac{V_0}{\kappa}$.

Gauss’s Law in Dielectrics
Polarization creates bound surface charges $\sigma_b = \mathbf{P} \cdot \hat{\mathbf{n}}$. Gauss’s law modifies to:

\oint \mathbf{D} \cdot d\mathbf{a} = Q_{\text{free}}}

where $\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} = \varepsilon \mathbf{E}$ is the electric displacement field, and Q_{\text{free}}} excludes bound charges.

Boundary Conditions
At a dielectric interface:

1. Normal component: $D_{1\perp} = D_{2\perp}$ (if no free surface charge).
2. Tangential component: $E_{1\parallel} = E_{2\parallel}$.

Key Implications

• Energy density in a dielectric: $u = \frac{1}{2} \varepsilon E^2$.
• Dielectric breakdown occurs at a critical field strength, limiting practical voltage.