Dielectrics

Source lecture(s): SC134 Lec6

Intuition

Insulating materials polarize in an electric field, reducing the internal field and increasing capacitance.

Formal Definition

A dielectric reduces the effective field by factor $\kappa$ (dielectric constant).

Mathematical Formulation

\[\vec{E} = \frac{\vec{E}_0}{\kappa}$$ $$C = \kappa C_0\]

Derivation

Polarization $\vec{P}$ produces bound surface charge $\sigma_b = \vec{P}\cdot\hat{n}$. Gauss's law with $D=\varepsilon_0 E + P$ gives $D = \varepsilon_0\kappa E$.

Worked Example

Inserting dielectric ($\kappa=5$) into isolated charged capacitor reduces $E$ by 5, increases $C$ by 5, and leaves $Q$ unchanged.

Common Mistakes

Forgetting $Q$ is fixed when capacitor is isolated.
Confusing $\kappa$ with resistivity.

Quiz

Q1. What happens to stored energy when dielectric is inserted with battery connected?

Answer

Energy increases; battery does work.