Electrodynamics

Electrostatic fields, conductors, and dielectrics

Static electric fields are sourced by charge; an equilibrium conductor has zero internal field, while a dielectric polarizes in an applied field.

Electrostatics is the time-independent limit of electrodynamics. Free charge sources electric displacement; bound charge is represented through material polarization.

Interpretation and consequences

Boundary conditions: tangential E is continuous; the normal component of D jumps by the free surface-charge density σ_f.

The conditions $E_t$ continuous and $D_{2n}-D_{1n}=\sigma_f$ determine fields at an interface. For a conductor, the interior field is zero, so just outside in vacuum $E_n=\sigma_f/\varepsilon_0$. This explains electrostatic shielding and why charge concentration at a sharp tip can produce a particularly intense local field.

In a dielectric, polarization $P$ produces bound volume charge $\rho_b=-\nabla\cdot P$ and surface charge $\sigma_b=P\cdot\hat n$. Gauss's law for $D$ contains only free charge because the microscopic dipole response is already encoded in $P$. This macroscopic bookkeeping differs from $\nabla\cdot E=\rho_{\rm total}/\varepsilon_0$, which counts all charge.

A parallel-plate capacitor of area $A$, gap $d$, and uniform dielectric illustrates the distinction between $E$ and $D$. Neglecting fringing, $C=\varepsilon A/d$, $E=V/d$, and $D=\varepsilon E=Q_f/A$. Removing the dielectric from an isolated capacitor leaves $Q_f$ fixed, so $D$ stays fixed while $E$ rises. With a voltage source attached, $E$ stays fixed and free charge changes instead. Comparisons must specify which electrical constraint is maintained.

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