Electric Fields in Material Space

Materials are graded by how freely charge moves through them — their conductivity \(\sigma\) (S/m). Conductors (copper, \(\sigma\approx 5.8\times10^7\)) have vast numbers of free electrons; dielectrics or insulators (glass, mica) have almost none; semiconductors sit between. Moving charge constitutes a current density \(\vec{J}\) (A/m²), the current per unit cross-sectional area.

Two mechanisms produce \(\vec{J}\). Convection current is charge \(\rho_v\) physically carried along at velocity \(\vec{u}\) — like a charged beam in vacuum; it needs no conductor and does not obey Ohm's law. Conduction current is the drift of free charges inside a conductor under a field, and it does.

Inside a conductor an applied field drives free electrons until they reach a steady drift, giving a conduction current proportional to the field. This is Ohm's law in point form — the local statement behind the familiar \(V=IR\):

In electrostatic equilibrium — no steady current — a conductor must have \(\vec{E}=0\) inside. If any field remained, it would keep pushing charges; they stop only when the field vanishes. The consequences are sharp and worth memorising:

Charge is conserved: it can move, but never appear or vanish. The current flowing out of a closed surface must equal the rate at which the enclosed charge drops. In point form, via the divergence theorem, this is the continuity equation:

Combining continuity with Ohm's law and Gauss's law shows that any charge placed inside a conductor decays exponentially to the surface with a characteristic relaxation time \(T_r = \varepsilon/\sigma\). For copper this is about \(10^{-19}\) seconds — effectively instantaneous, which is why we treat conductors as reaching equilibrium at once.

A dielectric has no free charge, but its bound charges shift. An applied field nudges each molecule's positive and negative centres apart, turning every molecule into a tiny dipole. Summed per unit volume, these dipoles give the polarization \(\vec{P}\) (C/m²). The aligned dipoles create their own field opposing the applied one, so the net field inside a dielectric is weaker.

Polarization adds to \(\varepsilon_0\vec{E}\) so that the flux density picks up the material's response. This generalises \(\vec{D}=\varepsilon_0\vec{E}\) to matter:

For most materials the polarization is proportional to the field: \(\vec{P}=\varepsilon_0\chi_e\vec{E}\), where \(\chi_e\) is the electric susceptibility. Substituting collapses everything back into the simple form \(\vec{D}=\varepsilon\vec{E}\), now with the material's permittivity:

The dimensionless dielectric constant \(\varepsilon_r\) (relative permittivity) tells you how many times stronger the material's response is than vacuum. It is always greater than one — vacuum is \(\varepsilon_r=1\):

A dielectric insulates only up to a point. Raise the field high enough and electrons are torn free, the material suddenly conducts, and it is often destroyed — this is dielectric breakdown. The maximum field a material withstands is its dielectric strength. This single number sets the voltage rating of every capacitor and the spacing of every high-voltage insulator, linking directly to the breakdown work studied across high-voltage engineering.

When a field crosses from one material into another, it bends. Two rules — derived by applying \(\oint\vec{E}\cdot d\vec{l}=0\) to a thin loop and \(\oint\vec{D}\cdot d\vec{S}=Q\) to a thin pillbox straddling the interface — govern the change. The tangential \(\vec{E}\) is continuous; the normal \(\vec{D}\) jumps only by any free surface charge present:

With no free charge on the interface (\(\rho_S=0\)), the normal component of \(\vec{D}\) is continuous too, and the field lines refract according to the ratio of permittivities — a "law of refraction" for field lines:

For each item, identify the material involved and whether the question is about conduction, polarization, or a boundary. Difficulty rises down the list.