= Solution
<Free charge> is charge assigned to externally supplied or mobile carriers in a macroscopic description. <Bound charge> describes the net charge associated with a material's <polarization density> $\mathbf P$. Its bulk and surface densities are
$$
\boxed{\rho_b=-\nabla\cdot\mathbf P,\qquad \sigma_b=\mathbf P\cdot\widehat{\mathbf n},}
$$
where the normal points out of the polarized material. Uniform bulk <polarization density> has no bulk <bound charge>, but it can still have surface <bound charge>.
The <electric field> is sourced by the total <electric charge>. Introducing the <electric displacement field> separates the free source in the macroscopic field equation:
$$
\nabla\cdot\mathbf E=\frac{\rho_f+\rho_b}{\epsilon_0},\qquad \boxed{\mathbf D=\epsilon_0\mathbf E+\mathbf P,\qquad\nabla\cdot\mathbf D=\rho_f.}
$$
This does not mean that <bound charges> are physically absent from $\mathbf D$: they are represented through $\mathbf P$. In a linear isotropic dielectric, $\mathbf P=\epsilon_0\chi_e\mathbf E$ and $\mathbf D=\epsilon\mathbf E$, with $\epsilon=\epsilon_0(1+\chi_e)$.
Across a stationary ordinary dielectric interface, electrostatics gives continuity of tangential <electric field>, while a free sheet charge gives
$$
\boxed{\widehat{\mathbf n}\times(\mathbf E_2-\mathbf E_1)=0,\qquad \widehat{\mathbf n}\cdot(\mathbf D_2-\mathbf D_1)=\sigma_f.}
$$
The normal <electric field> need not be continuous. For a fully filled <parallel-plate capacitor> at fixed free plate charge, polarization reduces the internal <electric field> by $\epsilon_r$ and increases <capacitance> by the same factor. If a battery instead holds the <voltage> fixed, extra free plate charge flows from the battery; the constraint must be stated before comparing stored <energies>.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[Chapter 10, §§10–1 to 10–4: dielectric capacitor, polarization charge and the field equations]{ref}
Back to article page
