A dielectric-filled parallel-plate capacitor has free surface charge density \(\sigma_{\mathrm f}=2.0\times10^{-6}\,\mathrm{C\,m^{-2}}\). Find the displacement field magnitude \(D\) between the plates.

For the same capacitor, the dielectric constant is \(\kappa=5.0\). Find the electric field magnitude \(E\).

A closed Gaussian surface in a dielectric has uniform outward displacement field magnitude \(D=1.5\times10^{-4}\,\mathrm{C\,m^{-2}}\) over area \(0.020\,\mathrm{m^2}\), with no flux through the rest of the surface. Find the enclosed free charge.

A parallel-plate capacitor of area \(A\) is filled by two dielectric layers in series, each of thickness \(d/2\), with dielectric constants \(\kappa_1\) and \(\kappa_2\). Neglect fringing. Use \(\vec D\) to derive the equivalent capacitance.

A Gaussian surface in a dielectric has total displacement flux \(4.0\times10^{-9}\,\mathrm{C}\). What free charge is enclosed?

A linear dielectric has \(\kappa=2.5\) and electric field \(E=6.0\times10^4\,\mathrm{V\,m^{-1}}\). Find the displacement field magnitude \(D\).

A dielectric-filled parallel-plate capacitor has \(D=5.0\times10^{-6}\,\mathrm{C\,m^{-2}}\) and \(\kappa=4.0\). Find \(E\).

A closed surface has \(\vec D\) everywhere normal and outward with magnitude \(2.0\times10^{-5}\,\mathrm{C\,m^{-2}}\) over area \(0.030\,\mathrm{m^2}\). Find the enclosed free charge.

A point free charge \(Q\) is embedded at the center of an infinite linear dielectric of constant \(\kappa\). Use Gauss's law for \(\vec D\) to find \(D(r)\) and \(E(r)\).

A spherical conductor of radius \(a\) carrying free charge \(Q\) is surrounded by a dielectric of constant \(\kappa\). Find the electric field just outside the conductor.

Across a dielectric boundary with no free surface charge, the normal component of \(\vec D\) is continuous. If \(\kappa_1=2.0\), \(\kappa_2=5.0\), and \(E_{1n}=1.0\times10^5\,\mathrm{V\,m^{-1}}\), find \(E_{2n}\).

At a boundary, the normal displacement changes from \(D_{1n}=3.0\times10^{-6}\,\mathrm{C\,m^{-2}}\) to \(D_{2n}=8.0\times10^{-6}\,\mathrm{C\,m^{-2}}\), both measured in the same normal direction. Find the free surface charge density on the boundary.

A linear dielectric has \(\kappa=3.0\) and electric field \(E=2.0\times10^5\,\mathrm{V\,m^{-1}}\). Find \(D\) and \(P\), and verify \(D=\epsilon_0E+P\).

A dielectric-filled parallel-plate capacitor has free surface charge density \(\sigma_f\). Derive the bound surface charge density magnitude using \(\vec D\), \(\vec P\), and \(\kappa\).

Two concentric conducting spherical shells of radii \(a\) and \(b\) are filled with a linear dielectric of constant \(\kappa\). Use \(\vec D\) to derive the capacitance.

A coaxial capacitor of length \(L\), inner radius \(a\), and outer radius \(b\) is filled with dielectric \(\kappa\). Use Gauss's law for \(\vec D\) to derive \(C/L\).

A parallel-plate capacitor is filled by two dielectric slabs side-by-side, each occupying half the area and spanning the full separation. Use \(\vec D\) or an equivalent argument to derive \(C\).

A dielectric interface has no free charge. Show that if the tangential component of \(\vec E\) is continuous and the normal component of \(\vec D\) is continuous, then field lines bend according to \(\tan\theta_2/\tan\theta_1=\kappa_2/\kappa_1\), where \(\theta\) is measured from the normal.

A parallel-plate capacitor has two dielectric layers in series with thicknesses \(d_1,d_2\) and constants \(\kappa_1,\kappa_2\). Derive the voltage drop across each layer in terms of free charge \(Q\).

Starting from ordinary Gauss's law for total charge and the relation \(\rho_b=-\nabla\cdot\vec P\), derive \(\nabla\cdot\vec D=\rho_f\) with \(\vec D=\epsilon_0\vec E+\vec P\).