A simple dipole consists of charges \(\pm 3.0\times10^{-19}\,\mathrm{C}\) separated by \(2.0\times10^{-10}\,\mathrm{m}\). Find the dipole moment magnitude.

A molecular dipole with \(p=4.0\times10^{-30}\,\mathrm{C\,m}\) is at \(60^\circ\) to an electric field of \(2.5\times10^6\,\mathrm{V\,m^{-1}}\). Find the torque magnitude.

A dielectric slab is placed between capacitor plates whose free-charge field points to the right. Identify the signs of bound charge on the left and right faces of the slab, and state the direction of the field due to bound charge.

Explain why a dielectric can reduce the electric field inside a capacitor while still allowing a nonzero field, whereas a conductor in electrostatic equilibrium has zero field inside its material.

A dipole has moment \(p=3.0\times10^{-30}\,\mathrm{C\,m}\) in a uniform electric field \(E=2.0\times10^6\,\mathrm{V\,m^{-1}}\). Find the potential energy when it is aligned with the field.

At what angle between \(\vec p\) and \(\vec E\) is the torque on an electric dipole largest?

A dielectric has uniform polarization magnitude \(P=4.0\times10^{-6}\,\mathrm{C\,m^{-2}}\). Find the bound surface charge density on a face whose outward normal is parallel to \(\vec P\), and on a face whose outward normal is opposite \(\vec P\).

A material contains \(2.0\times10^{28}\) identical molecular dipoles per cubic metre, each with average aligned dipole moment \(1.5\times10^{-31}\,\mathrm{C\,m}\). Estimate the polarization magnitude.

A dipole with \(p=5.0\times10^{-30}\,\mathrm{C\,m}\) is in a \(1.2\times10^6\,\mathrm{V\,m^{-1}}\) field. Find its energy at \(\theta=0^\circ\), \(90^\circ\), and \(180^\circ\).

A uniformly polarized dielectric block has \(\vec P\) pointing to the right. Identify the signs of bound charge on the left and right faces, and explain why the total bound charge on the whole block is zero.

Explain the difference between polarization of a nonpolar molecule and rotation of a permanent polar molecule in an applied electric field.

A dielectric fills the gap between capacitor plates. The free charge on the left plate is positive and the right plate is negative. Describe the direction of molecular dipole alignment and the signs of bound charge on the dielectric faces.

A dipole starts at \(\theta=90^\circ\) in a uniform electric field and is released from rest. Which way does it rotate, and what happens to its electric potential energy?

A linear dielectric has \(\kappa=4.0\) and field \(E=3.0\times10^5\,\mathrm{V\,m^{-1}}\). Use \(P=(\kappa-1)\epsilon_0E\) to estimate the polarization magnitude.

For a uniformly polarized slab with polarization \(P\) normal to its faces, derive the electric field produced by the two bound surface charge sheets alone.

A dielectric slab in a capacitor has free surface charge density \(\sigma_f\) on the plates and dielectric constant \(\kappa\). Derive the bound surface charge density magnitude on the dielectric faces in terms of \(\sigma_f\) and \(\kappa\).

Using \(U=-\vec p\cdot\vec E\), classify the stable and unstable orientations of a dipole in a uniform field and justify your answer from small angular changes.

A dielectric is placed in a nonuniform electric field. Explain qualitatively why it can experience a net force even if its net charge is zero.

For a simple dipole \(\pm q\) separated by vector \(\vec \ell\), derive the torque vector \(\vec\tau=\vec p\times\vec E\) in a uniform field.

A small dielectric molecule has induced dipole moment \(\vec p=\alpha\vec E\). Show that its interaction energy in a slowly applied field is \(U=-\frac12\alpha E^2\), and explain the origin of the factor \(1/2\).