+
+Let us briefly discuss how electric fields affect (charged) matter.
+
+
+
+Effect on fundamental particles: admittedly a complicated business, for which quantum mechanics
+is required. The assumption we make is that the charges of the particles we're dealing with are never
+changed by anything that happens to them. This is however not true in the real world, because the
+charges we measure are usually the remnants of some complicated intermingling of constituent particles.
+For example, the measured charge of an electron is a renormalized charge
+(it depends on how closely you probe the electron, in other words at what collision/scattering
+energies you probe it). This goes much beyond the current course, and needs the whole machinery
+of quantum electrodynamics.
+
+
+
+For us here, we will assume that fundamental particles remain 'unchanged' in the presence of an E field,
+irrespective of how strong the latter is.
+
+
+
+Simple atoms: for example Hydrogen.
+We simply have a nucleus (proton) with an orbiting electron.
+The mass ratio between these is about 1800 to 1. To treat this, we'd need to start from the
+nonrelativistic Schrödinger equation for the electron (assuming we're in the center-of-mass frame)
+in the presence of a constant (for simplicity) external electric field \({\bf E}\):
+\[
+-\frac{\hbar^2}{2m} {\boldsymbol \nabla}^2 \psi - \frac{e^2}{4\pi \varepsilon_0 r} \psi - e {\bf E} \cdot {\bf r} ~\psi = E \psi.
+\]
+As compared to the zero-field case, the energy levels are modified changed
+(by the Stark effect, linear and nonlinear (latter for case of hydrogen in fundamental level);
+see Landau Lifschitz, vol 3 nr 77).
+
+
+
+If the field is small, one can use perturbation theory. This gives an electric dipole moment of
+\[
+{\bf p} = \langle \psi | (-e {\bf r}) | \psi \rangle = ... = \frac{9}{2} (4\pi \varepsilon_0 a_B^3) {\bf E}
+\]
+where \(a_B\) is the Bohr radius, \(a_B = \hbar^2/m e^2\).
+
+
+
+Although the numerical factor is not guessable, the overall form is:
+Le Châtelier's principle tells us that the equilibrium position moves linearly with the strength of the
+perburtation.
+
+
+
+More complex atoms: we face a similar scenario.
+The nucleus is now relatively even heavier than each electron.
+At small fields, we can neglect nonlinear effects
+(e.g. a given electron orbital change leading to changes in other orbitals).
+We still expect to have some induced dipole moment which increases linearly with the external field,
+\[
+{\bf p} = \alpha {\bf E}
+\]
+except that now we have to solve a much more complicated QM problem.
+The factor \(\alpha\) is an atom-specific number called the atomic polarizability.
+
+
+
+Molecules: atoms can now 'share' electrons, so the charge distribution can become nontrivial.
+Example: carbon dioxide, \(O - C - O\). Higher polarizability along axis than perpendicular to axis.
+In totally non-symmetric case: expect
+\[
+p_i = \sum_{j = x,y,z} \alpha_{ij} E_j
+\]
+where \(\alpha_{ij}\) is the polarizability tensor of the molecule. Always possible to use 'principal' axes
+such that all but 3 of the terms cancel.
+
+
+
+Polar molecules: unlike individual atoms, molecules can have a permanent dipole moment. These are called
+polar molecules. Example: \(H Cl\) has elecronic density more closely bound on \(Cl\) than \(H\), so has a dipole
+moment pointing from \(Cl\) to \(H\). Other example: water, with \(105^\circ\) angle between the \(H^+\) and \(O^-\),
+dipole moment pointing from \(O^-\) along bisector.
+
+
+
+Torque on dipole: if field is uniform, overall force on dipole cancels, but torque remains:
+
+
+
+Torque on a dipole
+
+
+
+\[
+{\bf N} = {\bf p} \times {\bf E}
+\tag{di_N}\label{di_N}
+\]
+
+
+
+
+If field is non-uniform,
+\[
+{\bf F} = {\bf F}_+ + {\bf F}_- = q({\bf E}_+ - {\bf E}_-) = q ({\bf d} \cdot {\boldsymbol \nabla}) {\bf E}
+\]
+so
+
+
+
+
+\[
+{\bf F} = ({\bf p} \cdot {\boldsymbol \nabla}) {\bf E}
+\tag{di_F}\label{di_F}
+\]
+
+
+
+
+The energy of a dipole in an electric field can be shown to be
+
+
+
+\[
+U = -{\bf p} \cdot {\bf E}
+\tag{di_E}\label{di_E}
+\]
+
+
+
+Many atoms: gases and liquids: here, atoms or molecules are still more or less free from each other's influence
+as far as polarization is concerned. Random thermal motion,
+external field gives preferential direction to polarization.
+The relationship is still linear.
+
+
+
+Many atoms: solids: here, things can be more complicated. Material can be insulating or conducting.
+If conducting: external field makes charges move such that interior becomes equipotential.
+If insulating: each constituent atom/molecule can pick up an induced polarization, polar molecules can tend to
+line up, crystal structure can be deformed, …
+
+
+
+For zero field, the solid can have either zero or nonzero polarization. If nonzero: we call this spontaneous polarization,
+or rather ferroelectricity (after ferromagnetism, which is the correspdonding magnetic phenomenon happening with iron).
+
+