2.2.6. Polarization

2.2.6.1. Single Electric Dipole

A discrete electric dipole is formed by two equal and opposite electric charges \(q\), \(-q\), at points \(P_+\), \(P_- = P_+ \vec{l}\), in the limit \(q \rightarrow +\infty\), \(|\vec{l}| \rightarrow 0\) with \(q |\vec{l}|\) finite.

The electric field (stationary todo check what happens in the non-stationary case. Perhaps after deriving the general solution to the problem, as a solution to the wave equations in terms of EM potentials) generated at the point in space \(\vec{r}\) by an electric dipole at the point \(\vec{r}_0\) is calculated as the limit of the electric field generated by two equal and opposite charges \(q^{\mp}\) at the points \(\vec{r}_0 \mp \frac{\vec{l}}{2}\),

\[\vec{e}(\vec{r}) = -\frac{q}{4 \pi \varepsilon_0} \frac{\vec{r} - \left( \vec{r}_0 - \frac{\vec{l}}{2} \right)}{\left|\vec{r} - \left( \vec{r}_0 - \frac{\vec{l}}{2} \right)\right|^3} + \frac{q}{4 \pi \varepsilon_0} \frac{\vec{r} - \left( \vec{r}_0 + \frac{\vec{l}}{2} \right)}{\left|\vec{r} - \left( \vec{r}_0 + \frac{\vec{l}}{2} \right)\right|^3} \ .\]

Using the formula for the derivative of the terms

\[\partial_{\ell_k} \frac{x_i \pm \frac{\ell_i}{2}}{\left|\vec{x} \pm \frac{\vec{l}}{2} \right|^3} = \frac{1}{2} \left[ \pm \frac{\delta_{ik}}{r^3} - 3 r^{-4} \left( \pm \frac{x_k \pm \frac{\ell_k}{2}}{r} \right) \right]\]

\[\left. \partial_{\ell_k} \frac{x_i \pm \frac{\ell_i}{2}}{\left|\vec{x} \pm \frac{\vec{l}}{2} \right|^3} \right|_{\vec{l} = \vec{0}} = \mp \frac{1}{2} \left[ - \frac{\delta_{ik}}{|\vec{x}|^3} + 3 \left( \frac{x_k}{r^5} \right) \right] = \mp \frac{1}{2} \partial_{r_{0 k}} \frac{r_i - r_{0 i}}{|\vec{r} - \vec{r}_0|^3} = \mp \frac{1}{2} \nabla_{\vec{r}_0} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \]

we derive the first-order approximation in \(\vec{l}\) of the two terms

\[\begin{split}\begin{aligned} \frac{\vec{r} - \left( \vec{r}_0 \mp \frac{\vec{l}}{2} \right)}{\left|\vec{r} - \left( \vec{r}_0 \mp \frac{\vec{l}}{2} \right)\right|^3} & = \frac{\vec{r} - \vec{r}_0 }{\left|\vec{r} - \vec{r}_0 \right|^3} \pm \vec{l} \cdot \frac{1}{2} \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right) + o(|\vec{l}|)\\ \end{aligned}\end{split}\]

and, defining the dipole intensity \(\vec{P}_0 := q \vec{l}\) and taking the quantities to the desired limit, that of the electric field

\[\begin{split}\begin{aligned} \vec{e}(\vec{r}) & = - \frac{1}{4\pi \varepsilon_0} \, \vec{P}_0 \cdot \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right) = \\ & = - \frac{1}{4\pi \varepsilon_0} \left[ \frac{(\vec{r}-\vec{r}_0)(\vec{r}-\vec{r}_0)}{|\vec{r}-\vec{r}_0|^5} \cdot \vec{P}_0 - \frac{\vec{P}_0}{|\vec{r}-\vec{r}_0|^3} \right] = \\ & = - \frac{1}{4\pi \varepsilon_0} \left[ \frac{(\vec{r}-\vec{r}_0) \otimes (\vec{r}-\vec{r}_0)}{|\vec{r}-\vec{r}_0|^5} - \frac{\mathbb{I}}{|\vec{r}-\vec{r}_0|^3} \right] \cdot \vec{P}_0 \ . \end{aligned}\end{split}\]

todo In the general case, it would be necessary to pay attention to the order of the factors in the product between vectors and tensors, but in this case, the symmetry of the second-order tensor (or of the operations) can be exploited.

2.2.6.2. Continuous Distribution of Dipoles

A distribution of dipoles with volume density \(\vec{p}(\vec{r_0})\), which produces the elementary dipole \(\Delta \vec{P}(\vec{r}_0) = \vec{p}(\vec{r}_0) dV_0\) in the volume \(d V_0\), produces the electric field

\[\vec{e}(\vec{r}) = \int_{\vec{r}_0 \in V_0} \frac{1}{4 \pi \varepsilon_0} \vec{p}(\vec{r}_0) \cdot \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right) \ , \]

whose expression can be rewritten using the rules of integration by parts

\[\begin{split}\begin{aligned} \vec{e}(\vec{r}) & = \int_{\vec{r}_0 \in V_0} \frac{1}{4 \pi \varepsilon_0} \vec{p}(\vec{r}_0) \cdot \nabla_{\vec{r}_0} \left( \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \right) = \\ & = \oint_{\vec{r}_0 \in \partial V_0} \frac{1}{4 \pi \varepsilon_0} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \underbrace{ \hat{\vec{n}}(\vec{r}_0) \cdot \vec{p}(\vec{r}_0) }_{ =: \sigma_P(\vec{r}_0)} + \int_{\vec{r}_0 \in V_0} \frac{1}{4 \pi \varepsilon_0} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|^3} \underbrace{ \left( - \nabla_{\vec{r}_0} \cdot \vec{p}(\vec{r}_0) \right)}_{ =: \rho_P(\vec{r}_0) } \ , \\ \end{aligned}\end{split}\]

having defined the surface polarization charge density \(\sigma_P\) and the volume polarization charge density \(\rho_P\) as the intensities of the distributed sources of the electric field, in analogy with the expression of Coulomb’s law.

2.2.6.3. Reformulation of Maxwell’s Equations and Charge Continuity

Gauss’s equation determines the volume flux density of the electric field \(\vec{e}\),

\[\nabla \cdot \vec{e} = \frac{\rho}{\varepsilon_0} \ .\]

By decomposing the charge density as the sum of free charges \(\rho_f\) and polarization charges \(\rho_P := - \nabla \cdot \vec{p}\), we can rework Gauss’s equation,

\[\begin{split}\begin{aligned} & \nabla \cdot \vec{e} = \frac{\rho_f + \rho_P}{\varepsilon_0} \\ & \nabla \cdot \left( \varepsilon_0 \vec{e} + \vec{p} \right) = \rho_f \\ \\ & \nabla \cdot \vec{d} = \rho_f \ , \end{aligned}\end{split}\]

having introduced the displacement field, \(\vec{d} := \varepsilon_0 \vec{e} + \vec{p}\).

The decomposition of the electric current as the sum \(\vec{j} = \vec{j}_f + \vec{j}_P\) of the free current \(\vec{j}_f\) and the polarization current \(\vec{j}_P\), allows us to rework the continuity equation of electric charge

\[\begin{split}\begin{aligned} 0 & = \partial_t \rho + \nabla \cdot \vec{j} = \\ & = \partial_t (\rho_f + \rho_P) + \nabla \cdot \left( \vec{j}_f + \vec{j}_P \right) = \\ & = \partial_t \rho_f + \nabla \cdot \vec{j}_f + \partial_t \rho_P + \nabla \cdot \vec{j}_P \ , \end{aligned}\end{split}\]

and write the continuity equations for the two charge distributions (of different nature, it is assumed that both must satisfy charge continuity independently, if free charges remain free and polarization charges remain polarization charges),

\[\begin{split}\begin{aligned} & \partial_t \rho_f + \nabla \cdot \vec{j}_f = 0 \\ & \partial_t \rho_P + \nabla \cdot \vec{j}_P = 0 \qquad \rightarrow \qquad 0 = \nabla \cdot (-\partial_t \vec{p} + \vec{j}_P) \qquad \rightarrow \qquad \vec{j}_P = \partial_t \vec{p} \end{aligned}\end{split}\]

todo justify absence of constant field