2.2. Electromagnetism in Matter#

Electromagnetism in matter requires the description of the behavior of the matter involved in the process. In general, a medium immersed in an electromagnetic field may respond with local charge distributions, resulting in polarization and magnetization. Total electric \(\vec{e}(\vec{r},t)\) and magnetic field \(\vec{b}(\vec{r},t)\) can be written as the sum of contributions of free charges \(\rho_f\) and currents \(\vec{j}_f\) and bound charges \(\rho_b\) and currents \(\vec{j}_b\).

Bound charge density represents local separation of charges of molecules of dielectric media immersed in electric field, that can be represented as a volume distribution of charge dipole,

\[\rho = \rho_f + \rho_b = \rho_f + \rho_P \ .\]

Bound current density represents two effects: the variation of polarization charge and the orientation of Amperian currents - “non random” currents in the molecules of the medium, producing net contribution to the magnetic field, and can be represented as a volume distribution of elementary loop currents.

\[\begin{aligned} \vec{j} & = \vec{j}_f + \vec{j}_b = \vec{j}_f + \vec{j}_P + \vec{j}_M \ . \end{aligned}\]

As it will shown below, the bound current can be written as the divergence of the polarization field \(\vec{p}\), representing the volume density fo the dipole distribution, and the magnetization current as the curl of the magnetization field \(\vec{m}\),

\[\rho_p = - \nabla \cdot \vec{p} \quad , \quad \vec{j}_M = \nabla \times \vec{m} \ .\]

2.2.1. Equations of electromagnetism in matter#

Introducing the splitting of free and bound charge and current into the equations of the electromagnetism, namely electric charge continuity and Maxwell’s equations,

\[\begin{split} \partial_t \rho + \nabla \cdot \vec{j} = 0 \qquad , \qquad \begin{cases} \nabla \cdot \vec{e} = \frac{\rho}{\varepsilon_0} \\ \nabla \times \vec{e} + \partial_t \vec{b} = \vec{0} \\ \nabla \cdot \vec{b} = 0 \\ \nabla \times \vec{b} - \mu_0 \varepsilon_0 \partial_t \vec{e} = \mu_0 \vec{j} \end{cases}\end{split}\]

and more precisely

  • into Gauss’ law for the electric field

    \[\begin{split}\begin{aligned} & 0 = \nabla \cdot \vec{e} - \frac{\rho}{\varepsilon_0} = \nabla \cdot \vec{e} - \frac{\rho_f - \nabla \cdot \vec{p}}{\varepsilon_0} \\ \\ & \rightarrow \qquad \nabla \cdot \vec{d} = \rho_f \ , \end{aligned}\end{split}\]

    with \(\vec{d} = \varepsilon_0 \vec{e} + \vec{p}\) defined as the displacement field.

  • into continuity equation

    \[\begin{split} 0 & = \partial_t \rho + \nabla \cdot \vec{j} = \\ & = \partial_t \rho_f + \nabla \cdot \vec{j}_f + \partial_t \rho_b + \nabla \cdot ( \vec{j}_P + \nabla \times \vec{j}_M ) = \\ \end{split}\]

    since \(\nabla \cdot \nabla \times \vec{m} \equiv 0\), and keeping separated the contributions of free and bound charges,

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

  • and into Ampére-Maxwell’s law

    \[\begin{split}\begin{aligned} \vec{0} & = \nabla \times \vec{b} - \mu_0 \varepsilon_0 \partial_t \vec{e} - \mu_0 \vec{j} = \\ & = \nabla \times \vec{b} - \mu_0 \, \partial_t \left( \vec{d} - \vec{p} \right) - \mu_0 \vec{j}_f - \mu_0 \partial_t \vec{p} - \mu_0 \nabla \times \vec{m} \\ & = \nabla \times \left( \vec{b} - \mu_0 \vec{m} \right) - \ mu_0 \, \partial_t \vec{d} - \mu_0 \vec{j}_f \\ \\ & \rightarrow \qquad \nabla \times \vec{h} - \, \partial_t \vec{d} = \vec{j}_f \ , \end{aligned}\end{split}\]

    where \(\vec{h} := \vec{b} - \mu_0 \vec{m}\), the magnetic field strength.

2.2.2. Examples#

  • conductors

  • ferromagnetic and weak magnetism (para-, dia-, anti-)

2.2.3. Governing equations in differential form#

Differential form of Maxwell’s equations

\[\begin{split}\begin{cases} \nabla \cdot \vec{d} = \rho_f \\ \nabla \times \vec{e} + \partial_t \vec{b} = \vec{0} \\ \nabla \cdot \vec{b} = 0 \\ \nabla \times \vec{h} - \partial_t \vec{d} = \vec{j}_f \end{cases}\end{split}\]

todo continuity equation for charge

2.2.4. Governing equation in integral form#

Integral form of Maxwell’s equations

\[\begin{split}\begin{cases} \displaystyle \oint_{\partial V} \vec{d} \cdot \hat{\vec{n}} = \int_{V} \rho_f \\ \displaystyle \oint_{\partial S} \vec{e} \cdot \hat{\vec{t}} + \dfrac{d}{dt} \int_S \vec{b} \cdot \hat{\vec{n}} = 0 \\ \displaystyle \oint_{\partial V} \vec{b} \cdot \hat{\vec{n}} = 0 \\ \displaystyle \oint_{\partial S} \vec{h} \cdot \hat{\vec{t}} - \dfrac{d}{dt} \int_S \vec{d} \cdot \hat{\vec{n}} = \int_{S} \vec{j}_f \cdot \hat{\vec{n}} \\ \end{cases}\end{split}\]

  • control volume

  • arbitrary domain

  • low-speed relativity

2.2.5. Jump Conditions#

Letting \(V\) and \(S\) “collapse on a discontinuity”…

(2.1)#\[\begin{split}\begin{cases} [ d_n ] = \sigma_f \\ [ e_t ] = 0 \\ [ b_n ] = 0 \\ [ h_t ] = \iota_f \ , \end{cases}\end{split}\]

being \(\sigma_f\) and \(\iota_f\) surface charge and current density, with physical dimension \(\frac{\text{charge}}{\text{surface}}\), and \(\frac{\text{current}}{\text{surface}}\) respectively. These contributions can be thought of as Dirac delta contributions in volume density, namely

\[\rho(\vec{r},t) = \rho_0(\vec{r},t) + \sigma(\vec{r}_s,t) \delta_{1}(\vec{r}-\vec{r}_s) \ ,\]

being \(\rho(\vec{r},t)\) the regular part of the volume density in all the points of the domain \(\vec{r} \in V\), \(\sigma(\vec{r}_s,t)\) the surface density on 2-dimensional surfaces \(\vec{r}_s \in S\), \(\delta_1()\) the Dirac’s delta with physical dimension \(\frac{1}{\text{length}}\).

If there’s no free surface charge and currents, jump conditions for linear media become

(2.2)#\[\begin{split}\begin{cases} [ d_n ] = 0 \\ [ e_t ] = 0 \\ [ b_n ] = 0 \\ [ h_t ] = 0 \ , \end{cases} \qquad \rightarrow \qquad \begin{cases} d_{n,1} = d_{n,2} \quad \rightarrow \quad \varepsilon_1 e_{n,1} = \varepsilon_2 e_{n,2} \\ e_{t,1} = e_{t,2} \\ b_{n,1} = b_{n,2} \\ h_{t,1} = h_{t,2} \quad \rightarrow \quad \frac{1}{\mu_1} b_{t,1} = \frac{1}{\mu_2} b_{t,2} \\ \end{cases} \end{split}\]