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Electromagnetics

Maxwell’s equations

Maxwell’s equations are a system of equations that describe the electromagnetic field and its connection with electric charges and currents. Maxwell’s equations in matter can be written as follows:

\[\begin{split}\begin{align} \mathbf{\nabla} \cdot \mathbf{D} &= \rho && \quad , \quad \textit{(Gauss's law)}\\ \nabla \cdot \mathbf{B} &= 0 && \quad , \quad \textit{(Gauss's law for magnetism)}\\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} && \quad , \quad \textit{(Faraday's law)}\\ \nabla \times \mathbf{H} &= \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} && \quad , \quad\textit{(Ampère's law with Maxwell's correction)} \end{align}\end{split}\]

where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, \(\mathbf{D}\) is the electric displacement field, and \(\mathbf{H}\) is the magnetic field, with \(\rho\) being the density of free electric charges and \(\mathbf{J}\) being the electric current density.

The response of the medium to the electromagnetic field is described by the following constitutive relations:

\[\begin{split}\begin{align} \mathbf{D} &= \varepsilon_0 \mathbf{E} + \mathbf{P}_e &&\quad, \\ \mathbf{B} &= \mu_0 \mathbf{H} + \mathbf{P}_m &&\quad, \end{align}\end{split}\]

where \(\varepsilon_0\) is the vacuum permittivity, \(\mu_0\) is the vacuum permeability, \(\mathbf{P}_e\) is the electric polarization and \(\mathbf{P}_m\) is the magnetic polarization. Note that quite often instead of the magnetic polarization \(\mathbf{P}_m\) one uses the magnetization \(\mathbf{M}\) defined as \(\mathbf{M} = \mathbf{P}_m/\mu_0\).

For media whose response does not depend on the field strength (linear media) and on the frequency of the electromagnetic field (non-dispersive media), the electric and magnetic polarization can be written as \(\mathbf{P}_e = \varepsilon_0 \chi_e \mathbf{E}\) and \(\mathbf{P}_m = \mu_0 \chi_m \mathbf{H}\), where \(\chi_e\) and \(\chi_m\) are the electric and magnetic susceptibilities. Note that in anisotropic media, where the response of the medium depends on the direction of the electromagnetic field, the electric and magnetic susceptibilities are tensors, and the product of \(\chi_e\) or \(\chi_m\) with the corresponding field vector should be understood as a tensor product. Using these expressions for the polarizations we can rewrite the constitutive relations as follows:

\[\begin{split}\begin{align} \mathbf{D} &= \varepsilon_0\varepsilon_r \mathbf{E} &&\quad, \\ \mathbf{B} &= \mu_0\mu_r \mathbf{H} &&\quad, \end{align}\end{split}\]

where \(\varepsilon_r = 1 + \chi_e\) is the medium permittivity and \(\mu_r = 1 + \chi_m\) is the medium permeability.

Time-harmonic fields

Any time-dependent electromagnetic field can be represented as a superposition of harmonic waves, each of which oscillates at a its own frequency frequency. Mathematically, this statement can be expressed as the corresponding Fourier transform of the field vector function, which, using the example of the electric field \(\mathbf{E}\), can be written as

\[\mathbf{E}(\mathbf{r}, t) = \int_{-\infty}^{\infty} \tilde{\mathbf{E}}(\mathbf{r}, \omega) e^{-i\omega t} d\omega,\]

where \(\tilde{\mathbf{E}}(\mathbf{r}, \omega)\) is the complex spectral amplitude of a harmonic wave \(e^{-i\omega t}\) oscillating at the frequency \(f=\omega/2\pi\), with \(\omega\) being the angular frequency and \(\mathbf{r}\) is the spatial coordinate vector.

Taking into account that the spectral amplitude of the time derivative of a function is equal to the spectral amplitude of the function itself, multiplied by \(-i\omega\), we can rewrite the Maxwell’s equations in terms of complex spectral amplitudes as follows:

\[\begin{split}\begin{align} \mathbf{\nabla} \cdot \tilde{\mathbf{D}} &= \tilde{\rho} &&\quad, \\ \nabla \cdot \tilde{\mathbf{B}} &= 0 &&\quad, \\ \nabla \times \tilde{\mathbf{E}} &= i\omega \tilde{\mathbf{B}} &&\quad, \\ \nabla \times \tilde{\mathbf{H}} &= \tilde{\mathbf{J}} -i\omega \tilde{\mathbf{D}} &&\quad, \end{align}\end{split}\]

where the tilde symbol denotes the complex spectral amplitude of the corresponding vector function.

Similar to the electric field vectors, the constitutive relations for the spectral amplitudes can be written as

\[\begin{split}\begin{align} \tilde{\mathbf{D}}(\mathbf{r},\omega) &= \varepsilon (\mathbf{r},\omega) \tilde{\mathbf{E}}(\mathbf{r},\omega) &&\quad, \\ \tilde{\mathbf{B}}(\mathbf{r},\omega) &= \mu (\mathbf{r},\omega) \tilde{\mathbf{H}}(\mathbf{r},\omega) &&\quad, \end{align}\end{split}\]

where \(\varepsilon\) and \(\mu\) are the complex permittivity and permeability of the medium at a given frequency \(\omega\). The explicit dependence of the permittivity and permeability on the frequency in these expressions allows us to consider the dispersive media whose response depends on the frequency of electromagnetic field.

Models

Electric Models

Electrostatics

Phenomena associated with static electric fields are described by the following pair of equations:

\[\begin{split}\begin{align} \mathbf{\nabla} \cdot \mathbf{D} &= \rho &&\quad, \\ \nabla \times \mathbf{E} &= 0 &&\quad. \end{align}\end{split}\]

Here we see that not every time-independent electric field \(\mathbf{E}\) can represent an electrostatic field, but only one whose curl is zero. We can exploit this property of electrostatic electric fields to reduce a vector problem to a much simpler scalar problem. From the vector analysis we know that any vector function whose curl is zero is equal to the gradient of some scalar vector function. Therefore, we can write the electric field \(\mathbf{E}\) as

\[\mathbf{E} = -\nabla \phi \quad,\]

where \(\phi\) is known as the electric potential.

For a medium we have \(\mathbf{D} = \varepsilon \mathbf{E}\), which allows us to rewrite the first of the electrostatics equation above as

\[\nabla \cdot \left(\varepsilon \nabla \phi\right) = -\rho \quad.\]

This equation is known as Poisson’s equation.

With a proper choice of boundary conditions, Poisson’s equation allows to find the spatial distribution of the electric potential and, as the result, the electric field produced by a given distribution of electric charges in the region of interest.

Note

The model is not yet available.

Current Conduction

In classical electromagnetism, local charge conservation can be expressed as the following continuity equation:

\[\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0 \quad.\]

This equation can be derived from the Ampère’s law, if we apply the divergence to the both sides of the equation: \(\nabla \cdot (\nabla \times \mathbf{H}) = \nabla \cdot (\mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}) = \nabla \cdot \mathbf{J} + \frac{\partial \nabla \cdot \mathbf{D}}{\partial t}\), where we take into account that the divergence of curl of any vector function is zero and use the Gauss’s law for electric displacement field vector.

In the case of steady currents, when the charge density \(\rho\) does not change with time (the number of charges entering a given volume is equal to the number of charges leaving it), the continuity equation reduces to

\[\nabla \cdot \mathbf{J} = 0 \quad.\]

In turn, for a large class of conducting materials, the electric current density \(\mathbf{J}\) follows the Ohm’s law:

\[\mathbf{J} = \sigma \mathbf{E} \quad,\]

where \(\sigma\) is the electric conductivity of the medium. At the same time, the electric field \(\mathbf{E}\) can be expressed through the scalar electric potential \(\phi\) as \(\mathbf{E} = -\nabla \phi\). As a result, from the reduced continuity equation for the electric current density \(\mathbf{J}\) we obtain the following equation for the electric potential \(\phi\):

\[\nabla \cdot \left( \sigma \nabla \phi \right) = 0 \quad.\]

This equation is known as the Laplace’s equation and allows to find the spatial distribution of the electric field in a volume with a given distribution of conductivity under given boundary conditions.

Note

The model is not yet available.

Quasi-magnetostatic Models

Time-domain Magnetic

For low frequencies (usually below 1MHz) the wave effects can usually be neglected. If also the displacement currents can be neglected compared to the conduction currents and no free charges are present, the Maxwell’s equations can be written in the magneto-quasistatic limit as follows:

\[\begin{split}\begin{align} \nabla \times \mathbf{H} &= \mathbf{J} && \quad, \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} && \quad , \\ \nabla \cdot \mathbf{B} &= 0 && \quad. \end{align}\end{split}\]

Since the divergence of the curl of any vector function is always zero, the magnetic field vector \(\mathbf{B}\) can be represented through the corresponding vector potential \(\mathbf{A}\) as

\[\mathbf{B} = \nabla \times \mathbf{A} \quad.\]

For a medium we have \(\mathbf{B} = \mu \mathbf{H}\). Then, from the second of the magnetostatics equations above we obtain

\[\nabla \times \left(\frac{1}{\mu} \nabla \times \mathbf{A}\right) + \sigma \frac{\partial \mathbf{A}}{\partial t} = \mathbf{J} \quad.\]

Note

More details on the time-harmonic problems can be found in the Time-Domain Magnetics sections.

Maxwell Models

Time-harmonic Maxwell

We can rewrite the last two of Maxwell’s time-harmonic equations as

\[\begin{split}\begin{align} \nabla \times \tilde{\mathbf{E}} &= i\omega \mu \tilde{\mathbf{H}} &&\quad, \\ \nabla \times \tilde{\mathbf{H}} &= \tilde{\mathbf{J}} -i\omega \varepsilon \tilde{\mathbf{E}} &&\quad, \end{align}\end{split}\]

By taking the curl of the first equation we can eliminate the magnetic field amplitude and obtain the equation only for the electric field amplitude:

\[\nabla \times (\frac{1}{\mu_r} \nabla \times \tilde{\mathbf{E}}) - \varepsilon_0\mu_0 \varepsilon_r \omega^2 \tilde{\mathbf{E}} = i\omega \mu_0 \tilde{\mathbf{J}}.\]

Note

The model is not yet available.