1.5: Maxwell's Equations

In this and following sections, EM theory is presented in a form that aids in the understanding of distributed effects, such as propagation on transmission lines, coupling of transmission lines, and how transmission line effects can be used to realize components with unique functionality. While this is a review of material that most readers have previously learned, it is presented in a slightly different form than is usual. The treatment begins with Maxwell’s equations and not the static field laws. This is not the way EM theory is initially presented.

Maxwell’s equations are a remarkable insight and the early field laws can be derived from them. Most importantly, Maxwell’s equations describe the propagation of an EM field. Maxwell’s equations are presented in point form in Section 1.5.1 and in integral form in Section 1.5.5. From these, the early electric and magnetic field laws are derived. The effect of boundary conditions are introduced in Section 1.8 to arrive at implications for multimoding on transmission lines. Multimoding is almost always undesirable, and in designing transmission line structures so that multimoding is avoided, it is necessary to have rules that establish when multimoding can occur.

1.5.1 Point Form of Maxwell's Equaions

The characteristics of EM fields are described by Maxwell’s equations:

\[\label{eq:1}\nabla\times\overline{\mathcal{E}}=-\frac{\partial\overline{\mathcal{B}}}{\partial t}-\overline{\mathcal{J}}_{m} \]

\[\label{eq:2}\nabla\cdot\overline{\mathcal{D}}=\rho_{V} \]

\[\label{eq:3}\nabla\times\overline{\mathcal{H}}=\frac{\partial\overline{\mathcal{D}}}{\partial t}+\overline{\mathcal{J}} \]

\[\label{eq:4}\nabla\cdot\overline{\mathcal{B}}=\rho_{mV} \]

where \(\mu\) is called the permeability of the medium and \(\varepsilon\) is called the permittivity of the medium. (The left-hand side of Equation \(\eqref{eq:1}\) is read as “curl E” and the left-hand side of Equation \(\eqref{eq:2}\) as “div E.”) They are the property of a medium and describe the ability to store magnetic energy and electric energy. The other quantities in Equations \(\eqref{eq:1}\)–\(\eqref{eq:4}\) are

• \(\overline{\mathcal{E}}\), the electric field, with units of volts per meter (\(\text{V/m}\)), a time-varying vector
• \(\overline{\mathcal{D}}\), the electric flux density, with units of coulombs per square meter (\(\text{C/m}^{2}\))
• \(\overline{\mathcal{H}}\), the three-dimensional magnetic field, with units of amperes per meter (\(\text{A/m}\))
• \(\overline{\mathcal{B}}\), the magnetic flux density, with units of teslas (\(\text{T}\))
• \(\overline{\mathcal{J}}\), the electric current density, with units of amperes per square meter (\(\text{A/m}^{2}\))
• \(\rho_{V}\), the electric charge density, with units of coulombs per cubic meter (\(\text{C/m}^{3}\))
• \(\rho_{mV}\), the magnetic charge density, with units of webers per cubic meter (\(\text{Wb/m}^{3}\))
• \(\overline{\mathcal{J}}_{m}\) is the magnetic current density, with units of webers per second per square meter (\(\text{Wb}\cdot\text{s}^{-1}\cdot\text{m}^{-2}\)).

Magnetic charges do not exist, but their introduction in Maxwell’s equations through the magnetic charge density, \(\rho_{mV}\), and the magnetic current density, \(\overline{\mathcal{J}}_{m}\), introduce an aesthetically appealing symmetry. Maxwell’s equations are differential equations, and as with most differential equations, their solution is obtained with particular boundary conditions that here are imposed by conductors. Electric conductors (i.e., electric walls) support electric charges and hence electric current. By analogy, magnetic walls support magnetic charges and magnetic currents. Magnetic walls also provide boundary conditions to be used in the solution of Maxwell’s equations. The notion of magnetic walls is important in RF and microwave engineering, as they are approximated by the boundary between two dielectrics of different permittivity. The greater the difference in permittivity, the more closely the boundary approximates a magnetic wall. As a result, the analysis of many structures with different dielectrics can be simplified, aiding in intuitive understanding.

The fields in Equations \(\eqref{eq:1}\)–\(\eqref{eq:4}\) are three-dimensional fields, e.g.,

\[\label{eq:5}\overline{\mathcal{E}}=\mathcal{E}_{x}\hat{\mathbf{x}}+\mathcal{E}_{y}\hat{\mathbf{y}}+\mathcal{E}_{z}\hat{\mathbf{z}} \]

where \(\hat{\mathbf{x}},\:\hat{\mathbf{y}},\) and \(\hat{\mathbf{z}}\) are the unit vectors (having a magnitude of \(1\)) in the \(x,\: y\) and \(z\) directions, respectively. \(\mathcal{E}_{x},\:\mathcal{E}_{y},\) and \(\mathcal{E}_{z}\) are the electric field components in the \(x,\: y,\) and \(z\) directions, respectively.

The symbols and units used with the various field quantities and some of the other symbols to be introduced soon are given in Table \(\PageIndex{1}\). \(\overline{\mathcal{B}}\) and \(\overline{\mathcal{H}}\), and \(\overline{\mathcal{D}}\) and \(\overline{\mathcal{E}}\) are related to each other by the properties of the medium embodied

Symbol	SI Unit	SI Unit	Name and Base Units
\(E\)	volts per meter	\(\text{V/m}\)	electric field intensity base unit: \(\text{kg}\cdot\text{m}\text{s}^{-3}\cdot\text{A}^{-1}\)
\(H\)	amps per meter	\(\text{A/m}\)	magnetic field intensity
\(D\)	coulombs per square meter	\(\text{C/m}^{2}\)	\(D=\varepsilon E\), electric flux density base unit: \(\text{A}\cdot\text{s}\cdot\text{m}^{-2}\)
\(B\)	tesla, webers per square meter	\(\text{T}\)	\(B=\mu H\), magnetic flux density base unit: \(\text{kg}\cdot\text{s}^{-2}\cdot\text{A}^{-1}\)
\(I\)	amp	\(\text{A}\)	electric current
\(M\)	amps per meter	\(\text{A/m}\)	magnetization base unit: \(\text{A}\cdot\text{m}^{-1}\)
\(q_{e}\)	coulomb	\(\text{C}\)	electric charge base unit: \(\text{A}\cdot\text{s}\)
\(q_{m}\)	weber	\(\text{Wb}\)	magnetic charge base unit: \(\text{kg}\cdot\text{m}^{2}\cdot\text{s}^{-2}\cdot\text{A}^{-1}\)
\(\psi_{e}\)	coulomb	\(\text{C}\)	electric flux base unit: \(\text{A}\cdot\text{s}\)
\(\psi_{m}\)	weber	\(\text{Wb}\)	magnetic flux base unit: \(\text{kg}\cdot\text{m}^{2}\cdot\text{s}^{-2}\cdot\text{A}^{-1}\)
\(\rho_{V}\)	coulombs per cubic meter	\(\text{C}\cdot\text{m}^{-3}\)	charge density base unit: \(\text{A}\cdot\text{s}\cdot\text{m}^{-3}\)
\(\rho_{S}\)	coulombs per square emter	\(\text{C/m}^{2}\)	surface charge density base unit: \(\text{A}\cdot\text{s}\cdot\text{m}^{-2}\)
\(\rho_{mV}\)	webers per cubic meter	\(\text{Wb/m}^{3}\)	magnetic charge density base unit: \(\text{kg}\cdot\text{m}^{−1}\cdot\text{s}^{−2}\cdot\text{A}^{−1}\)
\(\rho_{mS}\)	webers per square meter	\(\text{Wb/m}^{2}\)	surface magnetic charge density base unit: \(\text{kg}\cdot\text{s}^{−2}\cdot\text{A}^{−1}\)
\(J\)	amps per square meter	\(\text{A/m}^{2}\)	electric current density
\(J_{S}\)	amps per meter	\(\text{A/m}\)	surface electric current density
\(J_{m}\)	webers per second per square meter	\(\text{Wb}\cdot\text{s}^{-1}\cdot\text{m}^{-2}\)	magnetic current density base unit: \(\text{kg}\cdot\text{s}^{−3}\cdot\text{A}^{−1}\)
\(J_{mS}\)	webers per second per meter	\(\text{Wb}\cdot\text{s}^{-1}\cdot\text{m}^{-1}\)	surface magnetic current density base unit: \(\text{kg}\cdot\text{m}^{−1}\cdot\text{s}^{−3}\cdot\text{A}^{−1}\)
\(S\)	square meters	\(\text{m}^{2}\)	surface
\(V\)	cubic meters	\(\text{m}^{3}\)	volume
\(\varepsilon\)	farads per meter	\(\text{F/m}\)	permittivity base unit: \(\text{kg}^{−1}\cdot\text{m}^{−3}\cdot\text{A}^{2}\cdot\text{s}^{4}\)
\(\mu\)	henry per meter	\(\text{H/m}\)	permeability base unit: \(\text{kg}\cdot\text{m}\cdot\text{s}^{−2}\cdot\text{A}^{−2}\)
\(dS\)	square meter	\(\text{m}^{2}\)	incremental area
\(d\ell\)	meter	\(\text{m}\)	incremental length
\(dV\)	cubic meter	\(\text{m}^{3}\)	incremental volume
\(\oint_{C}\)			integral around a closed contour
\(\oint_{S}\)			integral over a closed surface
\(\int_{V}\)			integral over a closed surface volume integral

Table \(\PageIndex{1}\): Quantities used in Maxwell’s equations. Magnetic charge and current are introduced in establishing boundary conditions, especially at dielectric interfaces. The SI units of other quantities used in RF and microwave engineering are given in Table 2.A.1.

in \(\mu\) and \(\varepsilon\):

\[\label{eq:6}\overline{\mathcal{B}}=\mu\overline{\mathcal{H}} \]

\[\label{eq:7}\overline{\mathcal{D}}=\varepsilon\overline{\mathcal{E}} \]

The quantity \(\mu\) is called the permeability and describes the ability to store magnetic energy in a region. The permeability in free space (or vacuum) is denoted \(\mu_{0}\) and the magnetic flux and magnetic field are related as (where \(\mu_{0} = 4\pi\times 10^{−7}\text{ H/m}\))

\[\label{eq:8}\overline{\mathcal{B}}=\mu_{0}\overline{\mathcal{H}} \]

1.5.2 Moments and Polarization

In this subsection the response of materials to electric and magnetic fields are described. In a later section there will be a more specific discussion about the interaction of dielectrics and metals to electric fields. The main purpose here is to define electric and magnetic polarization vectors.

Material Response to an Applied Electric Field

The generation of electric moments is the atomic-level response of a material to an applied electric field. With an atom the center of negative charge, the center of the electron cloud, and the center of the positive charge, the nucleus, overlap. When an electric field is applied the centers of positive and negative charge separate and electrical energy additional to that stored in free space is stored in a manner very similar to potential energy storage in a stretched spring. The separation of charge (charge by distance) forms an electric dipole which has an electric dipole moment \(\overline{p}\) having the SI units of \(\text{C}\cdot\text{m}\) or \(\text{A}\cdot\text{s}\cdot\text{m}\). The same occurs with solids but now the centers of positive and negative charge could be separated even without an external electric field and the material is said to be polarized. If there are \(n\) electric dipole moments \(\overline{p}\) per unit volume (in SI per \(\text{m}^{3}\)), then the polarization density (or electric polarization or simply polarization) is

\[\label{eq:9}\overline{\mathcal{P}}=n\overline{p} \]

which has the SI units of \(\text{C/m}^{2}\). \(\overline{\mathcal{P}}\) has the same units and has the same effect as the electric flux density \(\overline{\mathcal{D}}\). In a homogeneous, linear, isotropic material, \(\overline{\mathcal{P}}\) is proportional to the applied electric field intensity \(\overline{\mathcal{E}}\). A homogenous material has the same average properties everywhere and an isotropic material looks the same in all directions. Materials that are not isotropic are called anisotropic and most commonly these are crystals that have asymmetrical unit cells so that the separation of charge, the electric energy stored by this separation of charge (the strength of the spring) in response to an applied field depends on direction. As a result

\[\label{eq:10}\overline{\mathcal{P}}=\check{χ}_{e}\varepsilon_{0}\overline{\mathcal{E}} \]

where \(\check{χ}_{e}\) is called the electric susceptibility of the material. The electric susceptibility has been written as a tensor, a \(3\times 3\) matrix that relates the three directions of \(\overline{\mathcal{E}}\) to the three possible directions of \(\overline{\mathcal{P}}\). For an amorphous solid or a crystal with high unit cell symmetry, the nine elements of \(\check{χ}_{e}\) are the same and so the tensor notation is dropped and \(χ_{e}\) is used as a scalar.

Material Response to an Applied Magnetic Field

All materials contain electrons each having a charge and a spin. The charges produce an electric field and the spin produces a magnetic field. The magnetic field produced by an electron is almost that which would be produced by a mechanically spinning charge, although there is no actual mechanical rotation. Spin is a quantum mechanical property. Still it is very hard to avoid using the rotating charge analogy. The spin of an electron points in a particular direction and in most materials these spins are randomly arranged and cancel each other out so that there is no net magnetic effect. Some materials can be magnetized by a large externally applied magnetic field and the spins are then permanently aligned even when the external magnetic field is removed. Sometimes the alignment is established by crystal geometry.

Consider a magnetic material without permanent magnetization. When a magnetic field is applied to these materials the electron spin vector tends to rotate and energy is stored. The ability to store magnetic energy above that which would be stored in vacuum is described by the concept of relative permeability.

The response of a material to an applied magnetic field is not directly analogous to the response to an applied electric field because of the fundamental difference in the source of magnetic moments. In most materials the magnetic moment of one electron is paired with an electron occupying the same orbital with a magnetic moment in the opposite direction so that there is no net magnetic moment. In magnetic materials there are some orbitals that do not have a pair of electrons so that there is a net magnetic moment. Additional energy is stored when the net magnetic moments are rotated away from their preferred long-term direction by an externally applied magnetic field \(\overline{\mathcal{H}}\). This is like storing energy in a spring. It is as though there is an additional magnetic field called the magnetic polarization which has a particular density. The net magnetic polarization density is denoted \(\overline{\mathcal{M}}\) which has the SI units of \(\text{A/m}\) and

\[\label{eq:11}\overline{\mathcal{M}}=\check{χ}_{m}\overline{\mathcal{H}} \]

where \(\check{χ}_{m}\) is called the magnetic susceptibility of the material and in general can be a \(3\times 3\) matrix. The relative permeability is defined as

\[\label{eq:12}\check{\mu}_{r}=1+\check{χ}_{m} \]

Summary

A vacuum can store electric and magnetic energy. Materials can store additional energy and the propensity relative to vacuum is described by the relative permittivity and permeability, respectively, of the material.

1.5.3 Field Intensity and Flux Density

Permeability, \(\mu\), is generally a scalar, but in magnetic materials it can be a \(3\times 3\) matrix called a dyadic tensor with the relative permeability tensor being \(\check{\mu}_{r}\). Then

\[\label{eq:13}\overline{\mathcal{B}}=\mu_{0}(\overline{\mathcal{H}}+\overline{\mathcal{M}}) \]

where \(\overline{\mathcal{M}}\) is the magnetization of the material. Introducing phasors

\[\label{eq:14}\overline{\mathcal{B}}=\mu_{0}(\overline{\mathcal{H}}+\check{χ}_{m}\overline{\mathcal{H}})=\mu_{0}\check{\mu}_{r}\overline{\mathcal{H}} \]

\[\label{eq:15}\left[\begin{array}{c}{\mathcal{B}_{x}}\\{\mathcal{B}_{y}}\\{\mathcal{B}_{z}}\end{array}\right] =\check{\mu}_{r}\left[\begin{array}{c}{\mathcal{H}_{x}}\\{\mathcal{H}_{y}}\\{\mathcal{H}_{z}}\end{array}\right] =\left[\begin{array}{ccc}{\mu_{xx}}&{\mu_{xy}}&{\mu_{xz}}\\{\mu_{yx}}&{\mu_{yy}}&{\mu_{yz}}\\{\mu_{zx}}&{\mu_{zy}}&{\mu_{zz}}\end{array}\right] \left[\begin{array}{c}{\mathcal{H}_{x}}\\{\mathcal{H}_{y}}\\{\mathcal{H}_{z}}\end{array}\right] \]

The direction-dependent property indicated by this dyadic results from alignment of electron spins in a material. However, in most materials there is no alignment of spins and \(\mu = \mu_{0}\). The relative permeability, \(\mu_{r}\), refers to the ratio of permeability of a material to its value in a vacuum:

\[\label{eq:16}\mu_{r}=\frac{\mu}{\mu_{0}} \]

So \(\mu_{r} > 1\) indicates that a material can store more magnetic energy than can a vacuum in a given volume.

The other material quantity is the permittivity, \(\varepsilon\), which is denoted \(\varepsilon_{0}\), and in a vacuum

\[\label{eq:17}\mathcal{D}=\varepsilon_{0}\mathcal{E} \]

where \(\varepsilon_{0} = 8.854\times 10^{−12}\text{ F}\cdot\text{m}^{−1}\). The relative permittivity, \(\varepsilon_{r}\), refers to the ratio of permeability of a material to its value in a vacuum:

\[\label{eq:18}\varepsilon_{r}=\frac{\varepsilon}{\varepsilon_{0}} \]

So \(\varepsilon_{r} > 1\) indicates that a material can store more electric energy in a volume than can be stored in a vacuum. In some calculations it is useful to introduce an electric polarization, \(P_{e}\). Then

\[\label{eq:19}\mathcal{D}=\varepsilon\mathcal{E}=\varepsilon_{0}\mathcal{E}+\mathcal{P}_{e} \]

and so the polarization vector is

\[\label{eq:20}\mathcal{P}_{e}=(\varepsilon -\varepsilon_{0})\mathcal{E}=\varepsilon_{0}χ_{e}\mathcal{E} \]

where \(χ_{e}\) is called the electric susceptibility.

Some materials require a dyadic form of \(\varepsilon\). This usually indicates a dependence on crystal symmetry, and the relative movement of charge centers in different directions when an \(E\) field is applied. Some commonly used microwave substrates, such as sapphire, have permittivities that are direction dependent rather than having full dyadic permittivity. A material in which the permittivity is a function of direction is called an anisotropic material or is said to have dielectric anisotropy. In an isotropic material the permittivity is the same in all directions; the material has dielectric isotropy. Most materials are isotropic.

Maxwell’s original equations were put in the form of Equations \(\eqref{eq:1}\)–\(\eqref{eq:4}\) by the mathematician Oliver Heaviside from the less convenient form Maxwell originally used. The above equations are called the point form of Maxwell’s equations, relating the field components to each other and to charge and current density at a point.

Maxwell’s equations have three types of derivatives. First, there is the time derivative, \(\partial /\partial t\). Then there are two spatial derivatives: \(\nabla\times\), called curl, capturing the way a field circulates spatially (or the amount that it curls up on itself); and \(\nabla\cdot\), called the div operator, describing the spreading out of a field (i.e., its divergence). Curl and div have different forms in different coordinate systems, and in the rectangular system can be expanded as (\(\mathbf{A} = A_{x}\hat{\mathbf{x}} + A_{y}\hat{\mathbf{y}} + A_{z}\hat{\mathbf{z}})\)

\[\label{eq:21}\nabla\times\mathbf{A}=\left(\frac{\partial A_{z}}{\partial y}-\frac{\partial A_{y}}{\partial z}\right)\hat{\mathbf{x}}+\left(\frac{\partial A_{x}}{\partial z}-\frac{\partial A_{z}}{\partial x}\right)\hat{\mathbf{y}}+\left(\frac{\partial A_{y}}{\partial x}-\frac{\partial A_{x}}{\partial y}\right)\hat{\mathbf{z}} \]

and

\[\label{eq:22}\nabla\cdot\mathbf{A}=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z} \]

Curl and div in cylindrical and spherical coordinates are given in Equations (1.A.48), (1.A.49), (1.A.57), and (1.A.58). Studying Equation \(\eqref{eq:21}\) you will see that curl, \(\nabla\times\), describes how much a field circles around the \(x,\: y,\) and \(z\) axes. That is, the curl describes how a field circulates on itself. So Equation \(\eqref{eq:1}\) relates the amount an electric field circulates on itself to changes of the \(B\) field in time (and modified by the magnetic current). So a spatial derivative of electric fields is related to a time derivative of the magnetic field. Also in Equation \(\eqref{eq:3}\), the spatial derivative of the magnetic field is related to the time derivative of the electric field (and modified by the electric current). These are the key elements that result in self-sustaining propagation.

Div, \(\nabla\cdot\), describes how a field spreads out from a point. So the presence of net electric charge (say, on a conductor) will result in the electric field spreading out from a point (see Equation \(\eqref{eq:2}\)). The magnetic field (Equation \(\eqref{eq:4}\)) can never diverge from a point, which is a result of magnetic charges not existing. A magnetic wall approximation describes an open circuit and then effectively magnetic charges terminate the magnetic field. What actually happens in free space or on a transmission line depends on boundary conditions, and in the case of transmission lines, the dimensions involved.

How fast a field varies with time, \(\partial\overline{\mathcal{B}}/\partial t\) and \(\partial\overline{\mathcal{D}}/\partial t\), depends on frequency. The more interesting property is how fast a field can change spatially, \(\nabla\times\overline{\mathcal{E}}\) and \(\nabla\times\overline{\mathcal{H}}\)—this depends on wavelength relative to geometry. So if the cross-sectional dimensions of a transmission line are much less than a wavelength (say, less than \(\lambda /4\)), then it will be impossible for the fields to curl up on themselves and so there will be only one or, in some cases, no solutions to Maxwell’s equations.

Example \(\PageIndex{1}\): Energy Storage

Consider a material with a relative permittivity of \(65\) and a relative permeability of \(1000\). There is a static electric field \(E\) of \(1\text{ kV/m}\). How much energy is stored in the \(E\) field in a \(10\text{ cm}^{3}\) volume of the material?

Solution

Energy stored in a static electric field \(=\int_{V}\overline{D}\cdot\overline{E}\cdot dv\):

\[\begin{aligned}D&=\varepsilon E\text{ and field is constant (uniform)}\nonumber\\ \text{Energy stored }&=\varepsilon E^{2}\times (\text{volume})=65\varepsilon_{0}\cdot (10^{3})^{2}\cdot(10\cdot 10^{−6})= 5.75\cdot 10^{−9}\text{ J} = 5.75\text{ nJ}\nonumber\end{aligned} \nonumber \]

The complete calculation using SI units is

\[\begin{aligned}\text{Energy stored }&= 65(8.854\cdot 10^{−12}\cdot\text{F}\cdot\text{m}^{−1})(10^{3}\cdot\text{V}\cdot\text{m}^{-1})^{2}\cdot (10\cdot 10^{−6}\cdot\text{m}^{3})\nonumber \\ &=65(8.854\cdot 10^{−12}\cdot\text{kg}^{−1}\cdot\text{m}^{−3}\cdot\text{A}^{2}\cdot\text{s}^{4})\cdot (10^{3}\cdot\text{kg}\cdot\text{m}^{2}\cdot\text{A}^{−1}\cdot\text{s}^{−3}\cdot\text{m}^{−1})^{2}\cdot (10^{−5}\cdot\text{m}^{3})\nonumber \\ &= 65\cdot 8.854\cdot 10^{−12}\cdot 10^{6}\cdot 10^{−5}\cdot\text{kg}\cdot\text{m}^{2}\cdot\text{s}^{−2} = 5.75\cdot 10^{−9}\cdot\text{kg}\cdot\text{m}^{2}\cdot\text{s}^{-2}\nonumber \\ &= 5.75\cdot 10^{−9}\text{ J} = 5.75\text{ nJ}\nonumber\end{aligned} \nonumber \]

Example \(\PageIndex{2}\): Polarization Vector

A time-varying electric field in the \(x\) direction has a strength of \(100\text{ V/m}\) and a frequency of \(10\text{ GHz}\). The medium has a relative permittivity of \(10\). What is the polarization vector? Express this vector in the time domain?

Solution

\[\overline{\mathcal{D}}=\varepsilon\overline{\mathcal{E}}=\varepsilon_{0}\overline{\mathcal{E}}+\overline{\mathcal{P}}_{e}\nonumber \]

The polarization vector is

\[\overline{\mathcal{P}}_{e}=(\varepsilon -\varepsilon_{0})\overline{\mathcal{E}}=(10-1)\varepsilon_{0}\overline{\mathcal{E}}=9\varepsilon_{0}\overline{\mathcal{E}}\nonumber \]

Now \(\overline{\mathcal{E}}=100\cos(2\cdot\pi\cdot 10^{10}t)\hat{\mathbf{x}}\)

and \(\varepsilon_{0}=8\cdot 85\cdot 10^{12}\), so

\[\begin{align}\overline{\mathcal{P}}_{e}&= 9(8.85\times 10^{−12})\cdot (100)\cos(6.28\times 10^{10}t)\hat{\mathbf{x}}\nonumber \\ &= 79.7\times 10^{−10}\cdot\cos(6.28\times 10^{10}t)\hat{\mathbf{x}} \nonumber \\ \label{eq:23} &= 7.79\cos(6.28\times 10^{10}t)\hat{\mathbf{x}}\text{ nC/m}^{2} \end{align} \]

1.5.4 Maxwell's Equations in Phasor Form

Phasors reduce the dimensionality of Maxwell’s equations by replacing a time derivative by a complex scalar. The phasor form is used with a cosinusoidally varying quantity so, for example, an \(x\)-directed electric field in rectangular coordinates,

\[\label{eq:24}\overline{\mathcal{E}}=|E_{x}|\cos(\omega t+\phi)\hat{x} \]

is represented as the phasor

\[\label{eq:25}E_{x}=|E_{x}|e^{\jmath\phi}\hat{x} \]

The full three-dimensional electric field is

\[\label{eq:26}\overline{\mathcal{E}}=|E_{x}|\cos(\omega t+\phi_{x})\hat{\mathbf{x}}+|E_{y}|\cos(\omega t+\phi_{y})\hat{\mathbf{y}}+|E_{z}|\cos(\omega t+\phi_{z})\hat{\mathbf{z}} \]

and has the phasor form

\[\label{eq:27}\overline{E}=|E_{x}|e^{\jmath\phi_{x}}\hat{\mathbf{x}}+|E_{y}|e^{\jmath\phi_{y}}\hat{\mathbf{y}}+|E_{z}|e^{\jmath\phi_{z}}\hat{\mathbf{z}} \]

To express Maxwell’s equations in phasor form, the full complex form of Equation \(\eqref{eq:26}\) must be considered. Using Equation (1.A.8), Equation \(\eqref{eq:26}\) becomes

\[\begin{align}\overline{\mathcal{E}}&=\frac{1}{2}(|E_{x}|e^{\jmath\phi_{x}}\hat{\mathbf{x}}+|E_{y}|e^{\jmath\phi_{y}}\hat{\mathbf{y}}+|E_{z}|e^{\jmath\phi_{z}}\hat{\mathbf{z}})e^{\jmath\omega t}\nonumber \\ \label{eq:28} &\quad +\frac{1}{2}(|E_{x}|e^{-\jmath\phi_{x}}\hat{\mathbf{x}}+|E_{y}|e^{-\jmath\phi_{y}}\hat{\mathbf{y}}+|E_{z}|e^{-\jmath\phi_{z}}\hat{\mathbf{z}})e^{-\jmath\omega t}\end{align} \]

with the time derivative

\[\begin{align}\frac{\partial\mathbf{E}}{\partial t}&=\frac{1}{2}\jmath\omega(|E_{x}|e^{\jmath\phi_{x}}\hat{\mathbf{x}}+|E_{y}|e^{\jmath\phi_{y}}\hat{\mathbf{y}}+|E_{z}|e^{\jmath\phi_{z}}\hat{\mathbf{z}})e^{\jmath\omega t}\nonumber \\ \label{eq:29} &\quad -\frac{1}{2}\jmath\omega(|E_{x}|e^{-\jmath\phi_{x}}\hat{\mathbf{x}}+|E_{y}|e^{-\jmath\phi_{y}}\hat{\mathbf{y}}+|E_{z}|e^{-\jmath\phi_{z}}\hat{\mathbf{z}})e^{-\jmath\omega t}\end{align} \]

Equation \(\eqref{eq:29}\) has the phasor form (from the \(e^{\jmath\omega t}\) component)

\[\label{eq:30}\frac{2}{e^{\jmath\omega t}}\left\{\begin{array}{l}{e^{\jmath\omega t}} \\ {\text{component of }} \end{array} \frac{\partial\overline{\mathcal{E}}}{\partial t}\right\} =\jmath\omega (|E_{x}|e^{\jmath\phi_{x}}\hat{\mathbf{x}}+|E_{y}|e^{\jmath\phi_{y}}\hat{\mathbf{y}}+|E_{z}|e^{\jmath\phi_{z}}\hat{\mathbf{z}}) \]

Noting this, Maxwell’s equations in phasor form, from Equations \(\eqref{eq:1}\)–\(\eqref{eq:4}\), are

\[\begin{align} \label{eq:31}\nabla\times\overline{E}&=-\jmath\omega\overline{B}-\overline{J}_{m}\\ \label{eq:32}\nabla\times\overline{H} &=\jmath\omega\overline{D}+\overline{J} \\ \label{eq:33}\nabla\cdot\overline{D}&=\rho_{V} \\ \label{eq:34} \nabla\cdot\overline{B}&=\rho_{mV}\end{align} \]

1.5.5 Integral Form of Maxwell's Equations

It is sometimes more convenient to use the integral forms of Maxwell’s equations, and Equations \(\eqref{eq:1}\)–\(\eqref{eq:4}\) become

\[\begin{align} \label{eq:35}\oint_{s}\nabla\times\overline{\mathcal{E}}\cdot d\mathbf{s}=\oint_{C}\overline{\mathcal{E}}\cdot d\ell &=-\oint_{s}\frac{\partial\overline{\mathcal{B}}}{\partial t}\cdot d\mathbf{s}-\oint_{s}\overline{\mathcal{J}}_{m}\cdot d\mathbf{s} \\ \label{eq:36} \oint_{s}\nabla\times\overline{\mathcal{H}}\cdot d\mathbf{s} =\oint_{C}\overline{\mathcal{H}}\cdot d\ell &=\int_{s}\frac{\partial\overline{\mathcal{D}}}{\partial t}\cdot d\mathbf{s}+\overline{\mathcal{I}} \\ \label{eq:37} \int_{v}\nabla\cdot\overline{\mathcal{D}}dv =\oint_{s}\overline{\mathcal{D}}\cdot d\mathbf{s}&=\int_{V}\rho_{v}dv=Q_{\text{enclosed}} \\ \label{eq:38} \int_{v}\nabla\cdot\overline{\mathcal{B}}dv =\oint_{s}\overline{\mathcal{B}}\cdot d\mathbf{s}&=0\end{align} \]

where \(Q_{\text{enclosed}}\) is the total charge in the volume enclosed by the surface, \(S\). The subscript \(S\) on the integral indicates a surface integral and the circle on the integral sign indicates that the integral is over a closed surface. Two mathematical identities were used in developing Equations \(\eqref{eq:35}\)–\(\eqref{eq:38}\). The first identity is Stoke’s theorem, which states that for any vector field \(\mathbf{X}\),

\[\label{eq:39}\oint_{\ell}\mathbf{X}\cdot d\ell =\oint_{s}(\nabla\times\mathbf{X})\cdot d\mathbf{s} \]

So the contour integral of \(X\) around a closed contour, \(C\) (with incremental length vector \(d\ell\)), is the integral of \(\nabla\times X\) over the surface enclosed by the contour and \(ds\) is the incremental area multiplied by a unit vector normal to the surface. This identity is used in Equations \(\eqref{eq:35}\) and \(\eqref{eq:36}\). The divergence theorem is the other identity and is used in Equations \(\eqref{eq:37}\) and \(\eqref{eq:38}\). For any vector field \(\mathbf{X}\), the divergence theorem states that

\[\label{eq:40}\oint_{s}\mathbf{X}\cdot d\mathbf{s}=\int_{v}\nabla\cdot\mathbf{X}dv \]

That is, the volume integral of \(\nabla\cdot\mathbf{X}\) is equal to the closed surface integral of \(\mathbf{X}\).

Using phasors the integral-based forms of Maxwell’s equations become

\[\begin{align} \label{eq:41} \oint_{C}\overline{E}\cdot d\ell &=-\jmath\omega\oint_{s}\overline{B}\cdot d\text{s}-\oint_{s}\overline{J}_{m}\cdot d\mathbf{s} \\ \label{eq:42}\oint_{C}\overline{H}\cdot d\ell&=\jmath\omega\int_{s}\overline{D}\cdot d\mathbf{s}+\overline{I} \\ \label{eq:43}\oint_{s}\overline{D}\cdot d\mathbf{s}&=\int_{V}\rho_{v}dv=Q_{\text{enclosed}} \\ \label{eq:44}\oint_{s}\overline{B}\cdot d\mathbf{s}&=0\end{align} \]