Overview of Maxwell’s Equations

Purpose

Having provided the set of formative laws for electromagnetics, we present four common representations of Maxwell’s equations. This page is meant to serve as a quick guide. For specific problems, it may be beneficial to begin from less common forms of Maxwell’s equations. Please note however, that all forms can be derived from the expressions presented here.

Maxwell’s equations are comprised of the first four formative laws; i.e. Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law and the Ampere-Maxwell law. The equations can be written in various ways and characterize physical relationships between fields (e,h) and fluxes (b,d). Specific formulations can be obtained through use of the constitutive relations. Maxwell’s equations can be written in frequency or in time and in a differential or integral form:

1. Differential Form in the Time Domain

2. Differential Form in the Frequency Domain

3. Integral Form in the Time Domain

4. Integral Form in the Frequency Domain

This page is designed to be a quick access to the relevant equations with proper notation and units. The equations are appropriate for EM fields in matter. If the fields are in free space, then the same constititive relations are used but with \(\sigma = 0\), \(\mu_0\) and \(\varepsilon_0\). Constitutive equations are also written by assuming that the physical properties are isotropic and non-dispersive. Further elaboration about this can be found in [WH88] (pp. 133) or on the physical properties page.

Variables and Units

The variables and units for relevant quantities in Maxwell’s equations are given here.

Fields

Field	Frequency-Domain	Time-Domain	Units
Electric Field	\(\mathbf{E}\)	\(\mathbf{e}\)	V/m
Magnetic Field	\(\mathbf{H}\)	\(\mathbf{h}\)	A/m

Fluxes

Flux	Frequency-Domain	Time-Domain	Units
Current Density	\(\mathbf{J}\)	\(\mathbf{j}\)	A/m \(\! ^2\)
Electric Displacement	\(\mathbf{D}\)	\(\mathbf{d}\)	C/m \(\! ^2\)
Magnetic Flux Density	\(\mathbf{B}\)	\(\mathbf{b}\)	T

Physical Properties

Property	Symbol	Units
Conductivity	\(\sigma\)	S/m
Resistivity	\(\rho\)	\(\Omega m\)
Permeability	\(\mu\)	H/m
Permittivity	\(\varepsilon\)	F/m

Differential Form in the Time Domain

Here, we present differential forms for Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law and the Ampere-Maxwell equation in the time domain.

\[\begin{split}\begin{align} \textbf{Gauss for E-field:}\;\; &\nabla\cdot\mathbf{d}=\rho_f \\ \textbf{Gauss for B-field:}\;\; &\nabla\cdot\mathbf{b}=0 \\ \textbf{Faraday:} \;\; &\nabla\times\mathbf{e}=-\dfrac{\partial \mathbf{b}}{\partial t} \\ \textbf{Ampere-Maxwell:} \;\; &\nabla\times\mathbf{h}=\mathbf{j} + \dfrac{\partial \mathbf{d}}{\partial t} \end{align}\end{split}\]

where \(\rho_f\) is the free change density and \(\mathbf{j}\) is the free current density. The following constitutive relationships can be used to replace fields and fluxes:

\[\begin{split}\begin{align} \mathbf{j} &= \sigma \mathbf{e}\\ \mathbf{b} &= \mu \mathbf{h}\\ \mathbf{d} &= \varepsilon \mathbf{e} \end{align}\end{split}\]

Differential Form in the Frequency Domain

Here, we present differential forms for Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law and the Ampere-Maxwell equation in the frequency domain:

\[\begin{split}\begin{align} \textbf{Gauss for E-field:} \;\; &\nabla\cdot\mathbf{D}=\rho_f \\ \textbf{Gauss for B-field:} \;\; &\nabla\cdot\mathbf{B}=0 \\ \textbf{Faraday:} \;\; &\nabla\times\mathbf{E}=-i\omega\mathbf{B} \\ \textbf{Ampere-Maxwell:} \;\; &\nabla\times\mathbf{H}=\mathbf{J} + i\omega \mathbf{D} \end{align}\end{split}\]

where \(\rho_f\) is the free change density and \(\mathbf{J}\) is the free current density. The following constitutive relationships can be used to replace fields and fluxes:

\[\begin{split}\begin{align} \mathbf{J} &= \sigma \mathbf{E}\\ \mathbf{B} &= \mu \mathbf{H}\\ \mathbf{D} &= \varepsilon \mathbf{E} \end{align}\end{split}\]

Integral Form in the Time Domain

Here, we present integral forms for Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law and the Ampere-Maxwell equation in the time domain.

\[\begin{split}\begin{align} \textbf{Gauss for E-field:} \; & \int_V\nabla\cdot\mathbf{d}\; dV \! =\!\oint_S\mathbf{d}\cdot d\mathbf{a} \! = \! Q_f \\ \textbf{Gauss for B-field:} \; & \oint_S \mathbf{b} \cdot d \mathbf{a}=0 \\ \textbf{Faraday:} \; & \oint_C \mathbf{e}\cdot d\mathbf{l}=-\int_S\dfrac{\partial \mathbf{b}}{\partial t}\cdot d\mathbf{a} \\ \textbf{Ampere-Maxwell:} \; & \oint_C \!\mathbf{h} \cdot d\mathbf{l} = \! \int_S \!\Big ( \mathbf{j} \!+\! \dfrac{\partial \mathbf{d}}{\partial t} \Big ) \!\cdot d\mathbf{a} \end{align}\end{split}\]

where \(Q_f\) is the total enclose free electrical charge and \(\mathbf{j}\) is the free current density. \(d \mathbf{a}\) is an infinitessimal unit of surface area with vector direction normal to the surface \(S\). \(d \mathbf{l}\) is an infinitessimal length with vector direction along a closed path \(C\). The following constitutive relationships can be used to replace fields and fluxes:

\[\begin{split}\begin{align} \mathbf{j} &= \sigma \mathbf{e}\\ \mathbf{b} &= \mu \mathbf{h}\\ \mathbf{d} &= \varepsilon \mathbf{e} \end{align}\end{split}\]

Integral Form in the Frequency Domain

Here, we present integral forms for Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law and the Ampere-Maxwell equation in the frequency domain.

\[\begin{split}\begin{align} \textbf{Gauss for E-field:} \; &\int_V\nabla\cdot\mathbf{D}\; dV=\oint_S\mathbf{D}\cdot d \mathbf{a} \! =\! Q_f \\ \textbf{Gauss for B-field:} \; &\oint_S \mathbf{B} \cdot d\mathbf{a}=0 \\ \textbf{Faraday:} \; &\oint_C \mathbf{E}\cdot d\mathbf{l}=-i\omega\int_S\mathbf{B}\cdot d\mathbf{a} \\ \textbf{Ampere-Maxwell:} \; & \oint_C \!\mathbf{H} \cdot d\mathbf{l} = \! \int_S \!\big ( \mathbf{J} \!+\! i\omega \mathbf{D} \big ) \!\cdot d\mathbf{a} \end{align}\end{split}\]

where \(Q_f\) is the total enclose free electrical charge and \(\mathbf{J}\) is the free current density. \(d \mathbf{a}\) is an infinitessimal unit of surface area with vector direction normal to the surface \(S\). \(d \mathbf{l}\) is an infinitessimal length with vector direction along a closed path \(C\). The following constitutive relationships can be used to replace fields and fluxes:

\[\begin{split}\begin{align} \mathbf{J} &= \sigma \mathbf{E}\\ \mathbf{B} &= \mu \mathbf{H}\\ \mathbf{D} &= \varepsilon \mathbf{E} \end{align}\end{split}\]