Maxwell's Equations

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Maxwell’s electrodynamics, in many ways, was the crowning achievement of the scientific revolution started nearly two hundred years earlier by Isaac Newton. It yields a systematic mathematical approach to studying the otherwise mysterious phenomena of electromagnetism and, more philosophically important, showed that electricity and magnetism were really two manifestations of the same force. It is interesting to note that by trying to reconcile certain contradictions between Newton’s Mechanics and electromagnetism, scientists of the late 19th and early 20th century had to develop entirely new branches of physics. These culminated in Einstein’s Special and General Relativity Theories as well as Schroedinger and Heisenberg’s Quantum Mechanics.

All of classical electrodynamics is based on four famous partial differential equations: Maxwell’s Equations. These are commonly expressed as

\nabla\cdot\textbf{D} = \rho
\nabla\times\textbf{E} = -\frac{\partial\textbf{B}}{\partial t}
\nabla\cdot\textbf{B} = 0
\nabla\times\textbf{H} = \frac{\partial\textbf{D}}{\partial t}+\textbf{J}

At this point, some clarification of the variable definitions is in order. In the first equation, \textbf{D} represents the electric displacement vector and has units of Coulombs/meter^2. The term “electric displacement” is used mainly for historical reasons. It would be more appropriate, in some ways, to refer to this as “electric flux density.” On the right hand side of this first equation stands ρ, the charge density (a scalar) with units coulombs/meter^3. Note that units are expressed in the SI (standard metric or MKS) system. This system of units will be assumed throughout this article. Other units of measure exist – for example: the CGS and Gaussian system is often favoured by physicists as well as systems which assume the speed of light is normalised to one.

The second equation is the mathematical expression of Faraday’s Law: a changing magnetic field causes electric fields to be induced. \textbf{E} is the electric field vector (units:volts/meter). \textbf{B} is the magnetic flux density (in webers/meter^2 or Tesla).

The right-hand-side of equation 3 is zero because (as we will see in more detail later) magnetic monopoles have not been shown to exist.

The last equation contains Maxwell’s famous contribution – time varying electric fields create magnetic fields. \textbf{H} is the magnetic field intensity vector (in Amperes/meter) and \textbf{J} is the current density given in Amperes/meter^2.

In order to connect all the variables in Maxwell’s equations, some constitutive relationships are needed. These relationships contain information on the space (or materials) where the elecromagnetic fields are found. For a dielectric material, the electric displacement (or flux density) is related to the electric field by

\textbf{D}=\overline{\epsilon}_r \epsilon_0 \textbf{E}.

The relative dielectric permittivity of a material \overline{\epsilon}_r is usually a tensor and is a multiplying factor to the vacuum permittivity \epsilon_0=8.854\times 10^{-12}F/m in SI units.

The magnetic flux density is related to the magnetic field intensity by

\textbf{B}=\overline{\mu}_r \mu_0 \textbf{E},

where \overline{\mu}_r is the tensor multiplier that defines a relative magnetic permeability. The vacuum permeability \mu_0=4\pi \times 10^{-7}H/m.

More complicated relationships between electric and magnetic quantities exist (as in Faraday rotation and electro-optic effects, for example), but for now let us assume for simplicity and clarity that relative permeabilities and permittivities are scalar numbers.

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