Maxwell's equations

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Maxwell's equations on a plaque on his statue in Edinburgh

Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.

Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum, c (299792458 m/s). Known as electromagnetic radiation, these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays.

In partial differential equation form and a coherent system of units, Maxwell's microscopic equations can be written as (top to bottom: Gauss's law, Gauss's law for magnetism, Faraday's law, Ampère-Maxwell law) $${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =\frac{\rho }{{\epsilon }_0}\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\\ \mathrm{\nabla }\times \mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\\ \mathrm{\nabla }\times \mathbf{B}& ={\mu }_0\left(\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\end{array}}$$ With ${\displaystyle \mathbf{E}}$ the electric field, ${\displaystyle \mathbf{B}}$ the magnetic field, ${\displaystyle \rho }$ the electric charge density and ${\displaystyle \mathbf{J}}$ the current density. ${\displaystyle {\epsilon }_0}$ is the vacuum permittivity and ${\displaystyle {\mu }_0}$ the vacuum permeability.

The equations have two major variants:

• The microscopic equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale.
• The macroscopic equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic-scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials.

The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest. Maxwell's equations in curved spacetime, commonly used in high-energy and gravitational physics, are compatible with general relativity.^{[note 2]} In fact, Albert Einstein developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences.

The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics.

History of the equations

Main article: History of Maxwell's equations

Conceptual descriptions

Gauss's law

Main article: Gauss's law

Electric field from positive to negative charges

Gauss's law describes the relationship between an electric field and electric charges: an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of free space.

Gauss's law for magnetism

Main article: Gauss's law for magnetism

Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.

Gauss's law for magnetism states that electric charges have no magnetic analogues, called magnetic monopoles; no north or south magnetic poles exist in isolation. Instead, the magnetic field of a material is attributed to a dipole, and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field.

Faraday's law

Main article: Faraday's law of induction

In a geomagnetic storm, solar wind plasma impacts Earth's magnetic field causing a time-dependent change in the field, thus inducing electric fields in Earth's atmosphere and conductive lithosphere which can destabilize power grids. (Not to scale.)

The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds to the negative curl of an electric field. In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface.

The electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field and generates an electric field in a nearby wire.

Ampère–Maxwell law

Main article: Ampère's circuital law

Magnetic-core memory (1954) is an application of Ampère's circuital law. Each core stores one bit of data.

The original law of Ampère states that magnetic fields relate to electric current. Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.

Maxwell's modification of Ampère's circuital law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields. As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space.

The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.

Formulation in terms of electric and magnetic fields (microscopic or in vacuum version)

In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate law of nature, the Lorentz force law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. The vector calculus formalism below, the work of Oliver Heaviside, has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations in x, y and z components. The relativistic formulations are more symmetric and Lorentz invariant. For the same equations expressed using tensor calculus or differential forms (see § Alternative formulations).

The differential and integral formulations are mathematically equivalent; both are useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely local and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis.

Key to the notation

Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated. The equations introduce the electric field, E, a vector field, and the magnetic field, B, a pseudovector field, each generally having a time and location dependence. The sources are

• the total electric charge density (total charge per unit volume), ρ, and
• the total electric current density (total current per unit area), J.

The universal constants appearing in the equations (the first two ones explicitly only in the SI formulation) are:

• the permittivity of free space, ε₀, and
• the permeability of free space, μ₀, and
• the speed of light, ${\displaystyle c=({\epsilon }_0{\mu }_0{)}^{-1/2}}$

Differential equations

In the differential equations,

• the nabla symbol, ∇, denotes the three-dimensional gradient operator, del,
• the ∇⋅ symbol (pronounced "del dot") denotes the divergence operator,
• the ∇× symbol (pronounced "del cross") denotes the curl operator.

Integral equations

In the integral equations,

• Ω is any volume with closed boundary surface ∂Ω, and
• Σ is any surface with closed boundary curve ∂Σ,

The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law: $${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}{\iint }_{\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}={\iint }_{\mathrm{\Sigma }}\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\cdot \mathrm{d}\mathbf{S},}$$ Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss' and Stokes' theorems as appropriate.

• ${\displaystyle {\phantom{}}_{\mathrm{\partial }\mathrm{\Omega }}}$ is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed
• ${\displaystyle {\iiint }_{\mathrm{\Omega }}}$ is a volume integral over the volume Ω,
• ${\displaystyle {\oint }_{\mathrm{\partial }\mathrm{\Sigma }}}$ is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed.
• ${\displaystyle {\iint }_{\mathrm{\Sigma }}}$ is a surface integral over the surface Σ,
• The total electric charge Q enclosed in Ω is the volume integral over Ω of the charge density ρ (see the "macroscopic formulation" section below): $${\displaystyle Q={\iiint }_{\mathrm{\Omega }}\rho \mathrm{d}V,}$$ where dV is the volume element.
• The net magnetic flux Φ_B is the surface integral of the magnetic field B passing through a fixed surface, Σ: $${\displaystyle {\mathrm{\Phi }}_B={\iint }_{\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathbf{S},}$$
• The net electric flux Φ_E is the surface integral of the electric field E passing through Σ: $${\displaystyle {\mathrm{\Phi }}_E={\iint }_{\mathrm{\Sigma }}\mathbf{E}\cdot \mathrm{d}\mathbf{S},}$$
• The net electric current I is the surface integral of the electric current density J passing through Σ: $${\displaystyle I={\iint }_{\mathrm{\Sigma }}\mathbf{J}\cdot \mathrm{d}\mathbf{S},}$$ where dS denotes the differential vector element of surface area S, normal to surface Σ. (Vector area is sometimes denoted by A rather than S, but this conflicts with the notation for magnetic vector potential).

Formulation in the SI

Name	Integral equations	Differential equations
Gauss's law	${\displaystyle {\phantom{}}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{E}\cdot \mathrm{d}\mathbf{S}=\frac{1}{{\epsilon }_0}{\iiint }_{\mathrm{\Omega }}\rho \mathrm{d}V}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\frac{\rho }{{\epsilon }_0}}$
Gauss's law for magnetism	${\displaystyle {\phantom{}}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}=0}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$
Maxwell–Faraday equation (Faraday's law of induction)	${\displaystyle {\oint }_{\mathrm{\partial }\mathrm{\Sigma }}\mathbf{E}\cdot \mathrm{d}\mathit{\ell }=-\frac{\mathrm{d}}{\mathrm{d}t}{\iint }_{\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}$
Ampère–Maxwell law	${\displaystyle \begin{array}{rl}{\oint }_{\mathrm{\partial }\mathrm{\Sigma }}& \mathbf{B}\cdot \mathrm{d}\mathit{\ell }={\mu }_0\left({\iint }_{\mathrm{\Sigma }}\mathbf{J}\cdot \mathrm{d}\mathbf{S}+{\epsilon }_0\frac{\mathrm{d}}{\mathrm{d}t}{\iint }_{\mathrm{\Sigma }}\mathbf{E}\cdot \mathrm{d}\mathbf{S}\right)\end{array}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\left(\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)}$

Formulation in the Gaussian system

Main article: Gaussian units

The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of ε₀ and μ₀ into the units (and thus redefining these). With a corresponding change in the values of the quantities for the Lorentz force law this yields the same physics, i.e. trajectories of charged particles, or work done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the electromagnetic tensor: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension. Such modified definitions are conventionally used with the Gaussian (CGS) units. Using these definitions, colloquially "in Gaussian units", the Maxwell equations become:

Name	Integral equations	Differential equations
Gauss's law	${\displaystyle {\phantom{}}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{E}\cdot \mathrm{d}\mathbf{S}=4\pi {\iiint }_{\mathrm{\Omega }}\rho \mathrm{d}V}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=4\pi \rho }$
Gauss's law for magnetism	${\displaystyle {\phantom{}}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}=0}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$
Maxwell–Faraday equation (Faraday's law of induction)	${\displaystyle {\oint }_{\mathrm{\partial }\mathrm{\Sigma }}\mathbf{E}\cdot \mathrm{d}\mathit{\ell }=-\frac{1}{c}\frac{\mathrm{d}}{\mathrm{d}t}{\iint }_{\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{1}{c}\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}$
Ampère–Maxwell law	${\displaystyle \begin{array}{rl}{\oint }_{\mathrm{\partial }\mathrm{\Sigma }}& \mathbf{B}\cdot \mathrm{d}\mathit{\ell }=\frac{1}{c}\left(4\pi {\iint }_{\mathrm{\Sigma }}\mathbf{J}\cdot \mathrm{d}\mathbf{S}+\frac{\mathrm{d}}{\mathrm{d}t}{\iint }_{\mathrm{\Sigma }}\mathbf{E}\cdot \mathrm{d}\mathbf{S}\right)\end{array}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{B}=\frac{1}{c}\left(4\pi \mathbf{J}+\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)}$

The equations simplify slightly when a system of quantities is chosen in the speed of light, c, is used for nondimensionalization, so that, for example, seconds and lightseconds are interchangeable, and c = 1.

Further changes are possible by absorbing factors of 4π. This process, called rationalization, affects whether Coulomb's law or Gauss's law includes such a factor (see Heaviside–Lorentz units, used mainly in particle physics).

Relationship between differential and integral formulations

The equivalence of the differential and integral formulations are a consequence of the Gauss divergence theorem and the Kelvin–Stokes theorem.

Flux and divergence

Volume Ω and its closed boundary ∂Ω, containing (respectively enclosing) a source (+) and sink (−) of a vector field F. Here, F could be the E field with source electric charges, but not the B field, which has no magnetic charges as shown. The outward unit normal is n.

According to the (purely mathematical) Gauss divergence theorem, the electric flux through the boundary surface ∂Ω can be rewritten as

${\displaystyle {\phantom{}}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{E}\cdot \mathrm{d}\mathbf{S}={\iiint }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot \mathbf{E}\mathrm{d}V}$

The integral version of Gauss's equation can thus be rewritten as $${\displaystyle {\iiint }_{\mathrm{\Omega }}\left(\mathrm{\nabla }\cdot \mathbf{E}-\frac{\rho }{{\epsilon }_0}\right)\mathrm{d}V=0}$$ Since Ω is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied if and only if the integrand is zero everywhere. This is the differential equations formulation of Gauss equation up to a trivial rearrangement.

Similarly rewriting the magnetic flux in Gauss's law for magnetism in integral form gives

${\displaystyle {\phantom{}}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}={\iiint }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot \mathbf{B}\mathrm{d}V=0.}$

which is satisfied for all Ω if and only if ${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$ everywhere.

Circulation and curl

Surface Σ with closed boundary ∂Σ. F could be the E or B fields. Again, n is the unit normal. (The curl of a vector field does not literally look like the "circulations", this is a heuristic depiction.)

By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls) over a surface it bounds, i.e. $${\displaystyle {\oint }_{\mathrm{\partial }\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathit{\ell }={\iint }_{\mathrm{\Sigma }}(\mathrm{\nabla }\times \mathbf{B})\cdot \mathrm{d}\mathbf{S},}$$ Hence the Ampère–Maxwell law, the modified version of Ampère's circuital law, in integral form can be rewritten as $${\displaystyle {\iint }_{\mathrm{\Sigma }}\left(\mathrm{\nabla }\times \mathbf{B}-{\mu }_0\left(\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\right)\cdot \mathrm{d}\mathbf{S}=0.}$$ Since Σ can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that the integrand is zero if and only if the Ampère–Maxwell law in differential equations form is satisfied. The equivalence of Faraday's law in differential and integral form follows likewise.

The line integrals and curls are analogous to quantities in classical fluid dynamics: the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of the fluid is the curl of the velocity field.

Charge conservation

The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the Ampère–Maxwell law has zero divergence by the div–curl identity. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives: $${\displaystyle 0=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{B})=\mathrm{\nabla }\cdot \left({\mu }_0\left(\mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)\right)={\mu }_0\left(\mathrm{\nabla }\cdot \mathbf{J}+{\epsilon }_0\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\nabla }\cdot \mathbf{E}\right)={\mu }_0\left(\mathrm{\nabla }\cdot \mathbf{J}+\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}\right)}$$ i.e., $${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{J}=0.}$$ By the Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary:

${\displaystyle \frac{d}{dt}{Q}_{\mathrm{\Omega }}=\frac{d}{dt}{\iiint }_{\mathrm{\Omega }}\rho \mathrm{d}V=-}$ ${\displaystyle {\phantom{}}_{\mathrm{\partial }\mathrm{\Omega }}\mathbf{J}\cdot \mathrm{d}\mathbf{S}=-I_{\mathrm{\partial }\mathrm{\Omega }}.}$

In particular, in an isolated system the total charge is conserved.

Vacuum equations, electromagnetic waves and speed of light

Further information: Electromagnetic wave equation, Inhomogeneous electromagnetic wave equation, Sinusoidal plane-wave solutions of the electromagnetic wave equation, and Helmholtz equation

This 3D diagram shows a plane linearly polarized wave propagating from left to right, defined by E = E₀ sin(−ωt + k ⋅ r) and B = B₀ sin(−ωt + k ⋅ r) The oscillating fields are detected at the flashing point. The horizontal wavelength is λ. E₀ ⋅ B₀ = 0 = E₀ ⋅ k = B₀ ⋅ k

In a region with no charges (ρ = 0) and no currents (J = 0), such as in vacuum, Maxwell's equations reduce to: $${\displaystyle \begin{array}{rlr}\mathrm{\nabla }\cdot \mathbf{E}& =0,& \mathrm{\nabla }\times \mathbf{E}+\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}=0,\\ \mathrm{\nabla }\cdot \mathbf{B}& =0,& \mathrm{\nabla }\times \mathbf{B}-{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}=0.\end{array}}$$

Taking the curl (∇×) of the curl equations, and using the curl of the curl identity we obtain $${\displaystyle \begin{array}{r}{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{E}=0,\\ {\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{B}=0.\end{array}}$$

The quantity ${\displaystyle {\mu }_0{\epsilon }_0}$ has the dimension (T/L)². Defining ${\displaystyle c=({\mu }_0{\epsilon }_0{)}^{-1/2}}$, the equations above have the form of the standard wave equations $${\displaystyle \begin{array}{r}\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{E}=0,\\ \frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{B}=0.\end{array}}$$

Already during Maxwell's lifetime, it was found that the known values for ${\displaystyle {\epsilon }_0}$ and ${\displaystyle {\mu }_0}$ give ${\displaystyle c\approx 2.998\times {10}^8\text{m/s}}$, then already known to be the speed of light in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In the old SI system of units, the values of ${\displaystyle {\mu }_0=4\pi \times {10}^{-7}}$ and ${\displaystyle c=299792458\text{m/s}}$ are defined constants, (which means that by definition ${\displaystyle {\epsilon }_0=8.854187\mathrm{8...}\times {10}^{-12}\text{F/m}}$) that define the ampere and the metre. In the new SI system, only c keeps its defined value, and the electron charge gets a defined value.

In materials with relative permittivity, ε_r, and relative permeability, μ_r, the phase velocity of light becomes $${\displaystyle v_{\text{p}}=\frac{1}{\sqrt{{\mu }_0{\mu }_{\text{r}}{\epsilon }_0{\epsilon }_{\text{r}}}},}$$ which is usually less than c.

In addition, E and B are perpendicular to each other and to the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. In turn, that electric field creates a changing magnetic field through Maxwell's modification of Ampère's circuital law. This perpetual cycle allows these waves, now known as electromagnetic radiation, to move through space at velocity c.

Macroscopic formulation

The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping.

The microscopic version is sometimes called "Maxwell's equations in vacuum": this refers to the fact that the material medium is not built into the structure of the equations, but appears only in the charge and current terms. The microscopic version was introduced by Lorentz, who tried to use it to derive the macroscopic properties of bulk matter from its microscopic constituents.

"Maxwell's macroscopic equations", also known as Maxwell's equations in matter, are more similar to those that Maxwell introduced himself.

Name	Integral equations (SI)	Differential equations (SI)	Differential equations (Gaussian system)
Gauss's law	${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$ ${\displaystyle \mathbf{D}\cdot \mathrm{d}\mathbf{S}={\iiint }_{\mathrm{\Omega }}{\rho }_{\text{f}}\mathrm{d}V}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{D}={\rho }_{\text{f}}}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{D}=4\pi {\rho }_{\text{f}}}$
Ampère–Maxwell law	${\displaystyle \begin{array}{rl}{\oint }_{\mathrm{\partial }\mathrm{\Sigma }}& \mathbf{H}\cdot \mathrm{d}\mathit{\ell }=\\ & {\iint }_{\mathrm{\Sigma }}{\mathbf{J}}_{\text{f}}\cdot \mathrm{d}\mathbf{S}+\frac{d}{dt}{\iint }_{\mathrm{\Sigma }}\mathbf{D}\cdot \mathrm{d}\mathbf{S}\end{array}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{H}={\mathbf{J}}_{\text{f}}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{H}=\frac{1}{c}\left(4\pi {\mathbf{J}}_{\text{f}}+\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}\right)}$
Gauss's law for magnetism	${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$ ${\displaystyle \mathbf{B}\cdot \mathrm{d}\mathbf{S}=0}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$	${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$
Maxwell–Faraday equation (Faraday's law of induction)	${\displaystyle {\oint }_{\mathrm{\partial }\mathrm{\Sigma }}\mathbf{E}\cdot \mathrm{d}\mathit{\ell }=-\frac{d}{dt}{\iint }_{\mathrm{\Sigma }}\mathbf{B}\cdot \mathrm{d}\mathbf{S}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}$	${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{1}{c}\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}$

In the macroscopic equations, the influence of bound charge Q_b and bound current I_b is incorporated into the displacement field D and the magnetizing field H, while the equations depend only on the free charges Q_f and free currents I_f. This reflects a splitting of the total electric charge Q and current I (and their densities ρ and J) into free and bound parts: $${\displaystyle \begin{array}{rl}Q& =Q_{\text{f}}+Q_{\text{b}}={\iiint }_{\mathrm{\Omega }}\left({\rho }_{\text{f}}+{\rho }_{\text{b}}\right)\mathrm{d}V={\iiint }_{\mathrm{\Omega }}\rho \mathrm{d}V,\\ I& =I_{\text{f}}+I_{\text{b}}={\iint }_{\mathrm{\Sigma }}\left({\mathbf{J}}_{\text{f}}+{\mathbf{J}}_{\text{b}}\right)\cdot \mathrm{d}\mathbf{S}={\iint }_{\mathrm{\Sigma }}\mathbf{J}\cdot \mathrm{d}\mathbf{S}.\end{array}}$$

The cost of this splitting is that the additional fields D and H need to be determined through phenomenological constituent equations relating these fields to the electric field E and the magnetic field B, together with the bound charge and current.

See below for a detailed description of the differences between the microscopic equations, dealing with total charge and current including material contributions, useful in air/vacuum; and the macroscopic equations, dealing with free charge and current, practical to use within materials.

Bound charge and current

Main articles: Current density, Bound charge, and Bound current

Left: A schematic view of how an assembly of microscopic dipoles produces opposite surface charges as shown at top and bottom. Right: How an assembly of microscopic current loops add together to produce a macroscopically circulating current loop. Inside the boundaries, the individual contributions tend to cancel, but at the boundaries no cancelation occurs.

When an electric field is applied to a dielectric material its molecules respond by forming microscopic electric dipoles – their atomic nuclei move a tiny distance in the direction of the field, while their electrons move a tiny distance in the opposite direction. This produces a macroscopic bound charge in the material even though all of the charges involved are bound to individual molecules. For example, if every molecule responds the same, similar to that shown in the figure, these tiny movements of charge combine to produce a layer of positive bound charge on one side of the material and a layer of negative charge on the other side. The bound charge is most conveniently described in terms of the polarization P of the material, its dipole moment per unit volume. If P is uniform, a macroscopic separation of charge is produced only at the surfaces where P enters and leaves the material. For non-uniform P, a charge is also produced in the bulk.

Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the angular momentum of the components of the atoms, most notably their electrons. The connection to angular momentum suggests the picture of an assembly of microscopic current loops. Outside the material, an assembly of such microscopic current loops is not different from a macroscopic current circulating around the material's surface, despite the fact that no individual charge is traveling a large distance. These bound currents can be described using the magnetization M.

The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of P and M, which average these charges and currents on a sufficiently large scale so as not to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material. As such, Maxwell's macroscopic equations ignore many details on a fine scale that can be unimportant to understanding matters on a gross scale by calculating fields that are averaged over some suitable volume.

Auxiliary fields, polarization and magnetization

The definitions of the auxiliary fields are: $${\displaystyle \begin{array}{rl}\mathbf{D}(\mathbf{r},t)& ={\epsilon }_0\mathbf{E}(\mathbf{r},t)+\mathbf{P}(\mathbf{r},t),\\ \mathbf{H}(\mathbf{r},t)& =\frac{1}{{\mu }_0}\mathbf{B}(\mathbf{r},t)-\mathbf{M}(\mathbf{r},t),\end{array}}$$ where P is the polarization field and M is the magnetization field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density ρ_b and bound current density J_b in terms of polarization P and magnetization M are then defined as $${\displaystyle \begin{array}{rl}{\rho }_{\text{b}}& =-\mathrm{\nabla }\cdot \mathbf{P},\\ {\mathbf{J}}_{\text{b}}& =\mathrm{\nabla }\times \mathbf{M}+\frac{\mathrm{\partial }\mathbf{P}}{\mathrm{\partial }t}.\end{array}}$$

If we define the total, bound, and free charge and current density by $${\displaystyle \begin{array}{rl}\rho & ={\rho }_{\text{b}}+{\rho }_{\text{f}},\\ \mathbf{J}& ={\mathbf{J}}_{\text{b}}+{\mathbf{J}}_{\text{f}},\end{array}}$$ and use the defining relations above to eliminate D, and H, the "macroscopic" Maxwell's equations reproduce the "microscopic" equations.

Constitutive relations

Main article: Constitutive equation § Electromagnetism

In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between displacement field D and the electric field E, as well as the magnetizing field H and the magnetic field B. Equivalently, we have to specify the dependence of the polarization P (hence the bound charge) and the magnetization M (hence the bound current) on the applied electric and magnetic field. The equations specifying this response are called constitutive relations. For real-world materials, the constitutive relations are rarely simple, except approximately, and usually determined by experiment. See the main article on constitutive relations for a fuller description.

For materials without polarization and magnetization, the constitutive relations are (by definition)$${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E},\mathbf{H}=\frac{1}{{\mu }_0}\mathbf{B},}$$ where ε₀ is the permittivity of free space and μ₀ the permeability of free space. Since there is no bound charge, the total and the free charge and current are equal.

An alternative viewpoint on the microscopic equations is that they are the macroscopic equations together with the statement that vacuum behaves like a perfect linear "material" without additional polarization and magnetization. More generally, for linear materials the constitutive relations are $${\displaystyle \mathbf{D}=\epsilon \mathbf{E},\mathbf{H}=\frac{1}{\mu }\mathbf{B},}$$ where ε is the permittivity and μ the permeability of the material. For the displacement field D the linear approximation is usually excellent because for all but the most extreme electric fields or temperatures obtainable in the laboratory (high power pulsed lasers) the interatomic electric fields of materials of the order of 10¹¹ V/m are much higher than the external field. For the magnetizing field ${\displaystyle \mathbf{H}}$, however, the linear approximation can break down in common materials like iron leading to phenomena like hysteresis. Even the linear case can have various complications, however.

• For homogeneous materials, ε and μ are constant throughout the material, while for inhomogeneous materials they depend on location within the material (and perhaps time).
• For isotropic materials, ε and μ are scalars, while for anisotropic materials (e.g. due to crystal structure) they are tensors.
• Materials are generally dispersive, so ε and μ depend on the frequency of any incident EM waves.

Even more generally, in the case of non-linear materials (see for example nonlinear optics), D and P are not necessarily proportional to E, similarly H or M is not necessarily proportional to B. In general D and H depend on both E and B, on location and time, and possibly other physical quantities.

In applications one also has to describe how the free currents and charge density behave in terms of E and B possibly coupled to other physical quantities like pressure, and the mass, number density, and velocity of charge-carrying particles. E.g., the original equations given by Maxwell (see History of Maxwell's equations) included Ohm's law in the form $${\displaystyle {\mathbf{J}}_{\text{f}}=\sigma \mathbf{E}.}$$

Alternative formulations

For the equations in special relativity, see Classical electromagnetism and special relativity and Covariant formulation of classical electromagnetism.

For the equations in general relativity, see Maxwell's equations in curved spacetime.

For an overview, see Mathematical descriptions of the electromagnetic field.

For the equations in quantum field theory, see Quantum electrodynamics.

Following are some of the several other mathematical formalisms of Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the electrical potential φ and the vector potential A. Potentials were introduced as a convenient way to solve the homogeneous equations, but it was thought that all observable physics was contained in the electric and magnetic fields (or relativistically, the Faraday tensor). The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the electric and magnetic fields vanish (Aharonov–Bohm effect).

Each table describes one formalism. See the main article for details of each formulation.

The direct spacetime formulations make manifest that the Maxwell equations are relativistically invariant, where space and time are treated on equal footing. Because of this symmetry, the electric and magnetic fields are treated on equal footing and are recognized as components of the Faraday tensor. This reduces the four Maxwell equations to two, which simplifies the equations, although we can no longer use the familiar vector formulation. Maxwell equations in formulation that do not treat space and time manifestly on the same footing have Lorentz invariance as a hidden symmetry. This was a major source of inspiration for the development of relativity theory. Indeed, even the formulation that treats space and time separately is not a non-relativistic approximation and describes the same physics by simply renaming variables. For this reason the relativistic invariant equations are usually called the Maxwell equations as well.

Each table below describes one formalism.

Tensor calculus

Formulation	Homogeneous equations	Inhomogeneous equations
Fields Minkowski space	${\displaystyle {\mathrm{\partial }}_{[\alpha }{F}_{\beta \gamma ]}=0}$	${\displaystyle {\mathrm{\partial }}_{\alpha }{F}^{\alpha \beta }={\mu }_0{J}^{\beta }}$
Potentials (any gauge) Minkowski space	${\displaystyle F_{\alpha \beta }=2{\mathrm{\partial }}_{[\alpha }{A}_{\beta ]}}$	${\displaystyle 2{\mathrm{\partial }}_{\alpha }{\mathrm{\partial }}^{[\alpha }{A}^{\beta ]}={\mu }_0{J}^{\beta }}$
Potentials (Lorenz gauge) Minkowski space	${\displaystyle F_{\alpha \beta }=2{\mathrm{\partial }}_{[\alpha }{A}_{\beta ]}}$ ${\displaystyle {\mathrm{\partial }}_{\alpha }{A}^{\alpha }=0}$	${\displaystyle {\mathrm{\partial }}_{\alpha }{\mathrm{\partial }}^{\alpha }{A}^{\beta }={\mu }_0{J}^{\beta }}$
Fields any spacetime	${\displaystyle \begin{array}{rl}& {\mathrm{\partial }}_{[\alpha }{F}_{\beta \gamma ]}=\\ & {\mathrm{\nabla }}_{[\alpha }{F}_{\beta \gamma ]}=0\end{array}}$	${\displaystyle \begin{array}{rl}& \frac{1}{\sqrt{-g}}{\mathrm{\partial }}_{\alpha }(\sqrt{-g}{F}^{\alpha \beta })=\\ & {\mathrm{\nabla }}_{\alpha }{F}^{\alpha \beta }={\mu }_0{J}^{\beta }\end{array}}$
Potentials (any gauge) any spacetime (with §topological restrictions)	${\displaystyle F_{\alpha \beta }=2{\mathrm{\partial }}_{[\alpha }{A}_{\beta ]}}$	${\displaystyle \begin{array}{rl}& \frac{2}{\sqrt{-g}}{\mathrm{\partial }}_{\alpha }(\sqrt{-g}{g}^{\alpha \mu }{g}^{\beta \nu }{\mathrm{\partial }}_{[\mu }{A}_{\nu ]})=\\ & 2{\mathrm{\nabla }}_{\alpha }({\mathrm{\nabla }}^{[\alpha }{A}^{\beta ]})={\mu }_0{J}^{\beta }\end{array}}$
Potentials (Lorenz gauge) any spacetime (with topological restrictions)	${\displaystyle F_{\alpha \beta }=2{\mathrm{\partial }}_{[\alpha }{A}_{\beta ]}}$ ${\displaystyle {\mathrm{\nabla }}_{\alpha }{A}^{\alpha }=0}$	${\displaystyle {\mathrm{\nabla }}_{\alpha }{\mathrm{\nabla }}^{\alpha }{A}^{\beta }-R^{\beta }{}_{\alpha }{A}^{\alpha }={\mu }_0{J}^{\beta }}$

Differential forms

Formulation	Homogeneous equations	Inhomogeneous equations
Fields any spacetime	${\displaystyle \mathrm{d}F=0}$	${\displaystyle \mathrm{d}\star F={\mu }_{0}J}$
Potentials (any gauge) any spacetime (with topological restrictions)	${\displaystyle F=\mathrm{d}A}$	${\displaystyle \mathrm{d}\star \mathrm{d}A={\mu }_{0}J}$
Potentials (Lorenz gauge) any spacetime (with topological restrictions)	${\displaystyle F=\mathrm{d}A}$ ${\displaystyle \mathrm{d}\star A=0}$	${\displaystyle \star ◻A={\mu }_{0}J}$

• In the tensor calculus formulation, the electromagnetic tensor F_αβ is an antisymmetric covariant order 2 tensor; the four-potential, A_α, is a covariant vector; the current, J^α, is a vector; the square brackets, [ ], denote antisymmetrization of indices; ∂_α is the partial derivative with respect to the coordinate, x^α. In Minkowski space coordinates are chosen with respect to an inertial frame; (x^α) = (ct, x, y, z), so that the metric tensor used to raise and lower indices is η_αβ = diag(1, −1, −1, −1). The d'Alembert operator on Minkowski space is ◻ = ∂_α∂^α as in the vector formulation. In general spacetimes, the coordinate system x^α is arbitrary, the covariant derivative ∇_α, the Ricci tensor, R_αβ and raising and lowering of indices are defined by the Lorentzian metric, g_αβ and the d'Alembert operator is defined as ◻ = ∇_α∇^α. The topological restriction is that the second real cohomology group of the space vanishes (see the differential form formulation for an explanation). This is violated for Minkowski space with a line removed, which can model a (flat) spacetime with a point-like monopole on the complement of the line.
• In the differential form formulation on arbitrary space times, F = ⁠1/2⁠F_αβdx^α ∧ dx^β is the electromagnetic tensor considered as a 2-form, A = A_αdx^α is the potential 1-form, ${\displaystyle J=-J_{\alpha }\star \mathrm{d}{x}^{\alpha }}$ is the current 3-form, d is the exterior derivative, and ${\displaystyle \star }$ is the Hodge star on forms defined (up to its orientation, i.e. its sign) by the Lorentzian metric of spacetime. In the special case of 2-forms such as F, the Hodge star ${\displaystyle \star }$ depends on the metric tensor only for its local scale. This means that, as formulated, the differential form field equations are conformally invariant, but the Lorenz gauge condition breaks conformal invariance. The operator ${\displaystyle ◻=(-\star \mathrm{d}\star \mathrm{d}-\mathrm{d}\star \mathrm{d}\star )}$ is the d'Alembert–Laplace–Beltrami operator on 1-forms on an arbitrary Lorentzian spacetime. The topological condition is again that the second real cohomology group is 'trivial' (meaning that its form follows from a definition). By the isomorphism with the second de Rham cohomology this condition means that every closed 2-form is exact.

Other formalisms include the geometric algebra formulation and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation^[17][18] was used.

Solutions

Maxwell's equations are partial differential equations that relate the electric and magnetic fields to each other and to the electric charges and currents. Often, the charges and currents are themselves dependent on the electric and magnetic fields via the Lorentz force equation and the constitutive relations. These all form a set of coupled partial differential equations which are often very difficult to solve: the solutions encompass all the diverse phenomena of classical electromagnetism. Some general remarks follow.

As for any differential equation, boundary conditions and initial conditions are necessary for a unique solution. For example, even with no charges and no currents anywhere in spacetime, there are the obvious solutions for which E and B are zero or constant, but there are also non-trivial solutions corresponding to electromagnetic waves. In some cases, Maxwell's equations are solved over the whole of space, and boundary conditions are given as asymptotic limits at infinity. In other cases, Maxwell's equations are solved in a finite region of space, with appropriate conditions on the boundary of that region, for example an artificial absorbing boundary representing the rest of the universe, or periodic boundary conditions, or walls that isolate a small region from the outside world (as with a waveguide or cavity resonator).

Jefimenko's equations (or the closely related Liénard–Wiechert potentials) are the explicit solution to Maxwell's equations for the electric and magnetic fields created by any given distribution of charges and currents. It assumes specific initial conditions to obtain the so-called "retarded solution", where the only fields present are the ones created by the charges. However, Jefimenko's equations are unhelpful in situations when the charges and currents are themselves affected by the fields they create.

Numerical methods for differential equations can be used to compute approximate solutions of Maxwell's equations when exact solutions are impossible. These include the finite element method and finite-difference time-domain method. For more details, see Computational electromagnetics.

Overdetermination of Maxwell's equations

Maxwell's equations seem overdetermined, in that they involve six unknowns (the three components of E and B) but eight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampère's circuital laws). (The currents and charges are not unknowns, being freely specifiable subject to charge conservation.) This is related to a certain limited kind of redundancy in Maxwell's equations: It can be proven that any system satisfying Faraday's law and Ampère's circuital law automatically also satisfies the two Gauss's laws, as long as the system's initial condition does, and assuming conservation of charge and the nonexistence of magnetic monopoles. This explanation was first introduced by Julius Adams Stratton in 1941.

Although it is possible to simply ignore the two Gauss's laws in a numerical algorithm (apart from the initial conditions), the imperfect precision of the calculations can lead to ever-increasing violations of those laws. By introducing dummy variables characterizing these violations, the four equations become not overdetermined after all. The resulting formulation can lead to more accurate algorithms that take all four laws into account.

Both identities ${\displaystyle \mathrm{\nabla }\cdot \mathrm{\nabla }\times \mathbf{B}\equiv 0,\mathrm{\nabla }\cdot \mathrm{\nabla }\times \mathbf{E}\equiv 0}$, which reduce eight equations to six independent ones, are the true reason of overdetermination.

Equivalently, the overdetermination can be viewed as implying conservation of electric and magnetic charge, as they are required in the derivation described above but implied by the two Gauss's laws.

For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But in differential equations, and especially partial differential equations (PDEs), one needs appropriate boundary conditions, which depend in not so obvious ways on the equations. Even more, if one rewrites them in terms of vector and scalar potential, then the equations are underdetermined because of gauge fixing.

Maxwell's equations as the classical limit of QED

Maxwell's equations and the Lorentz force law (along with the rest of classical electromagnetism) are extraordinarily successful at explaining and predicting a variety of phenomena. However, they do not account for quantum effects, and so their domain of applicability is limited. Maxwell's equations are thought of as the classical limit of quantum electrodynamics (QED).

Some observed electromagnetic phenomena cannot be explained with Maxwell's equations if the source of the electromagnetic fields are the classical distributions of charge and current. These include photon–photon scattering and many other phenomena related to photons or virtual photons, "nonclassical light" and quantum entanglement of electromagnetic fields (see Quantum optics). E.g. quantum cryptography cannot be described by Maxwell theory, not even approximately. The approximate nature of Maxwell's equations becomes more and more apparent when going into the extremely strong field regime (see Euler–Heisenberg Lagrangian) or to extremely small distances.

Finally, Maxwell's equations cannot explain any phenomenon involving individual photons interacting with quantum matter, such as the photoelectric effect, Planck's law, the Duane–Hunt law, and single-photon light detectors. However, many such phenomena may be explained using a halfway theory of quantum matter coupled to a classical electromagnetic field, either as external field or with the expected value of the charge current and density on the right hand side of Maxwell's equations. This is known as semiclassical theory or self-field QED and was initially discovered by de Broglie and Schrödinger and later fully developed by E.T. Jaynes and A.O. Barut.

Variations

Popular variations on the Maxwell equations as a classical theory of electromagnetic fields are relatively scarce because the standard equations have stood the test of time remarkably well.

Magnetic monopoles

Main article: Magnetic monopole

Maxwell's equations posit that there is electric charge, but no magnetic charge (also called magnetic monopoles), in the universe. Indeed, magnetic charge has never been observed, despite extensive searches, and may not exist. If they did exist, both Gauss's law for magnetism and Faraday's law would need to be modified, and the resulting four equations would be fully symmetric under the interchange of electric and magnetic fields.

Electromagnetic pulse

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electromagnetic_pulse

An electromagnetic pulse (EMP), also referred to as a transient electromagnetic disturbance (TED), is a brief burst of electromagnetic energy. The origin of an EMP can be natural or artificial, and can occur as an electromagnetic field, as an electric field, as a magnetic field, or as a conducted electric current. The electromagnetic interference caused by an EMP can disrupt communications and damage electronic equipment. An EMP such as a lightning strike can physically damage objects such as buildings and aircraft. The management of EMP effects is a branch of electromagnetic compatibility (EMC) engineering.

The first recorded damage from an electromagnetic pulse came with the solar storm of August 1859, or the Carrington Event.

In modern warfare, weapons delivering a high energy EMP are designed to disrupt communications equipment, computers needed to operate modern warplanes, or even put the entire electrical network of a target country out of commission.

General characteristics

An electromagnetic pulse is a short surge of electromagnetic energy. Its short duration means that it will be spread over a range of frequencies. Pulses are typically characterized by:

• The mode of energy transfer (radiated, electric, magnetic or conducted).
• The range or spectrum of frequencies present.
• Pulse waveform: shape, duration and amplitude.

The frequency spectrum and the pulse waveform are interrelated via the Fourier transform which describes how component waveforms may sum to the observed frequency spectrum.

Types of energy

Main article: Electromagnetism

EMP energy may be transferred in any of four forms:

• Electric field
• Magnetic field
• Electromagnetic radiation
• Electrical conduction

According to Maxwell's equations, a pulse of electric energy will always be accompanied by a pulse of magnetic energy. In a typical pulse, either the electric or the magnetic form will dominate. It can be shown that the non-linear Maxwell's equations can have time-dependent self-similar electromagnetic shock wave solutions where the electric and the magnetic field components have a discontinuity.

In general, only radiation acts over long distances, with the magnetic and electric fields acting over short distances. There are a few exceptions, such as a solar magnetic flare.

Frequency ranges

A pulse of electromagnetic energy typically comprises many frequencies from very low to some upper limit depending on the source. The range defined as EMP, sometimes referred to as "DC [direct current] to daylight", excludes the highest frequencies comprising the optical (infrared, visible, ultraviolet) and ionizing (X and gamma rays) ranges.

Some types of EMP events can leave an optical trail, such as lightning and sparks, but these are side effects of the current flow through the air and are not part of the EMP itself.

Pulse waveforms

The waveform of a pulse describes how its instantaneous amplitude (field strength or current) changes over time. Real pulses tend to be quite complicated, so simplified models are often used. Such a model is typically described either in a diagram or as a mathematical equation.

Rectangular pulse

Double exponential pulse

Damped sinewave pulse

Most electromagnetic pulses have a very sharp leading edge, building up quickly to their maximum level. The classic model is a double-exponential curve which climbs steeply, quickly reaches a peak and then decays more slowly. However, pulses from a controlled switching circuit often approximate the form of a rectangular or "square" pulse.

EMP events usually induce a corresponding signal in the surrounding environment or material. Coupling usually occurs most strongly over a relatively narrow frequency band, leading to a characteristic damped sine wave. Visually it is shown as a high frequency sine wave growing and decaying within the longer-lived envelope of the double-exponential curve. A damped sinewave typically has much lower energy and a narrower frequency spread than the original pulse, due to the transfer characteristic of the coupling mode. In practice, EMP test equipment often injects these damped sinewaves directly rather than attempting to recreate the high-energy threat pulses.

In a pulse train, such as from a digital clock circuit, the waveform is repeated at regular intervals. A single complete pulse cycle is sufficient to characterise such a regular, repetitive train.

Types

An EMP arises where the source emits a short-duration pulse of energy. The energy is usually broadband by nature, although it often excites a relatively narrow-band damped sine wave response in the surrounding environment. Some types are generated as repetitive and regular pulse trains.

Different types of EMP arise from natural, man-made, and weapons effects.

Types of natural EMP events include:

• Lightning electromagnetic pulse (LEMP). The discharge is typically an initial current flow of perhaps millions of amps, followed by a train of pulses of decreasing energy.
• Electrostatic discharge (ESD), as a result of two charged objects coming into proximity or even contact.
• Meteoric EMP. The discharge of electromagnetic energy resulting from either the impact of a meteoroid with a spacecraft or the explosive breakup of a meteoroid passing through the Earth's atmosphere.
• Coronal mass ejection (CME), sometimes referred to as a solar EMP. A burst of plasma and accompanying magnetic field, ejected from the solar corona and released into the solar wind.

Types of (civil) man-made EMP events include:

• Switching action of electrical circuitry, whether isolated or repetitive (as a pulse train).
• Electric motors can create a train of pulses as the internal electrical contacts make and break connections as the armature rotates.
• Gasoline engine ignition systems can create a train of pulses as the spark plugs are energized or fired.
• Continual switching actions of digital electronic circuitry.
• Power line surges. These can be up to several kilovolts, enough to damage electronic equipment that is insufficiently protected.

Types of military EMP include:

• Nuclear electromagnetic pulse (NEMP), as a result of a nuclear explosion. A variant of this is the high altitude nuclear EMP (HEMP), which produces a secondary pulse due to particle interactions with the Earth's atmosphere and magnetic field.
• Non-nuclear electromagnetic pulse (NNEMP) weapons.

Lightning electromagnetic pulse (LEMP)

Main article: Lightning

Lightning is unusual in that it typically has a preliminary "leader" discharge of low energy building up to the main pulse, which in turn may be followed at intervals by several smaller bursts.

Electrostatic discharge (ESD)

Main article: Electrostatic discharge

ESD events are characterized by high voltages of many kV, but small currents sometimes cause visible sparks. ESD is treated as a small, localized phenomenon, although technically a lightning flash is a very large ESD event. ESD can also be man-made, as in the shock received from a Van de Graaff generator.

An ESD event can damage electronic circuitry by injecting a high-voltage pulse, besides giving people an unpleasant shock. Such an ESD event can also create sparks, which may in turn ignite fires or fuel-vapour explosions. For this reason, before refueling an aircraft or exposing any fuel vapor to the air, the fuel nozzle is first connected to the aircraft to safely discharge any static.

Switching pulses

The switching action of an electrical circuit creates a sharp change in the flow of electricity. This sharp change is a form of EMP.

Simple electrical sources include inductive loads such as relays, solenoids, and brush contacts in electric motors. These typically send a pulse down any electrical connections present, as well as radiating a pulse of energy. The amplitude is usually small and the signal may be treated as "noise" or "interference". The switching off or "opening" of a circuit causes an abrupt change in the current flowing. This can in turn cause a large pulse in the electric field across the open contacts, causing arcing and damage. It is often necessary to incorporate design features to limit such effects.

Electronic devices such as vacuum tubes or valves, transistors, and diodes can also switch on and off very quickly, causing similar issues. One-off pulses may be caused by solid-state switches and other devices used only occasionally. However, the many millions of transistors in a modern computer may switch repeatedly at frequencies above 1  GHz, causing interference that appears to be continuous.

Nuclear electromagnetic pulse (NEMP)

Main article: Nuclear electromagnetic pulse

A nuclear electromagnetic pulse is the abrupt pulse of electromagnetic radiation resulting from a nuclear explosion. The resulting rapidly changing electric fields and magnetic fields may couple with electrical/electronic systems to produce damaging current and voltage surges.

The intense gamma radiation emitted can also ionize the surrounding air, creating a secondary EMP as the atoms of air first lose their electrons and then regain them.

NEMP weapons are designed to maximize such EMP effects as the primary damage mechanism, and some are capable of destroying susceptible electronic equipment over a wide area.

A high-altitude electromagnetic pulse (HEMP) weapon is a NEMP warhead designed to be detonated far above the Earth's surface. The explosion releases a blast of gamma rays into the mid-stratosphere, which ionizes as a secondary effect and the resultant energetic free electrons interact with the Earth's magnetic field to produce a much stronger EMP than is normally produced in the denser air at lower altitudes.

Non-nuclear electromagnetic pulse (NNEMP)

Non-nuclear electromagnetic pulse (NNEMP) is a weapon-generated electromagnetic pulse without use of nuclear technology. Devices that can achieve this objective include a large low-inductance capacitor bank discharged into a single-loop antenna, a microwave generator, and an explosively pumped flux compression generator. To achieve the frequency characteristics of the pulse needed for optimal coupling into the target, wave-shaping circuits or microwave generators are added between the pulse source and the antenna. Vircators are vacuum tubes that are particularly suitable for microwave conversion of high-energy pulses.

NNEMP generators can be carried as a payload of bombs, cruise missiles (such as the CHAMP missile) and drones, with diminished mechanical, thermal and ionizing radiation effects, but without the consequences of deploying nuclear weapons.

The range of NNEMP weapons is much less than nuclear EMP. Nearly all NNEMP devices used as weapons require chemical explosives as their initial energy source, producing only one millionth the energy of nuclear explosives of similar weight. The electromagnetic pulse from NNEMP weapons must come from within the weapon, while nuclear weapons generate EMP as a secondary effect. These facts limit the range of NNEMP weapons, but allow finer target discrimination. The effect of small e-bombs has proven to be sufficient for certain terrorist or military operations. Examples of such operations include the destruction of electronic control systems critical to the operation of many ground vehicles and aircraft.

The concept of the explosively pumped flux compression generator for generating a non-nuclear electromagnetic pulse was conceived as early as 1951 by Andrei Sakharov in the Soviet Union, but nations kept work on non-nuclear EMP classified until similar ideas emerged in other nations.

Effects

Minor EMP events, and especially pulse trains, cause low levels of electrical noise or interference which can affect the operation of susceptible devices. For example, a common problem in the mid-twentieth century was interference emitted by the ignition systems of gasoline engines, which caused radio sets to crackle and TV sets to show stripes on the screen. CISPR 25 was established to set threshold standards that vehicles must meet for electromagnetic interference(EMI) emissions.

A demonstration of how Electromagnetic Radiation powers (and destroys) circuits.

At a high voltage level an EMP can induce a spark, for example from an electrostatic discharge when fuelling a gasoline-engined vehicle. Such sparks have been known to cause fuel-air explosions and precautions must be taken to prevent them.

A large and energetic EMP can induce high currents and voltages in the victim unit, temporarily disrupting its function or even permanently damaging it.

A powerful EMP can also directly affect magnetic materials and corrupt the data stored on media such as magnetic tape and computer hard drives. Hard drives are usually shielded by heavy metal casings. Some IT asset disposal service providers and computer recyclers use a controlled EMP to wipe such magnetic media.

A very large EMP event, such as a lightning strike or an air bursted nuclear weapon, is also capable of damaging objects such as trees, buildings and aircraft directly, either through heating effects or the disruptive effects of the very large magnetic field generated by the current. An indirect effect can be electrical fires caused by heating. Most engineered structures and systems require some form of protection against lightning to be designed in. A good means of protection is a Faraday shield designed to protect certain items from being destroyed.

Control

Main article: Electromagnetic compatibility

EMP simulator HAGII-C testing a Boeing E-4 aircraft.

EMPRESS I (antennas along shoreline) with USS Estocin (FFG-15) moored in the foreground for testing.

Like any electromagnetic interference, the threat from EMP is subject to control measures. This is true whether the threat is natural or man-made.

Therefore, most control measures focus on the susceptibility of equipment to EMP effects, and hardening or protecting it from harm. Man-made sources, other than weapons, are also subject to control measures in order to limit the amount of pulse energy emitted.

The discipline of ensuring correct equipment operation in the presence of EMP and other RF threats is known as electromagnetic compatibility (EMC).

Test simulation

To test the effects of EMP on engineered systems and equipment, an EMP simulator may be used.

Induced pulse simulation

Induced pulses are of much lower energy than threat pulses and so are more practicable to create, but they are less predictable. A common test technique is to use a current clamp in reverse, to inject a range of damped sine wave signals into a cable connected to the equipment under test. The damped sine wave generator is able to reproduce the range of induced signals likely to occur.

Threat pulse simulation

Sometimes the threat pulse itself is simulated in a repeatable way. The pulse may be reproduced at low energy in order to characterise the subject's response prior to damped sinewave injection, or at high energy to recreate the actual threat conditions. A small-scale ESD simulator may be hand-held. Bench- or room-sized simulators come in a range of designs, depending on the type and level of threat to be generated.

At the top end of the scale, large outdoor test facilities incorporating high-energy EMP simulators have been built by several countries. The largest facilities are able to test whole vehicles including ships and aircraft for their susceptibility to EMP. Nearly all of these large EMP simulators used a specialized version of a Marx generator. Examples include the huge wooden-structured ATLAS-I simulator (also known as TRESTLE) at Sandia National Labs, New Mexico, which was at one time the world's largest EMP simulator. Papers on this and other large EMP simulators used by the United States during the latter part of the Cold War, along with more general information about electromagnetic pulses, are now in the care of the SUMMA Foundation, which is hosted at the University of New Mexico. The US Navy also has a large facility called the Electro Magnetic Pulse Radiation Environmental Simulator for Ships I (EMPRESS I).

Safety

High-level EMP signals can pose a threat to human safety. In such circumstances, direct contact with a live electrical conductor should be avoided. Where this occurs, such as when touching a Van de Graaff generator or other highly charged object, care must be taken to release the object and then discharge the body through a high resistance, in order to avoid the risk of a harmful shock pulse when stepping away.

Very high electric field strengths can cause breakdown of the air and a potentially lethal arc current similar to lightning to flow, but electric field strengths of up to 200 kV/m are regarded as safe.

According to research from Edd Gent, a 2019 report by the Electric Power Research Institute, which is funded by utility companies, found that a large EMP attack would probably cause regional blackouts but not a nationwide grid failure and that recovery times would be similar to those of other large-scale outages. It is not known how long these electrical blackouts would last, or what extent of damage would occur across the country. It is possible that neighboring countries of the U.S. could also be affected by such an attack, depending on the targeted area and people.

According to an article from Naureen Malik, with North Korea's increasingly successful missile and warhead tests in mind, Congress moved to renew funding for the Commission to Assess the Threat to the U.S. from Electromagnetic Pulse Attack as part of the National Defense Authorization Act.

According to research from Yoshida Reiji, in a 2016 article for the Tokyo-based nonprofit organization Center for Information and Security Trade Control, Onizuka warned that a high-altitude EMP attack would damage or destroy Japan's power, communications and transport systems as well as disable banks, hospitals and nuclear power plants.

In popular culture

By 1981, a number of articles on electromagnetic pulse in the popular press spread knowledge of the EMP phenomenon into the popular culture. EMP has been subsequently used in a wide variety of fiction and other aspects of popular culture. Popular media often depict EMP effects incorrectly, causing misunderstandings among the public and even professionals. Official efforts have been made in the U.S. to remedy these misconceptions.

The novel One Second After by William R. Forstchen and the following books One Year After, The Final Day and Five Years After portrait the story of a fictional character named John Matherson and his community in Black Mountain, North Carolina that after the US loses a war and an EMP attack "sends our nation [the US] back to the Dark Ages".

Ballistics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Ballistics

Trajectories of three objects thrown at the same angle (70°).

  Stokes' drag

  Newtonian drag.

  No form of drag and moves along a parabola.

Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially weapon munitions such as bullets, unguided bombs, rockets and the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.

A ballistic body is a free-moving body with momentum, which can be subject to forces such as those exerted by pressurized gases from a gun barrel or a propelling nozzle, normal force by rifling, and gravity and air drag during flight.

A ballistic missile is a missile that is guided only during the relatively brief initial phase of powered flight, with the trajectory subsequently governed by the laws of classical mechanics, in contrast to (for example) a cruise missile, which is aerodynamically guided in powered flight like a fixed-wing aircraft.

History and prehistory

The earliest known ballistic projectiles were stones, spears, and the throwing stick.

Gaetano Marzagaglia, Del calcolo balistico, 1748

The oldest evidence of stone-tipped projectiles, which may or may not have been propelled by a bow (cf. atlatl), dating to c. 280,000 years ago, were found in Ethiopia, present day East Africa. The oldest evidence of the use of bows to shoot arrows dates to about 10,000 years ago; it is based on pinewood arrows found in the Ahrensburg valley north of Hamburg. They had shallow grooves on the base, indicating that they were shot from a bow. The oldest bow so far recovered is about 8,000 years old, found in the Holmegård swamp in Denmark.

Archery seems to have arrived in the Americas with the Arctic small tool tradition, about 4,500 years ago.

The first devices identified as guns appeared in China around 1000 AD, and by the 12th century the technology was spreading through the rest of Asia, and into Europe by the 13th century.

After millennia of empirical development, the discipline of ballistics was initially studied and developed by Italian mathematician Niccolò Tartaglia in 1531, although he continued to use segments of straight-line motion, conventions established by the Greek philosopher Aristotle and Albert of Saxony, but with the innovation that he connected the straight lines by a circular arc. Galileo established the principle of compound motion in 1638, using the principle to derive the parabolic form of the ballistic trajectory. Ballistics was put on a solid scientific and mathematical basis by Isaac Newton, with the publication of Philosophiæ Naturalis Principia Mathematica in 1687. This gave mathematical laws of motion and gravity which for the first time made it possible to successfully predict trajectories.

The word ballistics comes from the Greek βάλλειν ballein, meaning "to throw".

Projectiles

Main article: Projectile

A projectile is any object projected into space (empty or not) by the exertion of a force. Although any object in motion through space (for example a thrown baseball) is a projectile, the term most commonly refers to a weapon. Mathematical equations of motion are used to analyze projectile trajectory.

Examples of projectiles include balls, arrows, bullets, artillery shells, wingless rockets, etc.

Projectile launchers

Throwing

Main article: Throwing

Baseball throws can exceed 100 mph.

Throwing is the launching of a projectile by hand. Although some other animals can throw, humans are unusually good throwers due to their high dexterity and good timing capabilities, and it is believed that this is an evolved trait. Evidence of human throwing dates back 2 million years. The 90 mph throwing speed found in many athletes far exceeds the speed at which chimpanzees can throw things, which is about 20 mph. This ability reflects the ability of the human shoulder muscles and tendons to store elasticity until it is needed to propel an object.

Sling

Main article: Sling (weapon)

A sling is a projectile weapon typically used to throw a blunt projectile such as a stone, clay or lead "sling-bullet".

A sling has a small cradle or pouch in the middle of two lengths of cord. The sling stone is placed in the pouch. The middle finger or thumb is placed through a loop on the end of one cord, and a tab at the end of the other cord is placed between the thumb and forefinger. The sling is swung in an arc, and the tab released at a precise moment. This frees the projectile to fly to the target.

Bow

Main article: Bow and arrow

A bow is a flexible piece of material which shoots aerodynamic projectiles called arrows. The arrow is perhaps the first lethal projectile ever described in discussion of ballistics. A string joins the two ends and when the string is drawn back, the ends of the stick are flexed. When the string is released, the potential energy of the flexed stick is transformed into the velocity of the arrow. Archery is the art or sport of shooting arrows from bows.

Catapult

Main article: Catapult

Catapult 1 Mercato San Severino

A catapult is a device used to launch a projectile a great distance without the aid of explosive devices – particularly various types of ancient and medieval siege engines. The catapult has been used since ancient times, because it was proven to be one of the most effective mechanisms during warfare. The word "catapult" comes from the Latin catapulta, which in turn comes from the Greek καταπέλτης (katapeltēs), itself from κατά (kata), "against” and πάλλω (pallō), "to toss, to hurl". Catapults were invented by the ancient Greeks.

Gun

Main article: Gun

USS Iowa (BB-61) fires a full broadside, 1984.

A gun is a normally tubular weapon or other device designed to discharge projectiles or other material. The projectile may be solid, liquid, gas, or energy and may be free, as with bullets and artillery shells, or captive as with Taser probes and whaling harpoons. The means of projection varies according to design but is usually effected by the action of gas pressure, either produced through the rapid combustion of a propellant or compressed and stored by mechanical means, operating on the projectile inside an open-ended tube in the fashion of a piston. The confined gas accelerates the movable projectile down the length of the tube imparting sufficient velocity to sustain the projectile's travel once the action of the gas ceases at the end of the tube or muzzle. Alternatively, acceleration via electromagnetic field generation may be employed in which case the tube may be dispensed with and a guide rail substituted.

A weapons engineer or armourer who applies the scientific principles of ballistics to design cartridges are often called a ballistician.

Rocket

Main article: Rocket

SpaceX's Falcon 9 Full Thrust rocket, 2017

A rocket is a missile, spacecraft, aircraft or other vehicle that obtains thrust from a rocket engine. Rocket engine exhaust is formed entirely from propellants carried within the rocket before use. Rocket engines work by action and reaction. Rocket engines push rockets forward simply by throwing their exhaust backwards extremely fast.

While comparatively inefficient for low speed use, rockets are relatively lightweight and powerful, capable of generating large accelerations and of attaining extremely high speeds with reasonable efficiency. Rockets are not reliant on the atmosphere and work very well in space.

Rockets for military and recreational uses date back to at least 13th century China. Significant scientific, interplanetary and industrial use did not occur until the 20th century, when rocketry was the enabling technology for the Space Age, including setting foot on the Moon. Rockets are now used for fireworks, weaponry, ejection seats, launch vehicles for artificial satellites, human spaceflight, and space exploration.

Chemical rockets are the most common type of high performance rocket and they typically create their exhaust by the combustion of rocket propellant. Chemical rockets store a large amount of energy in an easily released form, and can be very dangerous. However, careful design, testing, construction and use minimizes risks.

Subfields

Ballistics can be studied using high-speed photography or high-speed cameras. A photo of a Smith & Wesson revolver firing, taken with an ultra high speed air-gap flash. Using this sub-microsecond flash, the bullet can be imaged without motion blur.

Ballistics is often broken down into the following four categories:

• Internal ballistics the study of the processes originally accelerating projectiles
• Transition ballistics the study of projectiles as they transition to unpowered flight
• External ballistics the study of the passage of the projectile (the trajectory) in flight
• Terminal ballistics the study of the projectile and its effects as it ends its flight

Internal ballistics

Main article: Internal ballistics

Internal ballistics (also interior ballistics), a sub-field of ballistics, is the study of the propulsion of a projectile.

In guns, internal ballistics covers the time from the propellant's ignition until the projectile exits the gun barrel. The study of internal ballistics is important to designers and users of firearms of all types, from small-bore rifles and pistols, to high-tech artillery.

For rocket propelled projectiles, internal ballistics covers the period during which a rocket engine is providing thrust.

Transitional ballistics

Main article: Transitional ballistics

Transitional ballistics, also known as intermediate ballistics, is the study of a projectile's behavior from the time it leaves the muzzle until the pressure behind the projectile is equalized, so it lies between internal ballistics and external ballistics.

External ballistics

Main article: External ballistics

External ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight.

External ballistics is frequently associated with firearms, and deals with the unpowered free-flight phase of the bullet after it exits the gun barrel and before it hits the target, so it lies between transitional ballistics and terminal ballistics.

However, external ballistics is also concerned with the free-flight of rockets and other projectiles, such as balls, arrows etc.

Terminal ballistics

Main article: Terminal ballistics

Terminal ballistics is the study of the behavior and effects of a projectile when it hits its target.

Terminal ballistics is relevant both for small caliber projectiles as well as for large caliber projectiles (fired from artillery). The study of extremely high velocity impacts is still very new and is as yet mostly applied to spacecraft design.

Applications

Apollo 11 – Astrodynamic calculations have permitted spacecraft to travel to and return from the Moon.

Forensic ballistics

Main article: Forensic firearm examination

Forensic ballistics involves analysis of bullets and bullet impacts to determine information of use to a court or other part of a legal system. Separately from ballistics information, firearm and tool mark examinations ("ballistic fingerprinting") involve analyzing firearm, ammunition, and tool mark evidence in order to establish whether a certain firearm or tool was used in the commission of a crime.

Astrodynamics

Main article: Orbital mechanics

Astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It is a core discipline within space mission design and control.

Psychodynamics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Psychodynamics 

Front row: Sigmund Freud, G. Stanley Hall, Carl Jung; Back row: Abraham A. Brill, Ernest Jones, Sándor Ferenczi, at: Clark University in Worcester, Massachusetts. Date: September 1909.

Psychodynamics, also known as psychodynamic psychology, in its broadest sense, is an approach to psychology that emphasizes systematic study of the psychological forces underlying human behavior, feelings, and emotions and how they might relate to early experience. It is especially interested in the dynamic relations between conscious motivation and unconscious motivation.

The term psychodynamics is sometimes used to refer specifically to the psychoanalytical approach developed by Sigmund Freud (1856–1939) and his followers. Freud was inspired by the theory of thermodynamics and used the term psychodynamics to describe the processes of the mind as flows of psychological energy (libido or psi) in an organically complex brain. However, modern usage differentiates psychoanalytic practice as referring specifically to the earliest forms of psychotherapy, practiced by Freud and his immediate followers, and psychodynamic practice as practice that is informed by psychoanalytic theory, but diverges from the traditional practice model.

In the treatment of psychological distress, psychodynamic psychotherapy tends to be a less intensive (once- or twice-weekly) modality than the classical Freudian psychoanalysis treatment (of 3–5 sessions per week) and typically relies less on the traditional practices of psychoanalytic therapy, such as the patient facing away from the therapist during treatment and free association. Psychodynamic therapies depend upon a psychoanalytic understanding of inner conflict, wherein unconscious thoughts, desires, and memories influence behavior and psychological problems are caused by unconscious or repressed conflicts.

Despite largely falling out of favor as the primary modality of psychotherapy and facing criticism as being "non-empirical", psychodynamic treatment has been shown to be effective at treating a number of psychological conditions in randomized controlled trials, more effectively than controls and to the same degree as other psychotherapy modalities.

Overview

In general, psychodynamics is the study of the interrelationship of various parts of the mind, personality, or psyche as they relate to mental, emotional, or motivational forces especially at the unconscious level. The mental forces involved in psychodynamics are often divided into two parts: (a) the interaction of the emotional and motivational forces that affect behavior and mental states, especially on a subconscious level; (b) inner forces affecting behavior: the study of the emotional and motivational forces that affect behavior and states of mind.

Freud proposed that psychological energy was constant (hence, emotional changes consisted only in displacements) and that it tended to rest (point attractor) through discharge (catharsis).

In mate selection psychology, psychodynamics is defined as the study of the forces, motives, and energy generated by the deepest of human needs.

In general, psychodynamics studies the transformations and exchanges of "psychic energy" within the personality. A focus in psychodynamics is the connection between the energetics of emotional states in the Id, ego and super-ego as they relate to early childhood developments and processes. At the heart of psychological processes, according to Freud, is the ego, which he envisions as battling with three forces: the id, the super-ego, and the outside world. The id is the unconscious reservoir of libido, the psychic energy that fuels instincts and psychic processes. The ego serves as the general manager of personality, making decisions regarding the pleasures that will be pursued at the id's demand, the person's safety requirements, and the moral dictates of the superego that will be followed. The superego refers to the repository of an individual's moral values, divided into the conscience – the internalization of a society's rules and regulations – and the ego-ideal – the internalization of one's goals. Hence, the basic psychodynamic model focuses on the dynamic interactions between the id, ego, and superego. Psychodynamics, subsequently, attempts to explain or interpret behaviour or mental states in terms of innate emotional forces or processes.

History

Ernst von Brücke, early developer of psychodynamics

Freud used the term psychodynamics to describe the processes of the mind as flows of psychological energy (libido) in an organically complex brain. The idea for this came from his first year adviser, Ernst von Brücke at the University of Vienna, who held the view that all living organisms, including humans, are basically energy-systems to which the principle of the conservation of energy applies. This principle states that "the total amount of energy in any given physical system is always constant, that energy quanta can be changed but not annihilated, and that consequently when energy is moved from one part of the system, it must reappear in another part." This principle is at the very root of Freud's ideas, whereby libido, which is primarily seen as sexual energy, is transformed into other behaviours. However, it is now clear that the term energy in physics means something quite different from the term energy in relation to mental functioning.

Psychodynamics was initially further developed by Carl Jung, Alfred Adler and Melanie Klein. By the mid-1940s and into the 1950s, the general application of the "psychodynamic theory" had been well established.

In his 1988 book Introduction to Psychodynamics – a New Synthesis, psychiatrist Mardi J. Horowitz states that his own interest and fascination with psychodynamics began during the 1950s, when he heard Ralph Greenson, a popular local psychoanalyst who spoke to the public on topics such as "People who Hate", speak on the radio at UCLA. In his radio discussion, according to Horowitz, he "vividly described neurotic behavior and unconscious mental processes and linked psychodynamics theory directly to everyday life."

In the 1950s, American psychiatrist Eric Berne built on Freud's psychodynamic model, particularly that of the "ego states", to develop a psychology of human interactions called transactional analysis which, according to physician James R. Allen, is a "cognitive-behavioral approach to treatment and that it is a very effective way of dealing with internal models of self and others as well as other psychodynamic issues.".

Around the 1970s, a growing number of researchers began departing from the psychodynamics model and Freudian subconscious. Many felt that the evidence was over-reliant on imaginative discourse in therapy, and on patient reports of their state-of-mind. These subjective experiences are inaccessible to others. Philosopher of science Karl Popper argued that much of Freudianism was untestable and therefore not scientific. In 1975 literary critic Frederick Crews began a decades-long campaign against the scientific credibility of Freudianism. This culminated in Freud: The Making of an Illusion which aggregated years of criticism from many quarters. Medical schools and psychology departments no longer offer much training in psychodynamics, according to a 2007 survey. An Emory University psychology professor explained, “I don’t think psychoanalysis is going to survive unless there is more of an appreciation for empirical rigor and testing.”

Freudian analysis

According to American psychologist Calvin S. Hall, from his 1954 Primer in Freudian Psychology:

Freud greatly admired Brücke and quickly became indoctrinated by this new dynamic physiology. Thanks to Freud's singular genius, he was to discover some twenty years later that the laws of dynamics could be applied to man's personality as well as to his body. When he made his discovery Freud proceeded to create a dynamic psychology. A dynamic psychology is one that studies the transformations and exchanges of energy within the personality. This was Freud’s greatest achievement, and one of the greatest achievements in modern science, It is certainly a crucial event in the history of psychology.

At the heart of psychological processes, according to Freud, is the ego, which he sees battling with three forces: the id, the super-ego, and the outside world. Hence, the basic psychodynamic model focuses on the dynamic interactions between the id, ego, and superego. Psychodynamics, subsequently, attempts to explain or interpret behavior or mental states in terms of innate emotional forces or processes. In his writings about the "engines of human behavior", Freud used the German word Trieb, a word that can be translated into English as either instinct or drive.

In the 1930s, Freud's daughter Anna Freud began to apply Freud's psychodynamic theories of the "ego" to the study of parent-child attachment and especially deprivation and in doing so developed ego psychology.

Jungian analysis

At the turn of the 20th century, during these decisive years, a young Swiss psychiatrist named Carl Jung had been following Freud's writings and had sent him copies of his articles and his first book, the 1907 Psychology of Dementia Praecox, in which he upheld the Freudian psychodynamic viewpoint, although with some reservations. That year, Freud invited Jung to visit him in Vienna. The two men, it is said, were greatly attracted to each other, and they talked continuously for thirteen hours. This led to a professional relationship in which they corresponded on a weekly basis, for a period of six years.

Carl Jung's contributions in psychodynamic psychology include:

1. The psyche tends toward wholeness.
2. The self is composed of the ego, the personal unconscious, the collective unconscious. The collective unconscious contains the archetypes which manifest in ways particular to each individual.
3. Archetypes are composed of dynamic tensions and arise spontaneously in the individual and collective psyche. Archetypes are autonomous energies common to the human species. They give the psyche its dynamic properties and help organize it. Their effects can be seen in many forms and across cultures.
4. The Transcendent Function: The emergence of the third resolves the split between dynamic polar tensions within the archetypal structure.
5. The recognition of the spiritual dimension of the human psyche.
6. The role of images which spontaneously arise in the human psyche (images include the interconnection between affect, images, and instinct) to communicate the dynamic processes taking place in the personal and collective unconscious, images which can be used to help the ego move in the direction of psychic wholeness.
7. Recognition of the multiplicity of psyche and psychic life, that there are several organizing principles within the psyche, and that they are at times in conflict.