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In classical electromagnetism, **polarization density** (or **electric polarization**, or simply **polarization**) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.^{ [1] }^{ [2] }

**Classical electromagnetism** or **classical electrodynamics** is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

In vector calculus and physics, a **vector field** is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

The **electric dipole moment** is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, the most commonly used unit in atomic physics and chemistry is the debye (D).

- Definition
- Other expressions
- Gauss's law for the field of P
- Differential form
- Relationship between the fields of P and E
- Homogeneous, isotropic dielectrics
- Anisotropic dielectrics
- Polarization density in Maxwell's equations
- Relations between E, D and P
- Time-varying polarization density
- Polarization ambiguity
- See also
- References and notes

Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization, which is the measure of the corresponding response of a material to a magnetic field in magnetism. The SI unit of measure is coulombs per square meter, and polarization density is represented by a vector **P**.^{ [2] }

In classical electromagnetism, **magnetization** or **magnetic polarization** is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector **M**.

A **magnetic field** is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field varies with location. As such, it is an example of a vector field.

**Magnetism** is a class of physical phenomena that are mediated by magnetic fields. Electric currents and the magnetic moments of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. The most familiar effects occur in ferromagnetic materials, which are strongly attracted by magnetic fields and can be magnetized to become permanent magnets, producing magnetic fields themselves. Only a few substances are ferromagnetic; the most common ones are iron, nickel and cobalt and their alloys such as steel. The prefix *ferro-* refers to iron, because permanent magnetism was first observed in lodestone, a form of natural iron ore called magnetite, Fe_{3}O_{4}.

An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements. These are elements which are bound to molecules and are not free to move around the material. Positive charged elements are displaced in the direction of the field, and negative charged elements are displaced opposite to the direction of the field. The molecules may remain neutral in charge, yet an electric dipole moment forms.^{ [3] }^{ [4] }

For a certain volume element in the material, which carries a dipole moment , we define the polarization density **P**:

In general, the dipole moment changes from point to point within the dielectric. Hence, the polarization density **P** of a dielectric inside an infinitesimal volume d*V* with an infinitesimal dipole moment d**p** is:

The net charge appearing as a result of polarization is called bound charge and denoted .

This definition of polarization as a "dipole moment per unit volume" is widely adopted, though in some cases it can lead to ambiguities and paradoxes.^{ [5] }

Let a volume d*V* be isolated inside the dielectric. Due to polarization the positive bound charge will be displaced a distance relative to the negative bound charge , giving rise to a dipole moment . Substitution of this expression in (1) yields

Since the charge bounded in the volume d*V* is equal to the equation for **P** becomes:^{ [3] }

where is the density of the bound charge in the volume under consideration.

For a given volume *V* enclosed by a surface *S*, the bound charge inside it is equal to the flux of **P** through *S* taken with the negative sign, or

Proof: Let a surface area *S*envelope part of a dielectric. Upon polarization negative and positive bound charges will be displaced. Let*d*and_{1}*d*be the distances of the bound charges and , respectively, from the plane formed by the element of area d_{2}*A*after the polarization. And let d*V*_{1}and d*V*_{2}be the volumes enclosed below and above the area d*A*.It follows that the negative bound charge moved from the outer part of the surface d

*A*inwards, while the positive bound charge moved from the inner part of the surface outwards.By the law of conservation of charge the total bound charge left inside the volume after polarization is:

Since

and (see image to the right)

The above equation becomes

By (2) it follows that , so we get:

And by integrating this equation over the entire closed surface

*S*we find thatwhich completes the proof.

By the divergence theorem, Gauss's law for the field **P** can be stated in *differential form* as:

- ,

where ∇ · **P** is the divergence of the field **P** through a given surface containing the bound charge density .

Proof: By the divergence theorem we have that - ,

for the volume

*V*containing the bound charge . And since is the integral of the bound charge density taken over the entire volume*V*enclosed by*S*, the above equation yields- ,

which is true if and only if

In a homogeneous, linear and isotropic dielectric medium, the **polarization** is aligned with and proportional to the electric field **E**:^{ [7] }

In physics, a **homogeneous** material or system has the same properties at every point; it is uniform without irregularities. A uniform electric field would be compatible with homogeneity. A material constructed with different constituents can be described as effectively homogeneous in the electromagnetic materials domain, when interacting with a directed radiation field.

A **dielectric** is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

In mathematics, two varying quantities are said to be in a relation of **proportionality**, if they are multiplicatively connected to a constant, that is, when either their ratio or their product yields a constant. The value of this constant is called the **coefficient of proportionality** or **proportionality constant**.

where ε_{0} is the electric constant, and χ is the electric susceptibility of the medium. Note that in this case χ simplifies to a scalar, although more generally it is a tensor. This is a particular case due to the *isotropy* of the dielectric.

Taking into account this relation between **P** and **E**, equation (3) becomes:^{ [3] }

The expression in the integral is Gauss's law for the field **E** which yields the total charge, both free and bound , in the volume *V* enclosed by *S*.^{ [3] } Therefore,

which can be written in terms of free charge and bound charge densities (by considering the relationship between the charges, their volume charge densities and the given volume):

Since within a homogeneous dielectric there can be no free charges , by the last equation it follows that there is no bulk bound charge in the material . And since free charges can get as close to the dielectric as to its topmost surface, it follows that polarization only gives rise to surface bound charge density (denoted to avoid ambiguity with the volume bound charge density ).^{ [3] }

may be related to **P** by the following equation:^{ [8] }

where is the normal vector to the surface *S* pointing outwards. (see charge density for the rigorous proof)

The class of dielectrics where the polarization density and the electric field are not in the same direction are known as * anisotropic * materials.

In such materials, the *i*th component of the polarization is related to the *j*th component of the electric field according to:^{ [7] }

This relation shows, for example, that a material can polarize in the x direction by applying a field in the z direction, and so on. The case of an anisotropic dielectric medium is described by the field of crystal optics.

As in most electromagnetism, this relation deals with macroscopic averages of the fields and dipole density, so that one has a continuum approximation of the dielectric materials that neglects atomic-scale behaviors. The polarizability of individual particles in the medium can be related to the average susceptibility and polarization density by the Clausius–Mossotti relation.

In general, the susceptibility is a function of the frequency *ω* of the applied field. When the field is an arbitrary function of time *t*, the polarization is a convolution of the Fourier transform of *χ*(*ω*) with the **E**(*t*). This reflects the fact that the dipoles in the material cannot respond instantaneously to the applied field, and causality considerations lead to the Kramers–Kronig relations.

If the polarization **P** is not linearly proportional to the electric field **E**, the medium is termed *nonlinear* and is described by the field of nonlinear optics. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), **P** is usually given by a Taylor series in **E** whose coefficients are the nonlinear susceptibilities:

where is the linear susceptibility, is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and is the third-order susceptibility (describing third-order effects such as the Kerr effect and electric field-induced optical rectification).

In ferroelectric materials, there is no one-to-one correspondence between **P** and **E** at all because of hysteresis.

The behavior of electric fields (**E** and **D**), magnetic fields (**B**, **H**), charge density (ρ) and current density (**J**) are described by Maxwell's equations in matter.

In terms of volume charge densities, the **free** charge density is given by

where is the total charge density. By considering the relationship of each of the terms of the above equation to the divergence of their corresponding fields (of the electric displacement field **D**, **E** and **P** in that order), this can be written as:^{ [9] }

This is known as the constitutive equation for electric fields. Here *ε _{0}* is the electric permittivity of empty space. In this equation,

In general, **P** varies as a function of **E** depending on the medium, as described later in the article. In many problems, it is more convenient to work with **D** and the free charges than with **E** and the total charge.^{ [1] }

Therefore, a polarized medium, by way of Green's Theorem can be split into four components.

- The bound volumetric charge density:
- The bound surface charge density:
- The free volumetric charge density:
- The free surface charge density:

When the polarization density changes with time, the time-dependent bound-charge density creates a *polarization current density * of

so that the total current density that enters Maxwell's equations is given by

where **J**_{f} is the free-charge current density, and the second term is the magnetization current density (also called the *bound current density*), a contribution from atomic-scale magnetic dipoles (when they are present).

The polarization inside a solid is not, in general, uniquely defined: It depends on which electrons are paired up with which nuclei.^{ [11] } (See figure.) In other words, two people, Alice and Bob, looking at the same solid, may calculate different values of **P**, and neither of them will be wrong. Alice and Bob will agree on the microscopic electric field **E** in the solid, but disagree on the value of the displacement field . They will both find that Gauss's law is correct (), but they will disagree on the value of at the surfaces of the crystal. For example, if Alice interprets the bulk solid to consist of dipoles with positive ions above and negative ions below, but the real crystal has negative ions as the topmost surface, then Alice will say that there is a negative free charge at the topmost surface. (She might view this as a type of surface reconstruction).

On the other hand, even though the value of **P** is not uniquely defined in a bulk solid, *variations* in **P***are* uniquely defined.^{ [11] } If the crystal is gradually changed from one structure to another, there will be a current inside each unit cell, due to the motion of nuclei and electrons. This current results in a macroscopic transfer of charge from one side of the crystal to the other, and therefore it can be measured with an ammeter (like any other current) when wires are attached to the opposite sides of the crystal. The time-integral of the current is proportional to the change in **P**. The current can be calculated in computer simulations (such as density functional theory); the formula for the integrated current turns out to be a type of Berry's phase.^{ [11] }

The non-uniqueness of **P** is not problematic, because every measurable consequence of **P** is in fact a consequence of a continuous change in **P**.^{ [11] } For example, when a material is put in an electric field **E**, which ramps up from zero to a finite value, the material's electronic and ionic positions slightly shift. This changes **P**, and the result is electric susceptibility (and hence permittivity). As another example, when some crystals are heated, their electronic and ionic positions slightly shift, changing **P**. The result is pyroelectricity. In all cases, the properties of interest are associated with a *change* in **P**.

Even though the polarization is *in principle* non-unique, in practice it is often (not always) defined by convention in a specific, unique way. For example, in a perfectly centrosymmetric crystal, **P** is usually defined by convention to be exactly zero. As another example, in a ferroelectric crystal, there is typically a centrosymmetric configuration above the Curie temperature, and **P** is defined there by convention to be zero. As the crystal is cooled below the Curie temperature, it shifts gradually into a more and more non-centrosymmetric configuration. Since gradual changes in **P** are uniquely defined, this convention gives a unique value of **P** for the ferroelectric crystal, even below its Curie temperature.

Another problem in the definition of **P** is related to the arbitrary choice of the "unit volume", or more precisely to the system's *scale*.^{ [5] } For example, at *microscopic* scale a plasma can be regarded as a gas of *free* charges, thus **P** should be zero. On the contrary, at a *macroscopic* scale the same plasma can be described as a continuous medium, exhibiting a permittivity and thus a net polarization **P** ≠ **0**.

- 1 2
*Introduction to Electrodynamics*(3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3 - 1 2
*McGraw Hill Encyclopaedia of Physics*(2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3 - 1 2 3 4 5 Irodov, I.E. (1986).
*Basic Laws of Electromagnetism*. Mir Publishers, CBS Publishers & Distributors. ISBN 81-239-0306-5 - ↑ Matveev. A. N. (1986).
*Electricity and Magnetism*. Mir Publishers. - 1 2 3 C.A. Gonano; R.E. Zich; M. Mussetta (2015). "Definition for Polarization P and Magnetization M Fully Consistent with Maxwell's Equations" (PDF).
*Progress In Electromagnetics Research B*.**64**: 83–101. - ↑ Based upon equations from Gray, Andrew (1888).
*The theory and practice of absolute measurements in electricity and magnetism*. Macmillan & Co. pp. 126–127., which refers to papers by Sir W. Thomson. - 1 2 Feynman, R.P.; Leighton, R.B. and Sands, M. (1964)
*Feynman Lectures on Physics: Volume 2*, Addison-Wesley, ISBN 0-201-02117-X - ↑ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
- ↑ Saleh, B.E.A.; Teich, M.C. (2007).
*Fundamentals of Photonics*. Hoboken, NJ: Wiley. p. 154. ISBN 978-0-471-35832-9. - ↑ A. Herczynski (2013). "Bound charges and currents" (PDF).
*American Journal of Physics*.**81**(3): 202–205. Bibcode:2013AmJPh..81..202H. doi:10.1119/1.4773441. - 1 2 3 4 Resta, Raffaele (1994). "Macroscopic polarization in crystalline dielectrics: the geometric phase approach" (PDF).
*Rev. Mod. Phys*.**66**: 899. Bibcode:1994RvMP...66..899R. doi:10.1103/RevModPhys.66.899. See also: D Vanderbilt,*Berry phases and Curvatures in Electronic Structure Theory*, an introductory-level powerpoint.

**Continuum mechanics** is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

In physics the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge *q* moving with a velocity *v* in an electric field **E** and a magnetic field **B** experiences a force of

**Maxwell's equations** are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.

An **electric field** is a vector field surrounding an electric charge that exerts force on other charges, attracting or repelling them. Mathematically the electric field is a vector field that associates to each point in space the force, called the Coulomb force, that would be experienced per unit of charge by an infinitesimal test charge at that point. The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m). Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

In electromagnetism, **absolute permittivity**, often simply called **permittivity**, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

In physics, **Gauss's law**, also known as **Gauss's flux theorem**, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.

In classical electromagnetism, **Ampère's circuital law** relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell derived it using hydrodynamics in his 1861 paper "On Physical Lines of Force" and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.

The **magnetic moment** is a quantity that represents the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current, permanent magnets, elementary particles, various molecules, and many astronomical objects.

In electromagnetism, **displacement current density** is the quantity ∂* D*/∂

In electricity (electromagnetism), the **electric susceptibility** is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field. The greater the electric susceptibility, the greater the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material. It is in this way that the electric susceptibility influences the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.

In physics, the **electric displacement field**, denoted by **D**, is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "**D**" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In SI, it is expressed in units of coulomb per metre squared (C⋅m^{−2}).

In electromagnetism, **charge density** is the amount of electric charge per unit length, surface area, or volume. *Volume charge density* is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C•m^{−3}), at any point in a volume. *Surface charge density* (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C•m^{−2}), at any point on a surface charge distribution on a two dimensional surface. *Linear charge density* (λ) is the quantity of charge per unit length, measured in coulombs per meter (C•m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

**Lorentz–Heaviside units** constitute a system of units within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant *ε*_{0} and magnetic constant *µ*_{0} do not appear, having been incorporated implicitly into the unit system and electromagnetic equations. Lorentz–Heaviside units may be regarded as normalizing *ε*_{0} = 1 and *µ*_{0} = 1, while at the same time revising Maxwell's equations to use the speed of light *c* instead.

The **method of image charges** is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem.

**Ewald summation**, named after Paul Peter Ewald, is a method for computing long-range interactions in periodic systems. It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system in order to calculate accurately the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.

In electromagnetism, **current density** is the electric current per unit area of cross section. The **current density vector** is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the charges at this point. In SI units, the electric current density is measured in amperes per square metre.

The **Optical Metric** was defined by German theoretical physicist Walter Gordon in 1923 to study the geometrical optics in curved space-time filled with moving dielectric materials. Let u_{a} be the normalized (covariant) 4-velocity of the arbitrarily-moving dielectric medium filling the space-time, and assume that the fluid’s electromagnetic properties are linear, isotropic, transparent, nondispersive, and can be summarized by two scalar functions: a dielectric permittivity ε and a magnetic permeability μ. Then **optical metric ** tensor is defined as

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