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Maxwell Equations

Maxwell's equations (English: Maxwell's equations) is a set of partial differential equations that describe the relationship between electric field, magnetic field, charge density, and current density. This set of equations consists of four equations, namely Gauss's law that describes how electric charges generate electric fields, Gauss's magnetic law that shows that magnetic monopoles do not exist, Faraday's law of induction that explains how time-varying magnetic fields produce electric fields, and explains current and time-varying Maxwell-Ampere's law of how an electric field produces a magnetic field. The Maxwell equations are named after the British physicist James Maxwell. Maxwell conceived an early form of this system of equations in the 1860s.

Different forms of Maxwell’s equations are used in different fields. For example, in high-energy physics and gravitational physics, the version of Maxwell's equations expressed in space-time is usually used. This expression is based on Einstein's concept of space-time, which combines time and space, rather than the Newtonian concept of absolute space-time, which is displayed independently of three-dimensional space and fourth-dimensional time. Einstein's representation of time and space clearly conforms to the special theory of relativity and the general theory of relativity. In quantum mechanics, the version of Maxwell's equations based on electric potential and magnetic potential is more popular.

Since the middle of the 20th century, physicists have understood that Maxwell’s equations are not precise laws. An accurate description requires the help of quantum electrodynamics theory that can better show the underlying physical foundation, and Maxwell’s equations are just one of them. The classical field theory is approximate. Nevertheless, for most cases involved in daily life, the difference between the solution obtained by Maxwell's equations and the exact solution is very small. For non-classical light, two-photon scattering, quantum optics, and many other phenomena related to photons or virtual photons, Maxwell's equations cannot give a solution close to the actual situation.

From Maxwell’s equations, it can be inferred that light waves are electromagnetic waves. Maxwell's equations and Lorentz force equations are the basic equations of classical electromagnetism. Thanks to this set of basic equations and related theories, many modern power technology and electronic technology have been invented and developed rapidly.


Maxwell’s equations are composed of four first-order linear partial differential equations. Although first-order and linear are both good mathematical properties, except for cases with a high degree of symmetry, its analytical solution is usually not found, so numerical methods must be used to find its numerical solution. But since electrodynamics is a linear theory, it can be solved by the principle of superposition.

Gauss's law

Gauss's law describes how electric fields are generated by electric charges. Electric field lines start with positive charges and end with negative charges. From the estimation of the number of electric field lines passing through a given closed surface, that is, the electric flux, the total charge contained in the closed surface can be known. In more detail, the law describes the relationship between the electric flux passing through an arbitrary closed surface and the amount of charge in the closed surface.

According to the Gaussian magnetic law, the magnetic field lines have no initial point or end point, but form a loop or extend to infinity. The schematic diagram shows the magnetic field lines formed by the current flowing in the toroidal conductor.

Gaussian magnetic law

Gaussian magnetic law shows that magnetic monopole (magnetic charge) does not exist in the universe. In terms of experiments, physicists have yet to find clear evidence of the existence of magnetic monopoles. The magnetic field generated by matter is generated by a configuration called a dipole. The magnetic dipole is best represented by a current loop. A magnetic dipole is like a positive magnetic charge and a negative magnetic charge that are inseparably bound together, and its net magnetic charge is zero. The magnetic field lines have no initial point and no end point. The magnetic field lines will loop or extend to infinity. In other words, the magnetic field lines that enter any area must also leave that area. In terms, the magnetic flux passing through any closed surface is equal to zero, and the magnetic field is a spiral vector field.

Faraday's law of induction

Faraday's law of induction describes how a time-varying magnetic field generates (induces) an electric field. Electromagnetic induction is the operating principle of many generators. For example, a rotating bar magnet generates a time-varying magnetic field, which in turn generates an electric field, causing the adjacent closed loop to induce current.

Maxwell-Ampere’s law

Maxwell-Ampere’s law states that a magnetic field can be generated in two ways: one is generated by electric current (the method originally described by Ampere’s law), and the other It is produced by the electric field that changes with time (the method described by Maxwell's correction term). In electromagnetics, Maxwell's correction term means that a time-varying electric field can generate a magnetic field, and because of Faraday's law of induction, a time-varying magnetic field can generate an electric field. In this way, if the time-varying electric field happens to produce a changing magnetic field, according to these two equations, the mutually generated electric and magnetic fields (ie electromagnetic waves) will continue to propagate in space by themselves (for more details, please refer to the article Electromagnetic waves Equation).

Summary of equations

Here are two equivalent expressions of Maxwell's equations: micro-representation and macro-representation.

The microscopic expression specifically calculates the electric and magnetic fields generated by finite source charges and finite source currents at the atomic scale in vacuum. Matter can be regarded as composed of point electrons and point atomic nuclei, and most of the other space inside is vacuum. However, due to the large number of electrons and nuclei, in practice, they cannot be included in the calculation. In fact, classical electromagnetism does not require overly precise answers. There are two main purposes for using the microscopic Maxwell equations. One is to derive the macroscopic Maxwell equations, and the other is to estimate the macroscopic material parameters from the atomic properties, such as permittivity, magnetic permeability, and so on. Micro-expression can give a lot of valuable information that macro-expression cannot.

The macro expression does not take into account the atomic structure of the substance, but regards the substance as a continuous medium whose properties are determined by the macroscopic substance parameters such as permittivity and magnetic permeability. From experiments, we can get the relationship between macroscopic material parameters and the nature, density, temperature, etc. of the material. The macroscopic Maxwell equations can be used to predict the average properties of charged particles, electric and magnetic fields. Using this expression will make various physical calculations in dielectric or magnetized substances easier.

Using different unit systems, the form of Maxwell's equations will be slightly changed, and the general form is still the same, but different constants will appear in different positions within the equation. The International System of Units (SI) is the most commonly used unit system, which is mostly used in engineering and chemistry fields, and almost all college physics textbooks also use this unit system. Other commonly used unit systems are Gauss, Lorenz-Heaviside, and Planck. The Gauss unit system derived from the centimeter-gram-second system is more suitable for teaching purposes and can make equations look simpler and easier to understand. The Gauss unit system will be elaborated later. The Lorenz-Heaviside unit system is also derived from the centimeter-gram-second system and is mainly used in particle physics. The Planck unit system is a natural unit system, and its units are defined according to the nature of nature, not set by humans. The Planck unit system is a very useful tool for studying theoretical physics and can give great enlightenment in theoretical discussions.

In this entry, unless otherwise specified, all equations are in the International System of Units.

Maxwell’s equations in vacuum

This form of Maxwell’s equations is also called "micro Maxwell’s equations", which can be used to derive macro Maxwell’s equations, or Used to find the relationship between atomic properties and macroscopic properties.

Micro-scale and macro-scale

In classical electromagnetics, the micro-scale refers to the scale range of the order of magnitude greater than 10 meters. To meet the microscopic scale, electrons and nuclei can be regarded as point charges, and the microscopic Maxwell equations are established; otherwise, the charge distribution inside the nucleus must be taken into consideration. The electric field and magnetic field calculated on the microscopic scale still change quite drastically, the distance of the spatial change is less than 10 meters, and the period of the time change is between 10 and 10 seconds. Therefore, from the microscopic Maxwell equations, the classical averaging operation must be performed to obtain the smooth, continuous, and slowly changing macroscopic electric field and macroscopic magnetic field. The lowest limit of the macro-scale is 10 meters. This means that the reflection and refraction behavior of electromagnetic waves can be described by the macro-Maxwell equations. Taking this minimum limit as the side length, a cube with a volume of 10 cubic meters contains about 10 nuclei and electrons. The physical behavior of so many nuclei and electrons, after classical averaging, is enough to smooth any violent fluctuations. According to reliable literature records, the classical averaging operation only requires averaging in space, not in time, and does not need to consider the quantum effects of atoms.

Classic averaging is a relatively simple averaging program. Given a function, the spatial average of this function is defined as

Among them, it is the average operation Space is a weight function.

Many functions can be selected as excellent weighting functions, Gaussian function is just one example:

The earliest Maxwell equations and related theories are macroscopic Material design is a kind of phenomenology. At that time, physicists did not know the basic cause of electromagnetic phenomena. Later, according to the particles of matter, the microscopic Maxwell equations were derived. In the first half of the twentieth century, breakthroughs and developments in the fields of quantum mechanics, relativity, and particle physics, combined with their new theories and the microscopic Maxwell equations, became the key cornerstone for the establishment of quantum electrodynamics. This is the most accurate theory in physics, and the calculated results can accurately match the experimental data.

Mathematical properties

Maxwell’s equations are like an overdetermined set: it only involves six unknowns (vector electric field and magnetic field each have three unknowns, current and charge are not unknowns , It is a physical quantity that is freely set and conforms to the conservation of charge), but it is composed of eight equations (two Gauss laws have two equations, and Faraday’s law and Maxwell-Ampere’s law have three equations each). After careful analysis, you can understand that it is actually not that simple.

The equations of Maxwell's equations have "independence"-from any one or more equations in the equations, no other equations in the equations can be derived. This means that Maxwell's equations are not overdetermined, and there are no equations that repeat any functions. The combination of Maxwell's equations, Lorentz force equations and Newton's second law of motion has "completeness". They can explain all the phenomena of classical electrodynamics without using any other equations. In a certain area, given appropriate initial and boundary conditions, the solution of Maxwell’s equations is "unique", that is, each dependent variable can only have one functional form, and it contains only constants or independent variables, and does not contain Any other dependent variables.

Faraday's law and Maxwell-Ampere's law jointly dominate the evolution of electromagnetic fields in space over time, while Gauss's law and Gauss's magnetic law are constraint equations. Electromagnetic fields must comply with this in all time and space. Two constraint equations. In theory, it can be assumed that a certain electromagnetic field obeys Faraday's law and Maxwell-Ampere's law in all spaces. On the contrary, if they do not abide by Gauss's law and Gauss's magnetic law, they cannot actually exist in the real world. In other words, Faraday's law and Maxwell-Ampere's law will give additional solutions, which do not meet the constraints of Gauss's law and Gauss's magnetic law.

Describe the electromagnetic properties in matter

Bound charge and Bound current

Main article: Current density, Bound charge and Bound current

A group of microscopic electric dipoles, the electric field generated by them can be regarded as the electric field generated by the surface charges located at the top and bottom respectively. A group of micro current loops together form a macro current loop. If the micro current loops are evenly distributed, the contributions of the current loops located inside will cancel each other, but the current loops located on the boundary will not be canceled, so a macro current loop will be formed.

Assume that an external electric field is applied to the dielectric. As a result of applying this electric field, the molecules of the dielectric form a microscopic electric dipole, which is accompanied by an electric dipole moment. The nucleus of the molecule will move slightly in the direction of the electric field, while the electron will move slightly in the opposite direction. This forms the electric polarization of the dielectric. Although, all the charges involved are still bound to their original molecules, the charge distribution caused by these tiny migrations becomes as if a thin layer of positive surface charges is formed on one side of the dielectric, and then on the other side. A thin layer of negative surface charge. The electric polarization is defined as the electric dipole moment density inside the dielectric, that is, the electric dipole moment per unit volume. In the dielectric, assuming that the electric polarization intensity is uniform, the macroscopic surface bound charge will only appear on the surface of the dielectric, that is, where it enters or leaves the dielectric; otherwise, if it is non-uniform, the dielectric Bound charges will also appear inside the mass.

Something similar to electrostatics is that in magnetostatics, it is assumed that an external magnetic field is applied to a substance. In response to this action, the substance will be magnetized, and the atomic component will show a magnetic moment. In essence, this magnetic moment is related to the angular momentum of each subatomic particle of the atom. Among them, the most notable response is electronics. This link of angular momentum can't help but reminiscent of a picture in which the magnetized matter becomes a group of microscopic bound current loops. Although each charge only moves in the microscopic circuits of its atoms, a group of microscopic bound current circuits gather together to form a macroscopic bound current circulating on the surface of the substance. These bound currents can be described by magnetization. The intensity of magnetization is defined as the density of the magnetic dipole moment in a magnetized substance, that is, the magnetic dipole moment per unit volume.

The physical behavior of these very complex and rough bound charges and bound currents can be expressed in terms of electrical polarization and magnetization on a macroscopic scale. The electric polarization and magnetization respectively average these bound charges and bound currents in space with appropriate scales. In this way, the uneven rough structure formed by individual atoms can be removed, but the physical properties that the intensity varies with position can be shown. Since all the involved vector fields have been averaged in the appropriate volume, the macro-Maxwell equations ignore many details on the micro-scale. However, these details may not be important for understanding the macro-scale properties of matter.

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