Classical electromagnetism
Encyclopedia
Classical electromagnetism (or classical electrodynamics) is a branch of theoretical physics
that studies consequences of the electromagnetic forces between electric charge
s and currents. It provides an excellent description of electromagnetic phenomena whenever the relevant length
scales and field strengths are large enough that quantum mechanical effects are negligible (see quantum electrodynamics
). Fundamental physical aspects of classical electrodynamics are presented e.g. by Feynman, Leighton and Sands, Panofsky and Phillips, and Jackson.
The theory of electromagnetism
was developed over the course of the 19th century, most prominently by James Clerk Maxwell
. For a detailed historical account, consult Pauli, Whittaker, and Pais. See also History of optics
, History of electromagnetism and Maxwell's equations
.
Ribarič and Šušteršič considered a dozen open questions in the current understanding of classical electrodynamics; to this end they studied and cited about 240 references from 1903 to 1989. The outstanding problem with classical electrodynamics, as stated by Jackson, is that researchers are able to obtain and study relevant solutions of its basic equations only in two limiting cases: »... one in which the sources of charges and currents are specified and the resulting electromagnetic fields are calculated, and the other in which external electromagnetic fields are specified and the motion of charged particles or currents is calculated... Occasionally, ..., the two problems are combined. But the treatment is a stepwise one -- first the motion of the charged particle in the external field is determined, neglecting the emission of radiation; then the radiation is calculated from the trajectory as a given source distribution. It is evident that this manner of handling problems in electrodynamics can be of only approximative validity.« As a consequence, there is not yet a physical understanding of those electromechanical systems where one cannot neglect the mutual interaction between electric charges and currents, and the electromagnetic field emitted by them. In spite of a century long effort, there is as yet no generally accepted classical equation of motion for charged particles, as well as no pertinent experimental data, cf.
particles:
where all boldfaced quantities are vectors: F is the force that a charge q experiences, E is the electric field
at the location of the charge, v is the velocity of the charge, B is the magnetic field
at the location of the charge.
The above equation illustrates that the Lorentz force is the sum of two vectors. One is the cross product of the velocity and magnetic field vectors. Based on the properties of the cross product, this produces a vector that is perpendicular to both the velocity and magnetic field vectors. The other vector is in the same direction as the electric field. The sum of these two vectors is the Lorentz force.
Therefore, in the absence of a magnetic field, the force is in the direction of the electric field, and the magnitude of the force is dependent on the value of the charge and the intensity of the electric field. In the absence of an electric field, the force is perpendicular to the velocity of the particle and the direction of the magnetic field. If both electric and magnetic fields are present, the Lorentz force is the sum of both of these vectors.
E is defined such that, on a stationary charge:
where q0 is what is known as a test charge. The size of the charge doesn't really matter, as long as it is small enough not to influence the electric field by its mere presence. What is plain from this definition, though, is that the unit of E is N/C (newtons per coulomb). This unit is equal to V/m (volt
s per meter), see below.
In electrostatics, where charges are not moving, around a distribution of point charges, the forces determined from Coulomb's law
may be summed. The result after dividing by q0 is:
where n is the number of charges, qi is the amount of charge associated with the ith charge, ri is the position of the ith charge, r is the position where the electric field is being determined, and ε0 is the electric constant
.
If the field is instead produced by a continuous distribution of charge, the summation becomes an integral:
where ρ(r) is the charge density
as a function of position,
is the unit vector pointing from dV to the point in space where E is being determined, and r is the distance from the point E is being calculated at to the point charge.
Both of the above equations are cumbersome, especially if one wants to determine E as a function of position. A scalar function called the electric potential
can help. Electric potential, also called voltage (the units for which are the volt), is defined by the line integral
where φE is the electric potential, and C is the path over which the integral is being taken.
Unfortunately, this definition has a caveat. From Maxwell's equations
, it is clear that is not always zero, and hence the scalar potential alone is insufficient to define the electric field exactly. As a result, one must add a correction factor, which is generally done by subtracting the time derivative of the A vector potential described below. Whenever the charges are quasistatic, however, this condition will be essentially met.
From the definition of charge, one can easily show that the electric potential of a point charge as a function of position is:
where q is the point charge's charge, r is the position at which the potential is being determined, and rq is the position of the point charge. The potential for a general distribution of charge ends up being:
where ρ(r) is the charge density as a function of position, and r is the distance from the volume element dV.
The scalar φ will add to other potentials as a scalar. This makes it relatively easy to break complex problems down in to simple parts and add their potentials. Taking the definition of φ backwards, we see that the electric field is just the negative gradient (the del
operator) of the potential. Or:
From this formula it is clear that E can be expressed in V/m (volts per meter).
. These waves travel in vacuum at the speed of light
and exist in a wide spectrum
of wavelength
s. Examples of the dynamic fields of electromagnetic radiation
(in order of increasing frequency): radio
waves, microwave
s, light
(infrared
, visible light and ultraviolet
), x-ray
s and gamma rays. In the field of particle physics
this electromagnetic radiation is the manifestation of the electromagnetic interaction between charged particles.
For the fields of general charge distributions, the retarded potentials can be computed and differentiated accordingly to yield Jefimenko's Equations
.
Retarded potentials can also be derived for point charges, and the equations are known as the Liénard-Wiechert potentials
. The scalar potential
is:
where q is the point charge's charge and r is the position. rq and vq are the position and velocity of the charge, respectively, as a function of retarded time. The vector potential
is similar:
These can then be differentiated accordingly to obtain the complete field equations for a moving point particle.
s of different degree of simplification and idealization to enhance our understanding of the specific electrodynamics phenomena, cf. An electrodynamics phenomenon is determined by the particular fields, specific densities of electric charges and currents, and the particular transmission medium. Since there are infinitely many of them, in modeling there is a need for some typical, representative electrical charges and currents, e.g. moving pointlike charges and electric and magnetic dipoles, electric currents in a conductor etc; electromagnetic fields, e.g. voltages, the Liénard-Wiechert potentials, the monochromatic plane waves , optical rays; radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays , gamma rays etc; transmission media, e.g. electronic components, antennas, electromagnetic waveguides, flat mirrors, mirrors with curved surfaces convex lenses, concave lenses; resistors, inductors, capacitors, switches; wires, electric and optical cables, transmission lines, integrated circuits etc;
which all have only few variable characteristics.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
that studies consequences of the electromagnetic forces between electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
s and currents. It provides an excellent description of electromagnetic phenomena whenever the relevant length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
scales and field strengths are large enough that quantum mechanical effects are negligible (see quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
). Fundamental physical aspects of classical electrodynamics are presented e.g. by Feynman, Leighton and Sands, Panofsky and Phillips, and Jackson.
The theory of electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
was developed over the course of the 19th century, most prominently by James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
. For a detailed historical account, consult Pauli, Whittaker, and Pais. See also History of optics
History of optics
Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, followed by theories on light and vision developed by ancient Greek and Indian philosophers, and the development of geometrical optics in the Greco-Roman world. The word optics is derived from the Greek term τα...
, History of electromagnetism and Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
.
Ribarič and Šušteršič considered a dozen open questions in the current understanding of classical electrodynamics; to this end they studied and cited about 240 references from 1903 to 1989. The outstanding problem with classical electrodynamics, as stated by Jackson, is that researchers are able to obtain and study relevant solutions of its basic equations only in two limiting cases: »... one in which the sources of charges and currents are specified and the resulting electromagnetic fields are calculated, and the other in which external electromagnetic fields are specified and the motion of charged particles or currents is calculated... Occasionally, ..., the two problems are combined. But the treatment is a stepwise one -- first the motion of the charged particle in the external field is determined, neglecting the emission of radiation; then the radiation is calculated from the trajectory as a given source distribution. It is evident that this manner of handling problems in electrodynamics can be of only approximative validity.« As a consequence, there is not yet a physical understanding of those electromechanical systems where one cannot neglect the mutual interaction between electric charges and currents, and the electromagnetic field emitted by them. In spite of a century long effort, there is as yet no generally accepted classical equation of motion for charged particles, as well as no pertinent experimental data, cf.
Lorentz force
The electromagnetic field exerts the following force (often called the Lorentz force) on chargedElectric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
particles:
where all boldfaced quantities are vectors: F is the force that a charge q experiences, E is the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
at the location of the charge, v is the velocity of the charge, B is the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
at the location of the charge.
The above equation illustrates that the Lorentz force is the sum of two vectors. One is the cross product of the velocity and magnetic field vectors. Based on the properties of the cross product, this produces a vector that is perpendicular to both the velocity and magnetic field vectors. The other vector is in the same direction as the electric field. The sum of these two vectors is the Lorentz force.
Therefore, in the absence of a magnetic field, the force is in the direction of the electric field, and the magnitude of the force is dependent on the value of the charge and the intensity of the electric field. In the absence of an electric field, the force is perpendicular to the velocity of the particle and the direction of the magnetic field. If both electric and magnetic fields are present, the Lorentz force is the sum of both of these vectors.
The electric field E
The electric fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
E is defined such that, on a stationary charge:
where q0 is what is known as a test charge. The size of the charge doesn't really matter, as long as it is small enough not to influence the electric field by its mere presence. What is plain from this definition, though, is that the unit of E is N/C (newtons per coulomb). This unit is equal to V/m (volt
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
s per meter), see below.
In electrostatics, where charges are not moving, around a distribution of point charges, the forces determined from Coulomb's law
Coulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
may be summed. The result after dividing by q0 is:
where n is the number of charges, qi is the amount of charge associated with the ith charge, ri is the position of the ith charge, r is the position where the electric field is being determined, and ε0 is the electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...
.
If the field is instead produced by a continuous distribution of charge, the summation becomes an integral:
where ρ(r) is the charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
as a function of position,
is the unit vector pointing from dV to the point in space where E is being determined, and r is the distance from the point E is being calculated at to the point charge.Both of the above equations are cumbersome, especially if one wants to determine E as a function of position. A scalar function called the electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
can help. Electric potential, also called voltage (the units for which are the volt), is defined by the line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
where φE is the electric potential, and C is the path over which the integral is being taken.
Unfortunately, this definition has a caveat. From Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
, it is clear that is not always zero, and hence the scalar potential alone is insufficient to define the electric field exactly. As a result, one must add a correction factor, which is generally done by subtracting the time derivative of the A vector potential described below. Whenever the charges are quasistatic, however, this condition will be essentially met.
From the definition of charge, one can easily show that the electric potential of a point charge as a function of position is:
where q is the point charge's charge, r is the position at which the potential is being determined, and rq is the position of the point charge. The potential for a general distribution of charge ends up being:
where ρ(r) is the charge density as a function of position, and r is the distance from the volume element dV.
The scalar φ will add to other potentials as a scalar. This makes it relatively easy to break complex problems down in to simple parts and add their potentials. Taking the definition of φ backwards, we see that the electric field is just the negative gradient (the del
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...
operator) of the potential. Or:
From this formula it is clear that E can be expressed in V/m (volts per meter).
Electromagnetic waves
A changing electromagnetic field propagates away from its origin in the form of a waveWave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
. These waves travel in vacuum at the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
and exist in a wide spectrum
Electromagnetic spectrum
The electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation. The "electromagnetic spectrum" of an object is the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object....
of wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s. Examples of the dynamic fields of electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
(in order of increasing frequency): radio
Radio
Radio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...
waves, microwave
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...
s, light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
(infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...
, visible light and ultraviolet
Ultraviolet
Ultraviolet light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV...
), x-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...
s and gamma rays. In the field of particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
this electromagnetic radiation is the manifestation of the electromagnetic interaction between charged particles.
General field equations
As simple and satisfying as Coulomb's equation may be, it is not entirely correct in the context of classical electromagnetism. Problems arise because changes in charge distributions require a non-zero amount of time to be "felt" elsewhere (required by special relativity).For the fields of general charge distributions, the retarded potentials can be computed and differentiated accordingly to yield Jefimenko's Equations
Jefimenko's equations
In electromagnetism, Jefimenko's equations describe the behavior of the electric and magnetic fields in terms of the charge and current distributions at retarded times....
.
Retarded potentials can also be derived for point charges, and the equations are known as the Liénard-Wiechert potentials
Liénard-Wiechert Potentials
Liénard-Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential...
. The scalar potential
Scalar potential
A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...
is:
where q is the point charge's charge and r is the position. rq and vq are the position and velocity of the charge, respectively, as a function of retarded time. The vector potential
Vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
is similar:
These can then be differentiated accordingly to obtain the complete field equations for a moving point particle.
Models
A branch of classical electromagnetisms such as optics, electrical and electronic engineering consist of a collection of relevant mathematical modelMathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
s of different degree of simplification and idealization to enhance our understanding of the specific electrodynamics phenomena, cf. An electrodynamics phenomenon is determined by the particular fields, specific densities of electric charges and currents, and the particular transmission medium. Since there are infinitely many of them, in modeling there is a need for some typical, representative electrical charges and currents, e.g. moving pointlike charges and electric and magnetic dipoles, electric currents in a conductor etc; electromagnetic fields, e.g. voltages, the Liénard-Wiechert potentials, the monochromatic plane waves , optical rays; radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays , gamma rays etc; transmission media, e.g. electronic components, antennas, electromagnetic waveguides, flat mirrors, mirrors with curved surfaces convex lenses, concave lenses; resistors, inductors, capacitors, switches; wires, electric and optical cables, transmission lines, integrated circuits etc;
which all have only few variable characteristics.
External links
- Electromagnetic Field Theory by Bo Thidé