Classical electromagnetism

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.

Fundamental physical aspects of classical electrodynamics are presented in many texts, such as those by Feynman, Leighton and Sands,[1] Griffiths,[2] Panofsky and Phillips,[3] and Jackson.[4]

History

The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of optics centuries before light was understood to be an electromagnetic wave. However, the theory of electromagnetism, as it is currently understood, grew out of Michael Faraday's experiments suggesting an electromagnetic field and James Clerk Maxwell's use of differential equations to describe it in his A Treatise on Electricity and Magnetism (1873). For a detailed historical account, consult Pauli,[5] Whittaker,[6] Pais,[7] and Hunt.[8]

Lorentz force

The electromagnetic field exerts the following force (often called the Lorentz force) on charged particles:

where all boldfaced quantities are vectors: F is the force that a particle with charge q experiences, E is the electric field at the location of the particle, v is the velocity of the particle, B is the magnetic field at the location of the particle.

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.

Although the equation appears to suggest that the Electric and Magnetic fields are independent, the equation can be rewritten in term of four-current (instead of charge) and a single tensor that represents the combined Electromagnetic field ()

The electric field E

The electric field 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 (volts 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 is the charge density and is the vector that points from the volume element to the point in space where E is being determined.

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 φ(r) 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 ∇ × E 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 ri is the position of each point charge. The potential for a continuous distribution of charge is:

where is the charge density, and is the distance from the volume element to point in space where φ is being determined.

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).

Electromagnetic waves

A changing electromagnetic field propagates away from its origin in the form of a wave. These waves travel in vacuum at the speed of light and exist in a wide spectrum of wavelengths. Examples of the dynamic fields of electromagnetic radiation (in order of increasing frequency): radio waves, microwaves, light (infrared, visible light and ultraviolet), x-rays and gamma rays. In the field of particle physics 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.

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.

Models

Branches of classical electromagnetism such as optics, electrical and electronic engineering consist of a collection of relevant mathematical models of different degrees of simplification and idealization to enhance the understanding of specific electrodynamics phenomena, cf.[9] 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

(a) electrical charges and currents, e.g. moving pointlike charges and electric and magnetic dipoles, electric currents in a conductor etc.;
(b) 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.;
(c) 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.;

all of which have only few variable characteristics.

See also

References

  1. ^ Feynman, R. P., R .B. Leighton, and M. Sands, 1965, The Feynman Lectures on Physics, Vol. II: the Electromagnetic Field, Addison-Wesley, Reading, Massachusetts
  2. ^ Griffiths, David J. (2013). Introduction to Electrodynamics (4th ed.). Boston, Mas.: Pearson. ISBN 978-0321856562.
  3. ^ Panofsky, W. K., and M. Phillips, 1969, Classical Electricity and Magnetism, 2nd edition, Addison-Wesley, Reading, Massachusetts
  4. ^ Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). New York: Wiley. ISBN 978-0-471-30932-1.
  5. ^ Pauli, W., 1958, Theory of Relativity, Pergamon, London
  6. ^ Whittaker, E. T., 1960, History of the Theories of the Aether and Electricity, Harper Torchbooks, New York.
  7. ^ Pais, A., 1983, »Subtle is the Lord...«; the Science and Life of Albert Einstein, Oxford University Press, Oxford
  8. ^ Bruce J. Hunt (1991) The Maxwellians
  9. ^ Peierls, Rudolf. Model-making in physics, Contemporary Physics, Volume 21 (1), January 1980, 3-17.

External links

Ampère's circuital law

In classical electromagnetism, Ampère's circuital law (not to be confused with Ampère's force law that André-Marie Ampère discovered in 1823) relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell (not Ampère) 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.

Classical Mechanics (Goldstein book)

Classical Mechanics is a textbook about that subject written by Herbert Goldstein. Intended for advanced undergraduate and beginning graduate students, it has been one of the standard references in its subject around the world since its first publication in 1951.In the second edition, Goldstein corrected all the errors that had been pointed out, added a new chapter on perturbation theory, a new section on Bertrand's theorem, and another on Noether's theorem. Other arguments and proofs were simplified and supplemented.Before the death of its primary author in 2005, a new (third) edition of the book was released, with the collaboration of Charles P. Poole and John L. Safko. In the third edition, the book discusses at length various mathematically sophisticated reformations of Newtonian mechanics, namely analytical mechanics, as applied to particles, rigid bodies and continua. In addition, it covers in some detail classical electromagnetism, special relativity, and field theory, both classical and relativistic. There is an appendix on group theory. New to the third edition include a chapter on nonlinear dynamics and chaos, a section on the exact solutions to the three-body problem obtained by Euler and Lagrange, a discussion of the damped driven pendulum that explains the Josephson junctions. This is counterbalanced by the reduction of several existing chapters motivated by the desire to prevent this edition from exceeding the previous one in length. For example, the discussions of Hermitian and unitary matrices were omitted because they are more relevant to quantum mechanics rather than classical mechanics, while those of Routh's procedure and time-independent perturbation theory were reduced.

Classical electromagnetism and special relativity

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. First of all, it gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another. Secondly, it sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity. Moreover, the apparent coincidences in which the same effect was observed due to different physical phenomena by two different observers would be shown to be not coincidental in the least by special relativity. In fact, half of Einstein's 1905 first paper on special relativity, "On the Electrodynamics of Moving Bodies," explains how to transform Maxwell's equations.

Covariant formulation of classical electromagnetism

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

This article uses the classical treatment of tensors and Einstein summation convention throughout and the Minkowski metric has the form diag (+1, −1, −1, −1). Where the equations are specified as holding in a vacuum, one could instead regard them as the formulation of Maxwell's equations in terms of total charge and current.

For a more general overview of the relationships between classical electromagnetism and special relativity, including various conceptual implications of this picture, see Classical electromagnetism and special relativity.

Electromagnetic tensor

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely.

Electromagnetism

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields, is responsible for electromagnetic radiation such as light, and is one of the four fundamental interactions (commonly called forces) in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force.

Electromagnetic phenomena are defined in terms of the electromagnetic force, sometimes called the Lorentz force, which includes both electricity and magnetism as different manifestations of the same phenomenon. The electromagnetic force plays a major role in determining the internal properties of most objects encountered in daily life. The electromagnetic attraction between atomic nuclei and their orbital electrons holds atoms together. Electromagnetic forces are responsible for the chemical bonds between atoms which create molecules, and intermolecular forces. The electromagnetic force governs all chemical processes, which arise from interactions between the electrons of neighboring atoms.

There are numerous mathematical descriptions of the electromagnetic field. In classical electrodynamics, electric fields are described as electric potential and electric current. In Faraday's law, magnetic fields are associated with electromagnetic induction and magnetism, and Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents.

The theoretical implications of electromagnetism, particularly the establishment of the speed of light based on properties of the "medium" of propagation (permeability and permittivity), led to the development of special relativity by Albert Einstein in 1905.

Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.

The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory.

Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. Local symmetry, the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same).

Gauge theories are important as the successful field theories explaining the dynamics of elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons.

Gauge theories are also important in explaining gravitation in the theory of general relativity. Its case is somewhat unusual in that the gauge field is a tensor, the Lanczos tensor. Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton. Gauge symmetries can be viewed as analogues of the principle of general covariance of general relativity in which the coordinate system can be chosen freely under arbitrary diffeomorphisms of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of general covariance with a true gauge principle with new gauge fields.

Historically, these ideas were first stated in the context of classical electromagnetism and later in general relativity. However, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below. Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields.

Magnetic potential

The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, or simply vector potential, A; and the magnetic scalar potential ψ. Both quantities can be used in certain circumstances to calculate the magnetic field B.

The more frequently used magnetic vector potential is defined so that its curl is equal to the magnetic field:

×

A

=

B

{\textstyle \nabla \times \mathbf {A} =\mathbf {B} \,}

. Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Therefore, many equations of electromagnetism can be written either in terms of the fields E and B, or equivalently in terms of the potentials A and φ. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

The magnetic scalar potential ψ is sometimes used to specify the magnetic H-field in cases when there are no free currents, in a manner analogous to using the electric potential to determine the electric field in electrostatics. One important use of ψ is to determine the magnetic field due to permanent magnets when their magnetization is known. With some care the scalar potential can be extended to include free currents as well.Historically, Lord Kelvin first introduced vector potential in 1851, along with the formula relating it to the magnetic field.

Magnetization

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.

Nonclassical light

Nonclassical light is light that cannot be described using classical electromagnetism; its characteristics are described by the quantized electromagnetic field and quantum mechanics.

Most common described forms of nonclassical light are the following:

Squeezed light exhibits reduced noise in one quadrature component. The most familiar kinds of squeezed light have either reduced amplitude noise or reduced phase noise, with increased noise of the other component.

Fock states (also called photon number states) have a well-defined number of photons (stored e.g. in a cavity), while the phase is totally undefined.

Nonradiation condition

Classical nonradiation conditions define the conditions according to classical electromagnetism under which a distribution of accelerating charges will not emit electromagnetic radiation. According to the Larmor formula in classical electromagnetism, a single point charge under acceleration will emit electromagnetic radiation, i.e. light. In some classical electron models a distribution of charges can however be accelerated so that no radiation is emitted. The modern derivation of these nonradiation conditions by Hermann A. Haus is based on the Fourier components of the current produced by a moving point charge. It states that a distribution of accelerated charges will radiate if and only if it has Fourier components synchronous with waves traveling at the speed of light.

Point particle

A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension: being zero-dimensional, it does not take up space.. A point particle is an appropriate representation of any object whose size, shape, and structure is irrelevant in a given context. For example, from far enough away, any finite-size object will look and behave as a point-like object. A point particle can also be referred in the case of a moving body in terms of physics

In the theory of gravity, physicists often discuss a point mass, meaning a point particle with a nonzero mass and no other properties or structure. Likewise, in electromagnetism, physicists discuss a point charge, a point particle with a nonzero charge.Sometimes, due to specific combinations of properties, extended objects behave as point-like even in their immediate vicinity. For example, spherical objects interacting in 3-dimensional space whose interactions are described by the inverse square law behave in such a way as if all their matter were concentrated in their centers of mass. In Newtonian gravitation and classical electromagnetism, for example, the respective fields outside a spherical object are identical to those of a point particle of equal charge/mass located at the center of the sphere.In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occupies a nonzero volume. For example, the atomic orbit of an electron in the hydrogen atom occupies a volume of ~10−30 m3. There is nevertheless a distinction between elementary particles such as electrons or quarks, which have no known internal structure, versus composite particles such as protons, which do have internal structure: A proton is made of three quarks. Elementary particles are sometimes called "point particles", but this is in a different sense than discussed above.

Raising and lowering indices

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.

Reciprocity (electromagnetism)

This page is about reciprocity theorems in classical electromagnetism. See also Reciprocity theorem (disambiguation) for unrelated reciprocity theorems, and Reciprocity (disambiguation) for more general usages of the term.In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints. Reciprocity is closely related to the concept of Hermitian operators from linear algebra, applied to electromagnetism.

Perhaps the most common and general such theorem is Lorentz reciprocity (and its various special cases such as Rayleigh-Carson reciprocity), named after work by Hendrik Lorentz in 1896 following analogous results regarding sound by Lord Rayleigh and light by Helmholtz (Potton, 2004). Loosely, it states that the relationship between an oscillating current and the resulting electric field is unchanged if one interchanges the points where the current is placed and where the field is measured. For the specific case of an electrical network, it is sometimes phrased as the statement that voltages and currents at different points in the network can be interchanged. More technically, it follows that the mutual impedance of a first circuit due to a second is the same as the mutual impedance of the second circuit due to the first.

Reciprocity is useful in optics, which (apart from quantum effects) can be expressed in terms of classical electromagnetism, but also in terms of radiometry.

There is also an analogous theorem in electrostatics, known as Green's reciprocity, relating the interchange of electric potential and electric charge density.

Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and antenna systems. For example, reciprocity implies that antennas work equally well as transmitters or receivers, and specifically that an antenna's radiation and receiving patterns are identical. Reciprocity is also a basic lemma that is used to prove other theorems about electromagnetic systems, such as the symmetry of the impedance matrix and scattering matrix, symmetries of Green's functions for use in boundary-element and transfer-matrix computational methods, as well as orthogonality properties of harmonic modes in waveguide systems (as an alternative to proving those properties directly from the symmetries of the eigen-operators).

Reciprocity theorem

Reciprocity theorem may refer to:

Quadratic reciprocity, a theorem about modular arithmetic

Cubic reciprocity

Quartic reciprocity

Artin reciprocity

Weil reciprocity for algebraic curves

Frobenius reciprocity theorem for group representations

Stanley's reciprocity theorem for generating functions

Reciprocity (electromagnetism), theorems relating sources and the resulting fields in classical electromagnetism

Tellegen's theorem, a theorem about the transfer function of passive networks

Reciprocity law for Dedekind sums

Relativistic electromagnetism

Relativistic electromagnetism is a physical phenomenon explained in electromagnetic field theory due to Coulomb's law and Lorentz transformations.

Surface integral

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.

Thomson scattering

Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is just the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequency do not change as a result of the scattering. This limit is valid as long as the photon energy is much smaller than the mass energy of the particle: , or equivalently, if the wavelength of the light is much greater than the Compton wavelength of the particle.

Vacuum permittivity

The physical constant ε0 (pronounced as “epsilon nought” or “epsilon zero”), commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has an exactly defined value that can be approximated as

ε0 = 8.8541878128(13)×10−12 F⋅m−1 (farads per metre).

It is the capability of the vacuum to permit electric field lines. This constant relates the units for electric charge to mechanical quantities such as length and force. For example, the force between two separated electric charges (in the vacuum of classical electromagnetism) is given by Coulomb's law:

The value of the constant fraction is approximately 9 × 109 N⋅m2⋅C−2, q1 and q2 are the charges, and r is the distance between them. Likewise, ε0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources.

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