Classical electromagnetism: Difference between revisions

{{short description|Branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents}}

'''Classical electromagnetism''' or '''classical electrodynamics''' is a branch of [[theoretical physics]] that studies the interactions between [[electric charge]]s and [[electrical current|currents]] using an extension of the [[classical Newtonian model]]. It is, therefore, a [[classical field theory]]. The theory provides a description of electromagnetic phenomena whenever the relevant [[length scale]]s 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]] which is a [[quantum field theory]].

The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of [[History of optics|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 the existence of an [[electromagnetic field]] and [[James Clerk Maxwell]]'s use of [[differential equation]]s to describe it in his ''[[A Treatise on Electricity and Magnetism]]'' (1873). The development of electromagnetism in Europe included the development of methods to measure [[voltage]], [[Electric current|current]], [[capacitance]], and [[Electrical resistance and conductance|resistance]]. ForDetailed ahistorical detailedaccounts historicalare account,given consultby [[Wolfgang Pauli]],<ref>Pauli, W., 1958, ''Theory of Relativity'', Pergamon, London</ref> [[E. T. Whittaker]],<ref>Whittaker, E. T., 1960, ''History of the Theories of the Aether and Electricity'', Harper Torchbooks, New York.</ref> [[Abraham Pais]],<ref>Pais, A., 1983, ''[[Subtle is the Lord: The Science and the Life of Albert Einstein]]'', Oxford University Press, Oxford</ref> and Bruce J. Hunt.<ref>Bruce J. Hunt (1991) [[The Maxwellians]]</ref>

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.

The [[electric field]] '''E''' is defined such that, on a stationary charge:

where ''q''₀ is what is known as a test charge and {{math|'''F'''}} is the [[Electrostatic force|force]] on that charge. The size of the charge doesn'tdoes not 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 {{math|'''E'''}} is N/C ([[newton (unit)|newtons]] per [[coulomb]]). This unit is equal to V/m ([[volt]]s per meter); see below.

where ''φ<math>\varphi(\textbf{r})''</math> 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 {{nowrap|∇ × '''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.

\varphi \mathbf{(r)} = \frac{1}{4 \pi \varepsilon_0 }

where <math>\rho(\mathbf{r'})</math> is the charge density, and <math>\mathbf{r}-\mathbf{r'}</math> is the distance from the volume element <math>\mathrm{d^3}\mathbf{r'}</math> 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 tointo 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:

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 [[electromagnetic spectrum|spectrum]] of [[wavelength]]s. Examples of the dynamic fields of [[electromagnetic radiation]] (in order of increasing frequency): [[radio wave]]s, [[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.

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 potential]]spotentials. The [[scalar potential]] is:

where ''<math>q''</math> is the point charge's charge and '''<math>\textbf{r'''}</math> is the position. '''r'''<submath>''\textbf{r}_{q''}</submath> and '''v''' <submath>''\textbf{v}_{q''}</submath> are the position and velocity of the charge, respectively, as a function of [[retarded time]]. The [[vector potential]] is similar:

Branches of classical electromagnetism such as optics, electrical and electronic engineering consist of a collection of relevant [[mathematical model]]s of different degrees of simplification and idealization to enhance the understanding of specific electrodynamics phenomena, cf.<ref>[[Rudolf Peierls|Peierls]], Rudolf. Model-making in physics, Contemporary Physics, Volume 21 (1), January 1980, 3-17.</ref> 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.