Classical electromagnetism

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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.

Theoretical physics Branch of physics

Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena.

Electric charge physical property that quantifies an objects interaction with electric fields

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two-types of electric charge; positive and negative. Like charges repel and unlike attract. An object with an absence of net charge is referred to as neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.

In physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple. The decoupling of different length scales makes it possible to have a self-consistent theory that only describes the relevant length scales for a given problem. Scientific reductionism says that the physical laws on the shortest length scales can be used to derive the effective description at larger length scales. The idea that one can derive descriptions of physics at different length scales from one another can be quantified with the renormalization group.

Contents

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]

Richard Feynman American theoretical physicist

Richard Phillips Feynman was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model. For contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga.

Robert Benjamin Leighton was a prominent American experimental physicist who spent his professional career at the California Institute of Technology (Caltech). His work over the years spanned solid state physics, cosmic ray physics, the beginnings of modern particle physics, solar physics, the planets, infrared astronomy, and millimeter- and submillimeter-wave astronomy. In the latter four fields, his pioneering work opened up entirely new areas of research that subsequently developed into vigorous scientific communities.

Matthew Sands American accelerator physicist

Matthew Linzee Sands was an American physicist and educator best known as a co-author of the Feynman Lectures on Physics. A graduate of Rice University, Sands served with the Naval Ordnance Laboratory and the Manhattan Project's Los Alamos Laboratory during World War II.

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]

Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, followed by theories on light and vision developed by ancient Greek philosophers, and the development of geometrical optics in the Greco-Roman world. The word optics is derived from the Greek term τα ὀπτικά meaning "appearance, look". Optics was significantly reformed by the developments in the medieval Islamic world, such as the beginnings of physical and physiological optics, and then significantly advanced in early modern Europe, where diffractive optics began. These earlier studies on optics are now known as "classical optics". The term "modern optics" refers to areas of optical research that largely developed in the 20th century, such as wave optics and quantum optics.

Electromagnetism Branch of science concerned with the phenomena of electricity and magnetism

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 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.

Michael Faraday English scientist

Michael Faraday FRS was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis.

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.

Electric field spatial distribution of vectors representing the force applied to a charged test particle

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. 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.

Magnetic field spatial distribution of vectors allowing the calculation of the magnetic force on a test particle

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. Magnetic fields are observed in a wide range of size scales, from subatomic particles to galaxies. The effects of magnetic fields are commonly 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 vary with location. As such, it is an example of a vector field.

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.

Cross product mathematical operation on two vectors

In mathematics and vector algebra, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol . Given two linearly independent vectors and , the cross product, , is a vector that is perpendicular to both and and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.

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

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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.