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Classical electromagnetism

From Wikipedia, the free encyclopedia
"Classical electrodynamics" redirects here. For the textbook by J. D. Jackson, seeClassical Electrodynamics (book).
Branch of theoretical physics
Electromagnetism
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Classical electromagnetism orclassical electrodynamics is a branch ofphysics focused on the study of interactions betweenelectric charges andcurrents using an extension of theclassical Newtonian model. It is, therefore, aclassical field theory. The theory provides a description of electromagnetic phenomena whenever the relevantlength scales and field strengths are large enough thatquantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described byquantum electrodynamics which is aquantum field theory.

History

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Main article:History of electromagnetism

The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field ofoptics centuries before light was understood to be an electromagnetic wave. However, the theory ofelectromagnetism, as it is currently understood, grew out ofMichael Faraday's experiments suggesting the existence of anelectromagnetic field andJames Clerk Maxwell's use ofdifferential equations to describe it in hisA Treatise on Electricity and Magnetism (1873). The development of electromagnetism in Europe included the development of methods to measurevoltage,current,capacitance, andresistance. Detailed historical accounts are given byWolfgang Pauli,[1]E. T. Whittaker,[2]Abraham Pais,[3] and Bruce J. Hunt.[4]

Lorentz force

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Main article:Lorentz force

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

F=q(E+v×B){\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}

where all boldfaced quantities arevectors:F is the force that a particle with chargeq experiences,E is theelectric field at the location of the particle,v is the velocity of the particle,B is themagnetic field at the location of the particle.

The above equation illustrates that the Lorentz force is the sum of two vectors. One is thecross 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.

Although the equation appears to suggest that the electric and magnetic fields are independent, the equationcan be rewritten in term offour-current (instead of charge) and a singleelectromagnetic tensor that represents the combined field (Fμν{\displaystyle F^{\mu \nu }}):

fα=FαβJβ.{\displaystyle f_{\alpha }=F_{\alpha \beta }J^{\beta }.\!}

Electric field

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Main article:Electric field

The electric fieldE is defined such that, on a stationary charge:

F=q0E{\displaystyle \mathbf {F} =q_{0}\mathbf {E} }

whereq0 is what is known as a test charge andF is theforce on that charge. The size of the charge does 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 ofE is N/C (newtons percoulomb). 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 fromCoulomb's law may be summed. The result after dividing byq0 is:

E(r)=14πε0∑i=1nqi(r−ri)|r−ri|3{\displaystyle \mathbf {E(r)} ={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {q_{i}\left(\mathbf {r} -\mathbf {r} _{i}\right)}{\left|\mathbf {r} -\mathbf {r} _{i}\right|^{3}}}}

wheren is the number of charges,qi is the amount of charge associated with theith charge,ri is the position of theith charge,r is the position where the electric field is being determined, andε0 is theelectric constant.

If the field is instead produced by a continuous distribution of charge, the summation becomes an integral:

E(r)=14πε0∫ρ(r′)(r−r′)|r−r′|3d3r′{\displaystyle \mathbf {E(r)} ={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho (\mathbf {r'} )\left(\mathbf {r} -\mathbf {r'} \right)}{\left|\mathbf {r} -\mathbf {r'} \right|^{3}}}\mathrm {d^{3}} \mathbf {r'} }

whereρ(r′){\displaystyle \rho (\mathbf {r'} )} is thecharge density andr−r′{\displaystyle \mathbf {r} -\mathbf {r'} } is the vector that points from the volume elementd3r′{\displaystyle \mathrm {d^{3}} \mathbf {r'} } to the point in space whereE is being determined.

Both of the above equations are cumbersome, especially if one wants to determineE as a function of position. A scalar function called theelectric potential can help. Electric potential, also called voltage (the units for which are the volt), is defined by theline integral

φ(r)=−∫CE⋅dl{\displaystyle \varphi \mathbf {(r)} =-\int _{C}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }

whereφ(r){\displaystyle \varphi ({\textbf {r}})} is the electric potential, andC is the path over which the integral is being taken.

This definition has a caveat. FromMaxwell'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 theA 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:

φ(r)=14πε0∑i=1nqi|r−ri|{\displaystyle \varphi \mathbf {(r)} ={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {q_{i}}{\left|\mathbf {r} -\mathbf {r} _{i}\right|}}}

whereq is the point charge's charge,r is the position at which the potential is being determined, andri is the position of each point charge. The potential for a continuous distribution of charge is:

φ(r)=14πε0∫ρ(r′)|r−r′|d3r′{\displaystyle \varphi \mathbf {(r)} ={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho (\mathbf {r'} )}{|\mathbf {r} -\mathbf {r'} |}}\,\mathrm {d^{3}} \mathbf {r'} }

whereρ(r′){\displaystyle \rho (\mathbf {r'} )} is the charge density, andr−r′{\displaystyle \mathbf {r} -\mathbf {r'} } is the distance from the volume elementd3r′{\displaystyle \mathrm {d^{3}} \mathbf {r'} } to point in space whereφ is being determined.

The scalarφ will add to other potentials as a scalar. This makes it relatively easy to break complex problems down into simple parts and add their potentials. Taking the definition ofφ backwards, we see that the electric field is just the negative gradient (thedel operator) of the potential. Or:

E(r)=−∇φ(r).{\displaystyle \mathbf {E(r)} =-\nabla \varphi \mathbf {(r)} .}

From this formula it is clear thatE can be expressed in V/m (volts per meter).

Electromagnetic waves

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Main article:Electromagnetic waves

A changing electromagnetic field propagates away from its origin in the form of awave. These waves travel in vacuum at thespeed of light and exist in a widespectrum ofwavelengths. Examples of the dynamic fields ofelectromagnetic radiation (in order of increasing frequency):radio waves,microwaves,light (infrared,visible light andultraviolet),x-rays andgamma rays. In the field ofparticle physics this electromagnetic radiation is the manifestation of theelectromagnetic interaction between charged particles.

General field equations

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Main articles:Jefimenko's equations andLiénard–Wiechert potential

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.[citation needed]

Retarded potentials can also be derived for point charges, and the equations are known as the Liénard–Wiechert potentials. Thescalar potential is:

φ=14πε0q|r−rq(tret)|−vq(tret)c⋅(r−rq(tret)){\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{\left|\mathbf {r} -\mathbf {r} _{q}(t_{\rm {ret}})\right|-{\frac {\mathbf {v} _{q}(t_{\rm {ret}})}{c}}\cdot (\mathbf {r} -\mathbf {r} _{q}(t_{\rm {ret}}))}}}

whereq{\displaystyle q} is the point charge's charge andr{\displaystyle {\textbf {r}}} is the position.rq{\displaystyle {\textbf {r}}_{q}} andvq{\displaystyle {\textbf {v}}_{q}}are the position and velocity of the charge, respectively, as a function ofretarded time. Thevector potential is similar:

A=μ04πqvq(tret)|r−rq(tret)|−vq(tret)c⋅(r−rq(tret)).{\displaystyle \mathbf {A} ={\frac {\mu _{0}}{4\pi }}{\frac {q\mathbf {v} _{q}(t_{\rm {ret}})}{\left|\mathbf {r} -\mathbf {r} _{q}(t_{\rm {ret}})\right|-{\frac {\mathbf {v} _{q}(t_{\rm {ret}})}{c}}\cdot (\mathbf {r} -\mathbf {r} _{q}(t_{\rm {ret}}))}}.}

These can then be differentiated accordingly to obtain the complete field equations for a moving point particle.

Models

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Branches of classical electromagnetism such as optics, electrical and electronic engineering consist of a collection of relevantmathematical models of different degrees of simplification and idealization to enhance the understanding of specific electrodynamics phenomena.[5] 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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Further reading

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Fundamental physical aspects of classical electrodynamics are presented in many textbooks. For the undergraduate level, textbooks likeThe Feynman Lectures on Physics,Electricity and Magnetism, andIntroduction to Electrodynamics are considered as classic references and for the graduate level, textbooks likeClassical Electricity and Magnetism,[6]Classical Electrodynamics, andCourse of Theoretical Physics are considered as classic references.

References

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  1. ^Pauli, W., 1958,Theory of Relativity, Pergamon, London
  2. ^Whittaker, E. T., 1960,History of the Theories of the Aether and Electricity, Harper Torchbooks, New York.
  3. ^Pais, A., 1983,Subtle is the Lord: The Science and the Life of Albert Einstein, Oxford University Press, Oxford
  4. ^Bruce J. Hunt (1991)The Maxwellians
  5. ^Peierls, Rudolf. Model-making in physics, Contemporary Physics, Volume 21 (1), January 1980, 3-17.
  6. ^Panofsky, W. K. H.;Phillips, M. (2005).Classical Electricity and Magnetism.Dover.ISBN 9780486439242.


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