Classical electromagnetism

Electromagnetism
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Electricity · Magnetism

Classical electromagnetism (or classical electrodynamics) is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges 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,[1] Panofsky and Phillips,[2] and Jackson.[3]

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,[4] Whittaker,[5] and Pais.[6]. See also History of optics, History of electromagnetism and Maxwell's equations.

Ribarič and Šušteršič[7] 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[3], is that we 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, we do not yet have physical understanding of those electromechanical systems where we 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.[8]

Contents

Lorentz force

Main article: Lorentz force

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

\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B}

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 electric field E

Main article: Electric field

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

\mathbf{F} = q_0 \mathbf{E}

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 as to not influence the electric field by its mere presence. What is plain from this definition, though, is that the unit of E is N/C, or newtons per coulomb. This unit is equal to V/m (volts per meter), see below.

The above definition seems a little bit circular but, in electrostatics, where charges are not moving, Coulomb's law works fine. The result is:

\mathbf{E} = \frac{1}{4 \pi \epsilon_0 } \sum_{i=1}^{n} \frac{q_i \left( \mathbf{r} - \mathbf{r}_i \right)} {\left| \mathbf{r} - \mathbf{r}_i \right|^3}

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.

Note: the above is just Coulomb's law, divided by q1, adding up multiple charges.

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

\mathbf{E} = \frac{1}{ 4 \pi \epsilon_0 } \int \frac{\rho(\mathbf{r}) \hat{\mathbf{r}}}{r^2} \mathrm{d}V

where ρ(r) is the charge density as a function of position, \hat{\mathbf{r}} is the unit vector pointing from dV to the point in space E is being calculated at, 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 calculate E as a function of position. There is, however, a scalar function called the electrical potential that can help. Electric potential, also called voltage (the units for which are the volt), which is defined by the line integral

\varphi_\mathbf{E} = - \int_C \mathbf{E} \cdot \mathrm{d}\mathbf{s} \, ,

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 ∇ × E is not always zero, and hence the scalar potential alone is insufficient to define the electric field exactly. As a result, one must resort to adding 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, so there will be few problems.

From the definition of charge, one can easily show that the electric potential of a point charge as a function of position is:

\varphi = \frac{q}{ 4 \pi \epsilon_0 \left| \mathbf{r} - \mathbf{r}_q \right|}

where q is the point charge's charge, r is the position, and rq is the position of the point charge. The potential for a general distribution of charge ends up being:

\varphi = \frac{1}{4 \pi \epsilon_0} \int \frac{\rho(\mathbf{r})}{r}\, \mathrm{d}V

where ρ(r) is the charge density as a function of position, and r is the distance from the volume element dV.

Note well that φ is a scalar, which means that it will add to other potential fields 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:

\mathbf{E} = -\nabla \varphi

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

Electromagnetic waves

Main article: 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). Disturbances of the electric field due to a charge propagate at the speed of light.

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:

\varphi = \frac{1}{4 \pi \epsilon_0} \frac{q}{\left| \mathbf{r} - \mathbf{r}_q(t_{ret}) \right|-\frac{\mathbf{v}_q(t_{ret})}{c} \cdot (\mathbf{r} - \mathbf{r}_q(t_{ret}))}

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:

\mathbf{A} = \frac{\mu_0}{4 \pi} \frac{q\mathbf{v}_q(t_{ret})}{\left| \mathbf{r} - \mathbf{r}_q(t_{ret}) \right|-\frac{\mathbf{v}_q(t_{ret})}{c} \cdot (\mathbf{r} - \mathbf{r}_q(t_{ret}))}

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

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, Mass.
  2. ↑ Panofsky, W.K., and M. Phillips, 1969, Classical Electricity and Magnetism, 2nd edition, Addison-Wesley, Reading, Mass.
  3. ↑ 3.0 3.1 Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). New York: Wiley. ISBN 0-471-30932-X. 
  4. ↑ Pauli, W., 1958, Theory of Relativity, Pergamon, London
  5. ↑ Whittaker, E.T., 1960, History of the Theories of the Aether and Electricity, Harper Torchbooks, New York.
  6. ↑ Pais, A., 1983, »Subtle is the Lord...«; the Science and Life of Albert Einstein, Oxford University Press, Oxford
  7. ↑ Ribarič, M., and L. Šušteršič, 1990, Conservation Laws and Open Questions of Classical Electrodynamics, World Scientific, Singapore
  8. ↑ Ribarič, M., and L. Šušteršič, 2005, Search for an equation of motion of a classical pointlike charge, arXiv:physics/0511033

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