Abstract
The (classical) electromagnetic fields in vacuo that satisfy given boundary conditions can be calculated through Maxwell’s equations. In the Gauss system they read as:
where \(\overrightarrow{j}\) and \(\rho \) are current density and charge density. In all, they are 4 functions of space and time. The fields can be computed and measured, however, it amazing that the Maxwell equations succeed in giving us 6 measurable quantities (3 components of \(\overrightarrow{E}\) and 3 of \(\overrightarrow{B}\)) having only 4 quantities in input. This is a most remarkable property of the electromagnetic field. Moreover, we can obtain the same field more easily by working out 4 quantities, namely the scalar potential \(\phi \) and the vector potential \(\overrightarrow{A},\) such that
Maxwell published his Treatise in 1873 and his equations have not been revised, continuing to be unharmed through the revolutionary changes produced by Relativity and Quantum Mechanics. Nevertheless, the quantum theory of light and the use of lasers have revealed many new phenomena. This might appear to be a contradiction, but it is the plain truth. I prepare the reader to appreciate how this came about by discussing some important facts about classical electromagnetism in this chapter.
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- 1.
An equivalent alternative which is in use is: \(\nabla ^{2} \phi (r) ={1 \over r^{2}}\frac{\partial }{\partial r}(r^{2}\frac{\partial }{\partial r} \phi (r))\).
- 2.
The Green’s function are named after the Englishman George Green, a solitary amateur genius who working in his mill in the Midlands invented mathematical methods essential for the theory of electromagnetism and all of the modern Theoretical Physics, although at his time the Dirac’s delta was not yet known.
- 3.
The quantum mechanical photons are emitted and adsorbed like particles but travel like waves, as we shall see below. However the description in terms of point particles works fine at short wavelengths.
- 4.
Pierre de Fermat stated it in 1662.
- 5.
This law is actually credited to the Arab mathematician Ibn Sahl in a manuscript of 984.
- 6.
Correlation between photons in two coherent beams of light, R. Hanbury Brown and R. Twiss, Nature 4497, 27 (1956).
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Cini, M. (2018). Some Consequences of Maxwell’s Equations. In: Elements of Classical and Quantum Physics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-71330-4_4
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DOI: https://doi.org/10.1007/978-3-319-71330-4_4
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