Maxwell’s equations of electromagnetism

In a strict sense, the classical Newtonian mechanics and the Maxwell s theory of electromagnetism are not compatible. The M-M-type experiments refuted the geometric optics completed by classical mechanics. In classical mechanics the inertial system was a basic concept, and the equation of motion must be invariant to the Galilean transformation Eq. (1). After the M-M experiments, Eq. (1) and so any equations of motion became invalid. Einstein realized that only the Maxwell equations are invariant for the Lorentz transformation. Therefore he believed that they are the authentic equations of motion, and so he created new concepts for the space, time, inertia, and so on. Within... 

A TREATISE ON ELECTRICITY AND MAGNETISM, James Clerk Maxwell. Important foundation work ol modern physics. Brings to final form Maxwell s theory of electromagnetism and rigorously derives his general equations of field theory. 1,084pp. 5b x 8b. 60636-8, 60637-6 Pa., Two-vol. set 19.00... 

In preparation for the thermodynamic analysis of radiation effects we study the pressure exerted by electromagnetic radiation, based on Maxwell s equations for electromagnetic fields. Readers not wishing to wade through the rather lengthy derivation may note the final result, Eq. (5.5.11), and proceed to the next section. 

The law of Stefan and Boltzmann is exactly valid for an absolutely black body in vacuo, as it is based on the two laws of thermodynamics and on Maxwell s equations of the electromagnetic field, which are exact under these conditions. The law of Stefan and Boltzmaun may therefore be used not only... 

One important macroscopic quantity related to the optical properties of non-metallic solids is their refractive index, which is closely related to their dielectric constant. Maxwell s equations for electromagnetic waves propagating in absorbing materials (see for instance [43]) lead to wave equations for the electric and magnetic fields in the material, and a solution for the amplitude of one component of these fields is ... 

Let us now study the nature of the electromagnetic wave as described by Maxwell s equations. The electromagnetic wave equations (without electric charge and current) in the vacuum are ... 

From Maxwell s theory of electromagnetic waves it follows that the relative permittivity of a material is equal to the square of its refractive index measured at the same frequency. Refractive index given by Table 1.2 is measured at the frequency of the D line of sodium. Thus it gives the proportion of (electronic) polarizability still effective at very high frequencies (optical frequencies) compared with polarizability at very low frequencies given by the dielectric constant. It can be seen from Table 1.2 that the dielectric constant is equal to the square of the refractive index for apolar molecules whereas for polar molecules the difference is mainly because of the permanent dipole. In the following discussion the Clausius-Mossoti equation will be used to define supplementary terms justifying the difference between the dielectric constant and the square of the refractive index (Eq. (29) The Debye model). 

Baeyer s speech has considerable interest. After the expected preliminaries, he began by asking the rhetorical question Is Kekule s benzene theory a true depiction of the molecule, or is it simply a heuristically useful fiction This question evoked a consideration of molecular models. Van t Hoff was not the first to suggest a tetrahedral shape for the carbon atom, Baeyer noted it was Kekule who had introduced tetrahedral carbon models in 1867. Of course, van t Hoff had taken the idea further than Kekule, in particular by affirming that the four valence bonds emanating from each carbon atom were relatively fixed and could therefore be studied chemically. In this sense Kekule s tetrahedral models were analogous to Heinrich Hertz s famous comment about James Clerk Maxwell s equations of the electromagnetic field that they have almost an independent life, that they can appear wiser even than their creator and can yield more than was ever invested in them. ... 

It follows from Maxwell s theory of electromagnetic radiation that e = n, where s is the dielectric constant measured at the frequency for which the refractive index is n. Equation (9.8) thus leads immediately to the Lorentz-Lorenz equation... 

Maxwell s theory of electromagnetic radiation fits within the classical doctrine because the electric and magnetic fields (and their rates of change with time) take on well-defined values at all times and the future values of the fields can be predicted with arbitr y precision fr MJjlSif iililial te usiiig the Maxwell equations. In Maxwell s theoryV] f upon the amplitude of the elec-... 

Sometimes mathematical expressions of principles apply almost universally. In physics, for example, the conservation laws indicate that in a closed system certain measurable quantities remain constant mass, momentum, energy, and mass-energy. Lastly, systems of equations are required to describe physical phenomena of various levels of complexity. Examples include English astronomer and mathematician Sir Isaac Newton s equations of motion, Scottish physicist and mathematician James Clerk Maxwell s equations for electromagnetic fields, and Swiss mathematician and physicist Leonhard Euler s and French engineer Claude-Louis Navier and British mathematician and physicist George Gabriel Stokes s (Navier-Stokes) equations in fluid mechanics. 

For the reasons discussed above we have to abandon the Galilean principle of relativity and accept that Newton s laws cannot be fundamental laws of Nature. We thus consider Maxwell s equations for electromagnetic fields to be valid in all inertial frames of reference and consequently obtain the relativity... 

Maxwell s theory of electromagnetism was actually the first theory that fulfilled the requirements of special relativity (i.e. the equations are invariant under a Lorentz transformation), even before special relativity was formulated by Einstein. 

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

The central equations of electromagnetic theory are elegantly written in the fonn of four coupled equations for the electric and magnetic fields. These are known as Maxwell s equations. In free space, these equations take the fonn ... 

In the previous sections we have described the interaction of the electromagnetic field with matter, that is, tlie way the material is affected by the presence of the field. But there is a second, reciprocal perspective the excitation of the material by the electromagnetic field generates a dipole (polarization) where none existed previously. Over a sample of finite size this dipole is macroscopic, and serves as a new source tenu in Maxwell s equations. For weak fields, the source tenu, P, is linear in the field strength. Thus,... 

All electromagnetic phenomena are governed by Maxwell s equations, and one of the consequences is that certain mathematical relationships can be determined when light encounters boundaries between media. Three important conclusions that result for ellipsometry are ... 

To illustrate the use of the vector operators described in the previous section, consider the equations of Maxwell. In a vacuum they provide the basic description of an electromagnetic field in terms of the vector quantifies the electric field and 9C the magnetic field The definition of the field in a dielectric medium requires the introduction of two additional quantities, the electric displacement SH and the magnetic induction. The macroscopic electromagnetic properties of the medium are then determined by Maxwell s equations, viz. 

We know from Maxwell s equations that whenever a charged particle undergoes acceleration, electromagnetic waves are generated. An electron in a circular orbit experiences an acceleration toward the center of the orbit and as a result emits radiation in an axis perpendicular to the motion. 

We shall assume light propagation along z-axis and electromagnetic field distribution independent of y coordinate. Solution of Maxwell s equations for such a structure can be assumed in the form ... 

The electromagnetic field in free space is described by the electric field vector E and the magnetic field vector H, which in the absence of charges satisfy Maxwell s equations... 

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