Maxwell s equations

These four fundamental equations of electromagnetic theory can be very elegantly expressed in terms of the vector operators divergence and curl. The first Maxwell s equation is a generalization of Coulomb s law for the electric field of a point charge  

The corresponding differential form is the first Maxwell s equation  

Free magnetic poles, the magnetic analog of electric charges, have never been observed. This implies that the divergence of the magnetic induction B equals zero, which serves as the second Maxwell s equation  

FIGURE 11.16 Faraday s law of electromagnetic induction, leading to third 

FIGURE 11.17 Magnetic field produced by electric current, described by Ampere s law V X B = /To J. 

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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... 

It is easy to calculate tire value of e as an application of the MAXWELL S equations in the case of a symmetric tube without defect. 

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,... 

The necessary boundary conditions required for E and //to satisfy Maxwell s equations give rise to tire well known wave equation for tire electromagnetic field ... 

Above we described tire nature of Maxwell s equations in free space in a medium, two more vector fields need to be... 

Maxwell s equations can be combined (61) to describe the propagation of light ia free space, yielding the following scalar wave equation ... 

These early results of Coulomb and his contemporaries led to the full development of classical electrostatics and electrodynamics in the nineteenth cenmry, culminating with Maxwell s equations. We do not consider electrodynamics at all in this chapter, and our discussion of electrostatics is necessarily brief. However, we need to introduce Gauss law and Poisson s equation, which are consequences of Coulomb s law. 

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 ... 

Spin 1, Mass Zero Particles. Photons.—For a mass zero, spin 1 particle, the set of relativistic wave equations describing the particle is Maxwell s equations. We adopt the vector 9(x) and the pseudovector (x) which are positive energy (frequency) solutions of... 

In the Lorentz gauge, Maxwell s equations when expressed in terms of the potentials assume the following form... 

If we apply Maxwell s equations to this boundary value problem we can derive a complete solution to the amplitude and phase of this field af every point in space. In general, however we can simplify the problem to describing the field at the entrance (or exit) aperture of a system and at the image plane (which is what we are really interested in the end). 

However, in Maxwell s days everyone assumed that there had to be a mechanical underpinning for the theory of EM. Many researchers worked on very detailed hidden variable theories for the EM field, in an attempt to prove that the laws of EM were in fact a theorem in NM, just like Kepler s laws are a theorem in NM. No one noticed that it was impossible to do this, since Maxwell s equations are not Galilei invariant and Newton s laws are. That includes Lorentz who discovered around 1900 that the Maxwell equations are invariant under another transformation that now bears his name. 

The Ether is not useful to teach MT. The EM field is most effectively viewed as an irreducible entity completely defined by Maxwell s equations. (If one wants to make the interaction with point charges in N.M or QM explicit, one can add the Lorentz force or the minimal coupling.) All physical properties of th EM field and its interaction with matter follow from Maxwell s equations and the matter equations. 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

Nearly two years ago, studying electrodynamics in curved space-time I found1 that Maxwell s equations impose on space-time a restriction which can be formulated by saying that a certain vector q determined by the curvature field must be the gradient of a scalar function, or... 

The following development is devoted to obtaining an expression for B in terms of more familiar quantities. To this end, consider the Fourier transform of Maxwell s equations (in the time independent case)... 

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