Maxwell’s wave equation

Basic Theories of Physics, Peter G. Bergmann A thorough coverage of the scientific method and conceptual framework of important topics in classical and modern physics, with concentration on physical ideas. Volume One is concerned with classical mechanics and electrodynamics, including Maxwell s wave equations volume Two is concerned with heat and quantum theory. Total of xxiii + 580pp. 5% x 8i/3. 

The interaction of laser radiation with the medium occurs through the third-order, non-linear, electric susceptibility denoted by and gives rise to an induced polarization field, which acts as a source term in Maxwell s wave equation. On solving the wave equation, one arrives at the following expression for the intensity of the CARS signal ... 

In summary for a planar texture and normal incidence we have outlined the derivation of the solution of Maxwell s wave equation and this has led to explicit expressions forp(=n Aq), AA(=pAn), and optical rotatory dispersion. We are now in a position to study special cases of normal and oblique incidence. 

Maxwell s wave equation can thus be considered an expression of traveling energy, which means that the characteristics of energy transmission can be analyzed as those of traveling... 

The next most familiar part of the picture is the upper right-hand corner. This i s the domain of classical applied mathematics and mathematical physics where the linear partial differential equations live. Here we find Maxwell s equations of electricity and magnetism, the heat equation, Schrodinger s wave equation in quantum mechanics, and so on. These partial differential equations involve an infinite continuum of variables because each point in space contributes additional degrees of freedom. Even though these systems are large, they are tractable, thanks to such linear techniques as Fourier analysis and transform methods. 

In fact, the better-known Faraday s law of induction (without which no transformer in the world would exist) is actually the hrst of the set of four Maxwell s unifying equations. So we learn that the E- and H-helds appear simultaneously, the moment the original magnetic or electric source has a hme variance. At some distance away, these helds combine to form an electromagnetic wave — that propagates out into space (at the speed of light). 

In the derivation of the molecular properties, which give rise to this effect, we have to take the spatial variation of the electric field vector into account and can thus not make the dipole approximation, contrary to the last section. This implies that we have to include a contribution from the interaction with the curl of the time-dependent electric-field, V xS r,t), to the expansion of the induced dipole moment of a molecule in Eq. (7.18). However, Maxwell s third equation, Eq. (2.37) relates the curl of the electric-field vector to the time derivative dB r,t)/dt of the magnetic induction and we can thus alternatively replace the spatial variation and expand the induced dipole moment instead in the electric field and the time derivative of the magnetic induction of a monochromatic wave (Buckingham, 1967) as... 

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 ... 

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

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... 

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. 

Invariance of the fields with respect to changes in potential is known as gauge invariance. It is used to simplify Maxwell s equations in regions where there is no free charge. In this case ip itself is a solution of the wave equation, so that it can be adjusted to cancel and eliminate the scalar potential. This means that in (13) V A= 0 and, as before... 

Incorrect conclusion 1 above is sometimes said to derive from the reciprocity principle, which states that light waves in any optical system all could be reversed in direction without altering any paths or intensities and remain consistent with physical reality (because Maxwell s equations are invariant under time reversal). Applying this principle here, one notes that an evanescent wave set up by a supercritical ray undergoing total internal reflection can excite a dipole with a power that decays exponentially with z. Then (by the reciprocity principle) an excited dipole should lead to a supercritical emitted beam intensity that also decays exponentially with z. Although this prediction would be true if the fluorophore were a fixed-amplitude dipole in both cases, it cannot be modeled as such in the latter case. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

We are trying to discover what waves are generated inside and outside a crystal when it is illuminated by X-rays. The propagation of any electromagnetic waves in any medium is accurately described by Maxwell s equations. These are, in vector notation. 

The electromagnetic fields (x,r) and H(x, t) associated with scattering from a microsphere satisfy Maxwell s equations. For a homogeneous, isotropic linear material the time-harmonic electrical held E and the magnetic held H satisfy vector wave equations, which in SI units are (Bohren and Huffman, 1983)... 

At the end of the nineteenth century classical physics assumed it had achieved a grand synthesis. The universe was thought of as comprising either matter or radiation as illustrated schematically in Fig. 2.1. The former consisted of point particles which were characterized by their energy E and momentum p and which behaved subject to Newton s laws of motion. The latter consisted of electromagnetic waves which were characterized by their angular frequency and wave vector and which satisfied Maxwell s recently discovered equations, ( = 2nv and — 2njX where v and X are the vibrational frequency... 

Bartlett and Corle [46] proposed modification of Maxwell s equations in the vacuum by assigning a small nonzero electric condictivity to the formalism. As pointed out by Harmuth [47], there was never a satisfactory concept of propagation velocity of signals within the framework of Maxwell s theory. Thus, the equations of the latter fail for waves with nonnegligible relative frequency bandwidth when propagating in a dissipative medium. To resolve this problem, a nonzero electric conductivity ct and a corresponding current density... 

These new potentials are solutions of wave equations including inside the sources. To obtain the general solution, one must add a particular solution of the inhomogenous potential equations. Usually, the electromagnetic helds Eo, Eo and the potentials , C are discarded for the following reasons. Either (1), they represent transient solutions of Maxwell s equations that decay rapidly to zero or... 

The electromagnetic field defined above is a solution of Maxwell s equations, provided the new potentials . C are solutions of wave equations and satisfy the Lorenz gauge. 

Therefore, questioning the physical significance of potential is not relevant here. The new formulation of Maxwell s equations [20-23], where potentials are directly coupled to fields clearly indicates that potentials, play a key role in particle behavior. To make a long story short, the difference in nature between potentials and fields stems from the fact that potentials relate to a state of equilibrium of stationary waves in the medium usually nonaccessible to an observer (except when potentials are used in a measurement process of the interferometric kind, at a given instant in time). Conversely, fields illustrate a nonequilibrium state of the medium as an observable progressive electromagnetic wave, since this wave induces the motion of material particles. 

The first attempt to formulate a theory of optical rotation in terms of the general equations of wave motion was made by MacCullagh17). His theory was extensively developed on the basis of Maxwell s electromagnetic theory. Kuhn 18) showed that the molecular parameters of optical rotation were elucidated in terms of molecular polarizability (J connecting the electric moment p of the molecule, the time-derivative of the magnetic radiation field //, and the magnetic moment m with the time-derivative of the electric radiation field E as follows ... 

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