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The uniqueness of the electromagnetic energy momentum tensor is established under general conditions.

A lumped parameter formulation for the electromechanics of a class of electret transducers is developed. An energy method is used to calculate the force of electrical origin and the technique is employed in the solution of two important electret transducer geometries. The results are checked by a surface integration of the Maxwell stress tensor.

Macroscopic field equations, boundary conditions and equations of state are derived for the non-linear, macroscopic elastic and dielectric response of an insulator. A centrosymmetric polynomial representation of order four is introduced for the energy density; the equations of state for the electric field and stress tensor are then deduced as polynomials of degree three in the displacement gradients and electric displacement field. The results are applied to the special case of m3m material symmetry. A finite, point-charge model of a centrosymmetric ionic crystal is introduced and used to determine 0°K microscopic expressions for the electric field and stress tensor equation of state coefficients introduced in the macroscopic analysis. The results are used to calculate the full set of second and third-order non-linear coefficients for NaI, based on a Born-Mayer potential and the 4·2°K elastic stiffness data of Claytor and Marshall.

The collision problem for a model of the neutron-deuteron system with a tensor force has been reduced to a set of coupled one-dimensional integral equations. The model uses the central plus tensor separable potential of Yamaguchi for the triplet two-body interaction, and a rank-one, S-wave, separable potential for the singlet interaction For J 1 2 the three-body problem is reduced to a set of four coupled equations for each parity, and for J = 1 2 to three coupled equations for each parity In the J = 1 2 even parity case the equations are equivalent to those obtained by previous workers

Summary THE problem of finding from Hamilton's Principle the most general field laws for an antisymmetric tensor of the second rank in five dimensions is solved. The tensor has 10 (6 + 4) components four of which are complex, and two scalar functions are introduced as a result of the variational problem ; in all there are sixteen functions. The sixteen equations obtained are those of radiation. For the case of free n)otiml and, to the first approximation, for an external elecfromagnetic field these sixteen equations can be combined into eight (C. G. Darwin’s equations)†. For the case of an electromagnetic field these equations are presented in a general tensor form, and the well-known operators appear quite naturally as terms in contravariant differentiation. It appears that the five-dlmensional continuum represents a natural system of reference for radiation phenomena.

In this paper it has been shown that all the quantities excepting a parameter σ and charge can be determined from the field equations of general relativity, when the electromagnetic tensor field is of the third class. The parameter σ and the charge can be determined from Maxwell equations. The electromagnetic tensor, stress tensor, pressure, mass and charge have been obtained as concomitants of gravitational tensorh λμ.

The classical equations of motion for electric and magnetic dipole atoms (or molecules) in an external electromagnetic field of force, treated in papers I and II on the basis of Moller's theory of the relativistic dynamics of systems with an internal angular momentum, are extended to the case that the reaction of radiation on the atoms is taken into account. To this end Moller's theory, which is valid only for finite systems (total energy-momentum tensor zero outside a finite region in space for arbitrary fixed time), is modified in order to be applicable to the case of radiating atoms (or molecules). Dirac's method of splitting retarded fields covariantly into self-parts (half sum of retarded and advanced fields) and “radiative” parts (half difference of retarded and advanced fields) is applied to the sub-atomic fields. It is proved that the sub-atomic self-force density can be written as minus the divergence of a symmetric four-tensor, which is zero outside the atomic system and which, added to the sub-atomic material energy-momentum tensor, may be interpreted as the total energy-momentum tensor of the finite atomic system. With the help of the latter tensor the atomic mass, intrinsic angular momentum and centre of gravity are defined, using Moller's theory. The influence of the “radiative” part of the field on the centre of gravity motion of the atoms and the change of their intrinsic angular momentum is then analysed. The equation of motion and the intrinsic angular momentum balance equation, obtained for radiating charged dipole atoms are used in order to derive the relativistic atomic energy-momentum tensor for a system consisting of a large number of these atoms. In contrast with the tensors in the previous papers, this tensor is no longer symmetrical. The treatment of the present paper could be extended to include the case in which the atoms also possess electric quadrupole moments.

From the atomic conservation laws of energy-momentum the corresponding macroscopic laws are derived with the help of a covariant averaging procedure. The total energy-momentum tensor is found as a statistical expression in terms of atomic quantities. It may be split into a field part T αβ (⨍) (α, β = 0, 1, 2, 3) containing the macroscopic fields and polarizations, which in the rest frame reads: T αβ (⨍) = 1 2 E 2 + 1 2 B 2 (E ∧ H) i (E∧H) i −E i D j −H i B j +( 1 2 E 2 + 1 2 B 2 −M⋅B)g ij (i,j =1,2,3) and a material part Tαβ(m) which forms the relativistic generalization of the usual energy and momentum expressions.

The mathematical equivalents of parallel plate openings are used to simulate real fractures dispersed in orientation, distributed in aperture, and of arbitrary spacing. With this idealization of jointed pervious rock, the discharge of each conductor is a second rank tensor proportional to the cube of aperture and to the projection of a hydraulic gradient generally parallel to no conductor. One may add components of discharge through each intersecting conductor and components of discharge through intervening pervious blocks. The reciprocal of specific surface of the fracture system like spacing serves as a weighting factor for the tensor sum, which is the anisotropic Darcy's law permeability of an equivalent continuous medium. Special cases of one, two, and three joint sets are modeled by applying Monte Carlo methodsto pair orientations of individual planes sampled from Fisher dispersions with apertures sampledfrom skewed normal distributions. Statistics of the orientation of principal axes and ofprincipal permeabilities are developed to show the relationship between joint geometry andanisotropy and to assess its variations.

We have developed a set of differential equations which describe the hydrodynamics of a conducting liquid crystal subjected to external electromagnetic fields. The mechanical forces resulting from the interaction of the fields with the liquid are expressed as operations on the Maxwell stress tensor. The use of this description allows a concise statement of the equations of motion. As an example of the validity of the formalism, we rigorously solve the boundary-value problem associated with the Williams domain (vortex) mode in nematic liquids. Using standard constitutive relations, physical boundary conditions, and experimentally measured material $p$-azoxylanisole constants, we quantitatively reproduce the significant experimental observations.

An elementary method of calculating that part of the tensor dielectric coefficient which determines the propagation of transverse electromagnetic radiation through a free‐electron gas in a uniform external magnetic field is presented. The method presented here is based on a particle‐orbit analysis and is somewhat analogous to a generalized version of the Kramers‐Heisenberg quantum theory of gaseous dispersion. It is shown that the elements of the transverse dielectric tensor can be obtained from a knowledge of the quantum‐mechanical transition probabilities for emission and absorption of photons (that is, from a knowledge of the Einstein A and B coefficients). The formal expression for the dielectric tensor thus obtained is shown to be valid for both the degenerate and the nondegenerate system of electrons. The dielectric tensor thus obtained is shown to reduce in the classical limit to the familiar results of the conventional classical hot plasma kinetic theory. The first quantum correction to the classi...

In an introductory paper, the vortex sponge was shown to be governed, in restricted cases, by Maxwell's free-space equations. In the present paper, analogs to electric and magnetic energies and Poynting's theorem are derived by simple mechanical considerations. Rotational stability, suggested originally by MacCullagh as a fundamental property of a luminiferous ether, turns out to be a quality of the medium, as do the stresses introduced by Faraday and Maxwell to explain the mechanical actions of electric and magnetic fields. A rudimentary model for the electrostatic field is suggested on this basis. A conventional definition of charge and the laws of Coulomb and Biot complete Maxwell's equations for cases including charges and currents. A model of the magnetic field based on the bulk rotation and the Faraday-Maxwell stresses, combined with the laws of Coulomb and Biot, permits the inference of the Lorentz force. Although numerous gaps occur in the treatment, it seems not unlikely that the vortex sponge has the qualities described by the electromagnetic field equations as well as the mechanical attributes required for a model of these fields.

Classical electric and magnetic fields have properties very much like those of an elastic medium: 1) they exert forces,viz. the divergence of the Maxwell stress tensor ∂T αβ/∂χ β yields the force per unit volume; 2) the fields have mass density given by Einstein’s mass-energy relation; 3) they have transverse compressibility as shown by the attainment of high magnetic-field intensities by lateral implosion. The first two properties enable us immediately to write a Newtonian equation of motion. The third property enables us to relate volume changes to stress changes. The linearized result, in simple cases in which field displacements are assumed to be very small, is a simple wave equation in which the velocity of propagation is √2c, notc. But this applies only to plane purely compressive stress waves, which would exist only under idealized conditions. The concept oftransverse tensile stress waves, analagous to the transverse vibrations of a stretched string or membrane, is here revived. Such waves would be generated by local compressive waves and thus play a role in the propagation of compressive waves. Evidence against the universal applicability of special relativity is discussed. The arguments supporting the existence of stress waves seem very cogent, but experimental verification seems not to be simple. Some possible applications of stress waves are indicated. The origin of the concept of stress waves is briefly reviewed in an Appendix. If the existence of stress waves here deduced in confirmed experimentally, it will constitute the first discovery of its kind since Maxwell.

It is shown that Maxwell's equations can be consistently symmetrized by the introduction of an additional vector 4-current as the source of the dual of the generalized electromagnetic tensor. The additional 4-current is related to a second type of electric charge which we shall call "m-electric charge," as distinguished from the conventional electric charge (denoted as "e-electric" charge). A Lagrangian formulation of this theory for classical point charges is constructed, yielding the symmetrized Maxwell equations, in which each particle is assumed to carry both an "e-electric" charge and an "m-electric" charge. We show that if the m-electric to e-electric charge ratio is the same for all particles in the model universe, then the predictions of the symmetrized Maxwell equations are the same as that of the unsymmetrized, conventional Maxwell equations. However, if all particles in a detector carry the same m-electric to e-electric charge ratio, not equal to zero, then a detected particle with different m-electric to e-electric charge ratio (than that of the detector) could appear to have only a fractional e-electric charge. This implies that fractionally charged particles could be generated even if only integral multiples of e-charge and m-charge were allowed in the symmetrized theory. This means that it might be experimentally difficult to distinguish between a differently "m-charged" particle, and an SU3-type "quark," in purely electromagnetic interactions alone.

A stationary plasma in the presence of a magnetic field may be represented by an equivalent dielectric tensor. The equivalent dielectric tensor of a moving plasma, valid within the quasi-static approximation, is found by a Lorentz transformation of the dielectric tensor from a reference frame in which the plasma is at rest to one in which the laboratory is at rest. The range of validity of the dielectric tensor is investigated, and the relation between this tensor and the equivalent dielectric tensor found from Maxwell's curl equation is determined. An infinite number of moving plasma streams with a Maxwellian velocity distribution are summed to obtain the dielectric tensor of a finite temperature plasma. The dielectric tensor leads directly to the dispersion relation for waves on a bounded plasma.

The equations of hydrodynamics are derived from the principles of classical statistical mechanics for a single component system of spherical molecules. Exact expressions for the stress tensor and heat flux are obtained without resorting to any methods of approximation. The derivations are carried out for a system of molecules which interact through a continuous pair potential, and for a system consisting solely of rigid spheres. The ``long‐wavelength'' expansion employed by Irving and Kirkwood [J. Chem Phys. 18, 817 (1950)] in their expressions for the stress tensor and heat flux is examined. It is demonstrated that this expansion converges only if the intermolecular potential goes to zero faster than any positive power of (1/r) in the limit of large r (r is the internuclear distance) and hence diverges for any realistic intermolecular potential. An elementary example is considered to demonstrate the effect of finite wavelength on the stress tensor.

Recently Moller has shown in a simple case that the energy and momentum of supplied heat form a four-vector. This result is extended to an arbitrary elastic system. The separation between work and heat is obtained from a division of the energy-momentum tensor into a mechanical tensor and a heat transfer tensor. No restrictions are imposed on the latter tensor, except that it should vanish in the absence of heat transfer.

The propagation of an electromagnetic wave in an anisotropic conductor in the presence of a static magnetic field is discussed. The medium is described by a general tensor relation between the current density and the electric field. In the isotropic, uncompensated limit the wave is known as the helicon. The infiniteslab boundary‐value problem is solved and the resistivity tensor elements are related to experimentally measurable quantities.

We derive expressions for the complex conductivity tensor of a homogeneous classical plasma in an external uniform magnetic field, in terms of electric field correlations, using the Kubo theory of transport phenomena. The main aim is to bring out explicitly the magnetic field dependence of the conductivity tensor. Exact relations between the conductivity tensor in the presence of the magnetic field and the same tensor in the absence of the magnetic field have been obtained.

Attention is drawn to a remarkable property of the geometrical representation of symmetrical second-rank tensors by means of Mohrdiagrams. Starting from the equation qi =Liipi , where q and p are three-dimensional vectors and Lii is a symmetrical second-rank tensor, it is shown that from a given vector p the vector q and its angular relationship to p may be obtained by a simple construction in the Mohr diagram.

With the aim of generalizing the relation between the fundamental tensor and the affine connexion of the real and the hermitian case of the unified field theory, a relation with eight arbitrary constants is contemplated; conditions for its general invariance and transposition invariance are imposed, and a relation with only two arbitrary coefficients is obtained; which, however, except in three particular cases, is substantially nothing else but the real or hermitian relation. One of the particular cases is noteworthy, as it imposes the first quadruplet of Maxwell's equations on the skew part of the fundamental tensor [relation (19) in the text].

The kinematic energy momentum tensor of the spinning point particle is determined. The equations of motion for translation and spin are derived from the fact that the tensor is divergencefree. The method is applied to the investigation of the equations of motion for an electrically charged particle with magnetic momentum.

The quantization of a massive spin one particle field, satisfying the first order system of equations due to Proca, is summarized. The interaction of such a field with an arbitrary field is then considered. Equivalent theories in terms of vector and antisymmetric second rank tensor spin one fields are then constructed in a simple manner, providing a generalization of an equivalence theorem due to the author. In addition, this approach, involving the use of first order systems of equations, allows a simple extension to the case of equivalence theorems for arbitrary integral spin. The nature of this extension is indicated.

A general vector potential derived for the Maxwell equations in their general form is used to find and analyze the tensor for the angular momentum density. With j = 0, the general Maxwell equations are invariant with respect to a four-parameter group of external displacement, whose addition to the group of external transformations leads to field equations which are invariant with respect to the ten-parameter group of external transformations. For integral quantities, the vector of the total field energy-momentum and the tensor of the total angular momentum, transformations from the inhomogeneous group of external transformations are equivalent to transformations from the inhomogeneous Lorentz group.

Moller's relativistic equations of motion for systems with an internal angular momentum in an arbitrary (non-gravitational) external field of force are applied to the special model of electric and magnetic dipole point atoms in an external electromagnetic field of force. The resulting equations of motion are used in order to derive the relativistic atomic energy-momentum tensor for a system, consisting of these dipole atoms. The tensor, found in this way, has exactly the same form as the energy-momentum tensor, obtained recently by de Groot and Suttorp for dipole atoms, although these authors use a different definition for the centre of gravity of the atoms as the one used here, and also make several approximations in their calculations. The only difference is, that some of the quantities appearing in their final expression for the atomic energy-momentum tensor, are approximations of the corresponding quantities, found in the present paper.

The classical equations of motion for electric and magnetic dipole atoms (or molecules) in an external electromagnetic field of force, derived in a previous paper on the basis of Moller's theory of the relativistic dynamics of systems with an internal angular momentum, are simplified by showing that certain terms, which contain an unphysical trembling motion (“Zitterbewegung”), are completely negligible with respect to the other terms in these equations. The resulting equations are used in order to derive the relativistic atomic energy-momentum tensor for a system, consisting of these dipole atoms. The field part of this tensor has exactly the same form as obtained before, but the material part is slightly different as a consequence of the simplification in the equations of motion. The same symmetrization procedure, as used in the preceding paper, can be applied to the total energy-momentum tensor. Radiative effects are neglected throughout the theory.

The high-frequency expansion of a vacuum gravitational field in powers of its small wavelength is continued. We go beyond the previously discussed linearization of the field equations to consider the lowest-order nonlinearities. These are shown to provide a natural, gauge-invariant, averaged stress tensor for the effective energy localized in the high-frequency gravitational waves. Under the assumption of the WKB form for the field, this stress tensor is found to have the same algebraic structure as that for an electromagnetic null field. A Poynting vector is used to investigate the flow of energy and momentum by gravitational waves, and it is seen that high-frequency waves propagate along null hypersurfaces and are not backscattered by the lowest-order nonlinearities. Expressions for the total energy and momentum carried by the field to flat null infinity are given in terms of coordinate-independent hypersurface integrals valid within regions of high field strength. The formalism is applied to the case of spherical gravitational waves where a news function is obtained and where the source is found to lose exactly the energy and momentum contained in the radiation field. Second-order terms in the metric are found to be finite and free of divergences of the $\frac{(\mathrm{ln}r)}{r}$ variety.

The low-energy levels of the od-odd nucleus Y/sup 90/ are calculated with finite-range central and tensor forces to first order by means of the j-j coupled odd-group model. The two-body matrix elements for the central and tensor forces are expressed in the j-j representution, from which a generalization to off-diagonal matrix elements is obtained in the limit of the zero range. A phenomenological Gaussian potential without a hard core, estimated from the free two-nucleon potentials of Jackson-Blatt and BruecknerGammel- Thaler, is used for the residual interaction. The effects of the tensor force are analyzed in detail as a function of the force range. The numerical results of the calculation are in reasonably good agreement with available experimental spectra. (auth)

The possibility of asymmetry in a variety of second‐rank tensor quantities of importance in magnetic resonance phenomena is discussed. A classical model demonstrating this effect in the case of shielding is described. The present methods for the measurement of anisotropy of tensor quantities in crystals are discussed, and the contributions of the antisymmetric part of such a tensor are shown to occur in terms quadratic in the shielding (or g) tensor components.

The average stress tensor, power-flow density, momentum density, and energy density for electromagnetic waves in a linear stationary anisotropic medium with space and time dispersion are clarified with the help of the second-order nonlinear terms in the equation of motion for polarization. The wave fields are connected with the medium-mass motion through the nonlinear terms; the net force between the fields and the medium mass plays a crucial role in interpreting the stress tensor and the wave momentum. Asymmetry in the space-space part of the energy-momentum tensor, which is quadratic in the field fluctuations, is ascribed to anisotropic restoring forces of the polarizations; asymmetry in the space-time part is attributed to nonvanishing average forces between the fields and the medium-mass motion.

It is frequently stated that the electric field of the macroscopic Maxwell equations is the mean of that of the microscopic Maxwell equations. By “mean” is meant either a volume average or a statistical average, the result being the same. In this paper, the electrostatic field is considered, and it is shown that the mean microscopic field is not appropriate for use in the concept of dielectric constant. A suitable definition of macroscopic field is discussed, and it is shown that it differs from the mean microscopic field in a nonuniform medium.

A general method is set up for modifying the energy-momentum tensor so as to remove the singularities in the flow of energy and momentum into the world-line of a particle without affecting the equations of motion of the particle. It is shown how the singularities of different order may be removed one by one. In the case of the electromagnetic and meson fields it is shown that the modified tensor leads to a finite integral of energy and momentum over any space-like surface. In other cases the corresponding result may be secured by making a further modification in the tensor.

It is shown that the postulate of indistinguishability of the Maxwell field tensor from its dual leads to the concept of the electromagnetic field tensor as a spinor component in dual space. The demand for algebraic consistency dictates a unique connection with the gravitational field. The Maxwell field must be viewed as a set of potentials, and the necessity for a duality gauge condition excludes the existence of magnetic monopoles.

A three-body formalism with separable potentials is developed, with the inclusion of tensor forces in addition to the usual s-wave forces in triplet and singlet N-N states. For completeness, the formalism is generalized to include the effect of a hard core term in the singlet s-state, which is shown to require an extra spectator function, over and above three others already associated with the tensor and s-wave forces. Numerical results are presented for the doublet n-d scattering length, which, unlike its quartet counterpart, is rather sensitive to variations in the two-body potential parameters. While it has not yet been possible to obtain a quantitative estimate of the hard-core effect (because of the enormous computational problems associated with four spectators) the effect of the tensor force has been investigated in some detail with several sets of input parameters as are currently available. It is found that while the inclusion of tensor force greatly improves the magnitude of the scattering length, its sign is still negative, contrary to the experimental situation. The assumption of a smaller magnitude for the (negative) n-n scattering length, than its charge-independent value, (for which there is some recent evidence), yields a further improvement in the value of a 1 2 , but not enough to flip its sign. This discrepancy is briefly discussed in the context of similar results obtained by other authors.

Four types of stress functions are known for solving elasticity problems: the components of the displacement vector, the components of the Galerkin vector, the Maxwell stress functions, and the Morera stress functions. For problems with stress type boundary conditions, the Maxwell stress functions are, in many respects, the simplest to use, but they lack the simple transformation properties of vectors. It was shown by C. Weber (1) 2 that the Maxwell and Morera functions supplement each other, and that together they are the components of a second order symmetric Cartesian tensor. In this paper, the compatibility equations for an isotropic Hookean body that is subjected to boundary stresses and temperature gradients are developed in terms of the Maxwell stress functions, and their general solution is presented for steady temperature fields.It is shown that, when the complementary energy of a homogeneous body with arbitrary elastic properties is expressed in terms of the components of the Maxwell-Morera tensor, the Euler equations for the integral of the complementary energy density are the complete set of compatibility equations in terms of the stress components. The Maxwell-Morera tensor is generalized, so that it represents the general solution of the equilibrium equations in any curvilinear coordinates. As an application, the general solution of the equilibrium equations in cylindrical coordinates is derived.

From the electromagnetic equations for fields, produced by point particles (electrons and nuclei) the field equations, taking into account the existence of stable atoms (or molecules, or ions) are derived in covariant form. These 'atomic field equations' involve, besides the field tensor (e,b) an atomic polarization tensor (p,m), or alternatively the field tensor (d, h) = (e + p, b - m). The atomic polarization tensor is given (up to second order in the internal atomic variables) as a function of atomic positions and velocities; proper electric dipole, quadrupole and magnetic dipole moments; and time derivatives of these quantities. (The 'proper atomic multipole moments' are defined in Lorentz frames in which the atoms are momentarily at rest). (Author)

We present a classical theory of the interaction of a spinor field with the electromagnetic field in which the potentials enter only implicitly through the Maxwell tensor. This theory differs from previous ones in that it is both manifestly covariant under Lorentz transformations and local in time. We then carry out the quantization without a need to introduce special terms or an indefinite metric, and proceed to formulate the perturbation expansion of theS-matrix.

The triton wave function in presence of tensor forces is investigated, when full symmetry of the three-nucleon system in configuration, spin and isobaric spaces is taken into account assuming separable potentials to operate between pairs. The wave function for the three-particle system involves only single parameter functions, two in the case of central forces and three in the case of tensor forces, resulting in a corresponding number of coupled one-dimensional integral equations.

A solution is found for Maxwell's equations which are associated with a radial flow of electric charge moving with the speed of light.

The traceTof the metrical energy-momentum tensorTklof fields associated with particles of zero rest mass may be zero, either identically or as a consequence of the field equations. This property ofTklis correlated here with the behaviour of the Lagrangian of the field under arbitrary conformal transformations. Certain classes of special fields are considered explicitly. It is shown in particular thatTvanishes for all non-zero spin fields which correspond respectively to the two-component neutrino field or the photon field.

The second‐rank tensor field φμν is decomposed into its various subspaces under the Lorentz group and the appropriate projection operators are exhibited explicitly. The most general local, Hermitian, free‐field Lagrangian which can be formed from this field is written down, and the corresponding equations of motion and subsidiary conditions are derived by means of a variational principle. Finally some possible applications of this theory are discussed (in particular spin‐2 boson theory), and all the possible couplings of this field to a Dirac particle are listed in full.

A study is made of the effect of the tensor part of the neutronproton interaction in hypertriton. It is found that the corrections obtained for the volume integral of the A-nucleon interaction result in appreciable variation of the parameters of the λ-nucleon potential as derived from studies of light hypernuclei.

The processing of magnetotelluric data involves concepts from electromagnetic theory, time series analysis and linear systems theory for reducing natural electric and magnetic field variations recorded at the earth's surface to forms suitable for studying the electrical properties of the earth's interior. The electromagnetic field relations lead to either a scalar transfer impedance which couples an electric component to an orthogonal magnetic component at the surface of a plane-layered earth, or a tensor transfer impedance which couples each electric component to both magnetic components in the vicinity of a lateral inhomogeneity. A number of time series spectral analysis methods can be used for estimating the complex spectral coefficients of the various field quantities. These in turn are used for estimating the nature of the transfer function or tensor impedance. For two dimensional situations, the tensor impedance can be rotated to determine the principal directions of the electrical structure. In general for real data, estimates of the apparent resistivity are more stable when calculated from the tensor elements rather than from simple orthogonal field ratios (Cagniard estimates), even when the fields are measured in the principal coordinates.

A surface of the second degree is constructed from the five components of the second rank tensor which describes the alinement of a spin-assembly undergoing magnetic resonance. The functions which characterize alinement are given a simple geometric interpretation in terms of radii vectores of this ellipsoid. The time-dependence of the different resonance functions at frequencies 0, ω and 2 ω is easily understood in terms of the rotation of the ellipsoid. In the absence of an r.f. field, and with pumping and relaxation processes only, the ellipsoid is uniaxial with its axes in the direction of the static field ( Z axis). With a weak r.f. field the shape of the ellipsoid is unchanged, but it is tilted and precesses round the Z axis at the frequency of the driving field. With stronger r.f. fields the shape of the ellipsoid changes, but at resonance one of the principal axes is always in the direction of the r.f. field and the length of this axis is independent of the field strength. At resonance also, the tilt increases to a limiting value of 1/4π with increasing r.f. field strength and the lengths of the axes in the plane perpendicular to the r.f. field tend to equality.

The Maxwell equations are derived in covariant manner from the microscopic equations for the electromagnetic field in the presence of point charges. The polarization tensor is given as an expansion to all orders in the atomic electromagnetic moments, defined in atomic rest frames.

Herein is presented a student-teacher oriented examination of how one might arrive at Maxwell's equations via a number of reasonable assumptions.

Conservation equations are derived for a weakly turbulent plasma in the quasi-linear approximation. Our results are a generalization of Tidman's to include magnetic fields. The usual conservation equations are modified by extra terms, due to fluctuating fields, in the energy density of the electromagnetic field, the Maxwell stress tensor, and the Poynting vector. Our steady-state conservation equations disagree with those of Kennel and Sagdeev for a shock wave in high-β plasmas. They keep the time-dependent terms arising from the fluctuating fields that vanish in the steady state assumed for a shock wave.