Maxwell’s derivation

Exercise. Maxwell s derivation of the velocity distribution in a gas was based on the assumptions that it could only depend on the speed v, and that the cartesian components are statistically independent. Show that this leads to Maxwell s law. 

Maxwell s derivation of velocity slip and temperature jump boundary condition is based on kinetic theory of gases. A similar boundary condition can be derived by an approximate analysis of the motion of gas in an isothermal condition, which has been presented in this section. 

In 1879 Lord Kelvin introduced the term nwtivity for the possession, the waste of which is called dissipation at constant temperature this is identical with Maxwell s available energy. He showed in a paper On Thermodynamics founded on Motivity and Energy Phil. Mag., 1898), that all the thermodynamic equations could be derived from the properties of motivity which follow directly from Carnot s theorem, without any explicit introduction of the entropy. 

It may be noted that equation 10.86 is identical to equation 10.30. (Stefan s Law) and. Stefan s law can therefore also be derived from Maxwell s Law of Diffusion. 

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). 

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. 

This result leads to one of Maxwell s relations [Eq. (2-53)]L The three remaining relations are found by analogous derivations. 

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. 

Maxwell s equations, as well as the Lorentz force, can be derived from the Lagrangian density... 

Because partial derivatives are used so prominently in thermodynamics (See Maxwell s Relationships), we briefly consider the properties of partial derivatives for systems having three variables x, y, and z, of which two are independent. In this case, z = z(x,y), where x and y are treated as independent variables. If one deals with infinitesimal changes in x and y, the corresponding changes in z are described by the partial derivatives ... 

The space-charge current density in vacuo expressed by Eqs. (3) and (4) constitutes the essential part of the present extended theory. To specify the thus far undetermined velocity C, we follow the classical method of recasting Maxwell s equations into a four-dimensional representation. The divergence of Eq. (1) can, in combination with Eq. (4), be expressed in terms of a fourdimensional operator, where (j, 7 p) thus becomes a 4-vector. The potentials A and are derived from the sources j and p, which yield... 

A set of first-order field equations was proposed by Hertz [53-55], who substituted the partial time derivatives in Maxwell s equations by total time derivatives... 

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 ... 

Equation (7) is called Maxwell s distribution law after Clerk Maxwell (1831-79), Professor of Physics at Cambridge (England), who obtained it in 1860, before Boltzmann in 1871 obtained his wider distribution law, Maxwell derived his distribution lav/ from the conservation of energy together with the assumption that the motion is separable in three mutually orthogonal directions. The latter assumption was violently attacked by mathematicians, but we now iccognize that the assumption is both reasonable and true,... 

The derivation of Eq. (13) means that the equation of continuity is a mere mathematical consequence of only two of Maxwell s equations that is, the condition of continuity does not add additional physical information to Maxwell s equations. 

The relations derived up to this point are not sufficient to solve the problem of the dispersion of the optical branch for small k values. The retardation effect must also be taken into account. Because of the finite velocity of electromagnetic waves, the forces at a certain point of time and space in a crystal are determined by the states of the whole crystal at earlier times. A precise description of the dispersion effect therefore requires the introduction of Maxwell s equations. With a harmonic ansatz for P and P which is analogous to Eq. (II.15) they lead to the relation... 

Based on the value that takes (0 =) and Fig. 1.16 we can see that the unsymmetrical distribution is in question. The x -distribution is derived from Maxwell s distribution of molecular velocities in gases [5],... 

The propagation of light in a nonlinear medium is governed by the wave equation, which was derived from Maxwell s equations for an arbitrary homogeneous dielectric medium,... 

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