Deduction of generalised Kirchhoff's Laws from the basic principles of electromagnetism

The deduction of the so well-renowned and established Laws of Kirchhoff relating to the currents flowing in a network, which are considered to be almost axiomatic in electrical engineering sciences, seems to be preposterous and non-sensical at first sight. However, on a closer examination it will appear that these laws are based on two principles, viz, the steady-state condition under which accumulation of free electrical charge is precluded, and, the experimentalOhm's Law, which propounds a linear relation between the difference of potential across a conductor and the total current flowing through the same. But, there are two things, which are normally overlooked. The conductors inKirchhoff's Laws are “wires”, which are one-dimensional lines, and theOhm's Law is the macroscopicOhm's Law for total currents applied to these lumped resistors. Strictly speaking, the “Laws” are to be deduced from the more fundamental electromagnetic equations for continuous media and the microscopicOhm's Law. It is to be noted that theKirchhoff Nodal Law is but a consequence of the steady state condition derivable from these basic equations, viz, the current-density is divergence-free. For a continuous medium, what the form of theKirchhoff's Laws will be, is difficult to guess unless deduced from the basic equations. Once these are established from the basic equations of electromagnetism for a continuous medium, the usual form ofKirchhoff's Laws will follow as corollary of the general case, as has been shown here.     

Oxidation of naphthalene and phenanthrene by pyridinium fluorochromate (PFC)—a kinetic and mechanistic study

The kinetics of the oxidation of naphthalene and phenanthrene by pyridinium fluorochromate, C₅H₅NHCrO₃F, PFC, has been studied. The main product of the oxidation is the corresponding quinone. While each oxidation, studied in aqueous acetic acid (80–95%, v/v) medium, is first order with respect to the oxidant, the rate is almost independent of the substrate concentration. The reactions are catalyzed by acid, the acid-catalyzed reactions being very fast, which precluded determination of their order in acid medium. The kinetic isotope effect, k_H/k_D is 5.50 at 303 K for naphthalene. The reaction does not induce polymerization of acrylonitrile. The effects of temperature and solvent compositions were studied and activation parameters evaluated. Probable mechanisms are discussed.     

The matrix equations AX = C, XB = D

For the pair of matrix equations AX = C, XB = D this paper gives common solutions of minimum possible rank and also other feasible specified ranks.     

Ion-exchange separation of metal ions with thiosulphate

Summary In a cationic column of Amberlite IR-120 with thiosulphate as the eluting agent it has been possible to separate metal ions from several two to five component mixtures.

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Zusammenfassung Mischungen von 2–5 Metallionen konnten durch Ionenaustausch-chromatographie an Amberlite IR-120 mit Thiosulfat als Eluierungsmittel getrennt werden.

     

On the calculation of average magnetic susceptibility for anistropic paramagnetic systems

It is shown that the expression $\text{x}$=$\text{1}\text{3}$x_|+$\text{2}\text{3}$x_⊥ gives fairly incorrect results when used in the saturation region for highly paramagnetic anistropic systems. Explicit expressions for the average magnetic susceptibility in the saturation region are suggested, which besides being extremely accurate, involve computation of the susceptibility only in 3 or 4 fixed directions.     

Carbon-13 resonance assignments of some friedelanes and taraxasteranes

Carbon-13 signal assignments of taraxasteryl acetate, taraxasteryl palmitate, taraxasterol and epi-friedelinyl acetate are reported. Some signal assignments of the friedelanones, reported recently by others, are inconsistent and require revision.     

Flows in a rotating elastico-viscous fluid induced by torsional oscillations of a plate acting for a finite time

Unsteady flows in an electrically conducting rotating elastico-viscous liquid in the presence of a uniform magnetic field and induced by small amplitude torsional oscillations, acting for a finite time, of an infinite, rigid, non-conducting plate, are discussed. This analysis indicates the general features of the steady and unsteady velocity field, and the structure of the associated boundary layers on the plate including the effect of rotation, hydromagnetic and elastic parameters involved in the problem. The velocity field related to small elastic parameter is calculated with physical significance. It is further shown that the Ekman suction velocity, which is responsible for the generation of an axial inflow toward the bounadry layer, does not appear unless the plate is subject to oscillations for an indefinite period. Several limiting results are found to follow as special cases of this analysis.