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On the Bending of Electromagnetic Micro-Waves below the Horizon

An interesting phase in the development of the modern radio technique are the experiments conducted during the last few years with very short wave-lengths. Marchese Marconi [1] reported about an extensive series of successful radio connections over distances up to 260 km., in which waves of from 50 cm. to 60 cm. were used, concentrated with the help of a parabolic reflector. Clavier and Gallant [2] went even to still shorter waves of only 17.4 cm. which they sent over a distance of 61 km. also concentrating them with a reflector of 3.8 m. in diameter. The most remarkable feature of Marchese Marconi's results is that the distances covered by him exceed several times the range of rectilinear visibility from the sending station. The memory is still fresh of the great surprise which was caused among physicists by the unusually long range of long wave radio-reception. The explanation of these puzzling facts about long waves was traced, in the meantime [3], to the influence of the Kennelly-Heaviside layer of the upper atmosphere, and the question, naturally, arises to what extent atmospheric influences are responsible for the phenomena observed by Marchese Marconi with micro-waves. The first step in answering this question must be an investigation of how much bending is to be expected from the point of view of the wave theory completely neglecting the atmosphere. Such an investigation is the subject of this paper. The simple method which we propose is based on Huyghens' principle and treats the surface of the earth as a perfectly absorbing screen. As far as we know, it was not used heretofore and there are good reasons for this: In the case of long waves, the properties of the soil play an important part both in their production and their propagation. The height of the receiving station is only a fraction of the wave-length, so that only the so-called "surface wave" is of practical interest. On the other hand, the micro-waves are produced away from the soil and independently from it and, after they strike the earth, the surface wave is so thin as to be entirely unimportant. The transmission is, in this case, a matter of space propagation on which the physical properties of the earth surface have no material influence. It is therefore, perfectly permissible to replace it by a perfectly absorbing screen. The results of our calculations and their comparison with Marchese Marconi's observations are summarized in the last section

On the Magnetic Energy of Supraconductors

A long stretched supraconductive body (wire), placed longitudinally in a homogeneous magnetic field of slowly increasing strength H0 loses its supraconductivity when the field strength reaches a certain critical value Ho = Hc. Measurements by de Haas and Voogd [1] disclosed that the case is different when the direction of the wire is transverse: then the transition takes place at a value of the applied field H0 lower than Hc. By H0 is meant the field as it would be if the supraconductor were absent. When it is present, the strength H is in general different from H0 and varies from point to point, because the field cannot penetrate into the supraconductive material, so that the lines of induction are crowded at its surface. These conditions caused Von Laue [2] to enunciate the following hypothesis: A supraconductive body begins its transition into the normal state when the strongest part of the magnetic field H at its surface reaches the critical value H = Hc. Subsequent measurements by de Haas and co-workers on circular and elliptic cylinders [3] and on spheres4 confirmed Von Laue's rule, so that it must be regarded as firmly established experimentally. Recently it has gained added theoretical importance, having been made by Landau [5] the starting point of interesting considerations on the structure of supraconductors during the process of transition. The theoretical reasons for the dependence of the critical field Hc on the temperature were cleared up in a significant paper by Gorter and Casimir [6], in which these authors also attempted a theoretical derivation of Von Laue's rule. However, this part of their work is, in our opinion, unconvincing and open to certain objections. In view of the great importance of the subject it seemed desirable to study it more closely in order to remove all doubts which may arise. The purpose of the present paper is to supply this want and to modify the derivation of Von Laue's rule in such a way as to meet the objections

Electronic charge density in helium in the first order shielding approximation

Electronic charge density for ground state of helium as function of nuclear charg

Some Remarks on the Use of the Variational Principle for the Second Order Energy

Approximate ground state wave function used in variational principle for second order energ

The variational method 2 - Perturbation theory

Variational method to approximate solutions of perturbation equations, perturbation analyses, and use of perturbation theory to infer variational principl