Physics

Using the standard formalism of Lorentz transformation of the special theory of relativity, we derive the exact expression of the Thomas precession rate for an electron in a classical circular orbit around the nucleus of a hydrogen-like atom.

Garay-Avendaño \& Zamboni-Rached (2014) defined two classes of axisymmetric solutions of the free Maxwell equations. We prove that the linear combinations of these two classes of solutions cover all totally propagating time-harmonic axisymmetric free Maxwell fields -- and hence, by summation on frequencies, all totally propagating axisymmetric free Maxwell fields. It provides an explicit representation for these fields. This will be important, e.g., to have the interstellar radiation field in a disc galaxy modelled as an exact solution of the free Maxwell equations. Keywords: Maxwell equations; axial symmetry; exact solutions; electromagnetic duality

In this short note we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (see \cite{rajagopal2003implicit}), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as constitutive relations that imply limiting strain, as special subclasses.

Expositions of the Euler equations for the rotation of a rigid body often invoke the idea of a specially damped system whose energy dissipates while its angular momentum magnitude is conserved in the body frame. An attempt to explicitly construct such a damping function leads to a more general, but still integrable, system of cubic equations whose trajectories are confined to nested sets of quadric surfaces in angular momentum space. For some choices of parameters, the lines of fixed points along both the largest and smallest moment of inertia axes can be simultaneously attracting. Limiting cases are those that conserve either the energy or the magnitude of the angular momentum. Parallels with rod mechanics, micromagnetics, and particles with effective mass are briefly discussed.

If a particle has to fall first vertically 1 m from A and then move horizontally 1 m to B, it takes a time t(= τ 1 + τ 2 = τ 3 =3/ 2g − − √ )=0.67 s. Under gravity and without friction, if it sides down on a linear track inclined at 45 0 between two points A and B of 1 m height, it takes time t(= τ 4 =2/ g √ )=0.63 s. Between these two extremes, historically, Bernoulli (1718) proved that the fastest track between these points A and B is cycloid with the least time of descent t= τ B =0.58 s. Apart from other interesting cases, here we study the frictionless motion of a particle/bead on an interesting track/wire between A and B given by y(x)=(1− x ν ) 1/ν . For ν>1 the track becomes convex and t>> τ 4 , and when ν>1.22 , the motion with zero initial speed is not possible. We find that when ν∈(0.09653,0.31749), τ 4 <t< τ 3 and when ν∈(0.31749,1), τ B <t< τ 4 . But most remarkably, the concave curve becomes very steep/deep if ν∈(0, ν c =0.09653) , then t=0.2258 s < τ B , this is as though a particle would travel 1 meter horizontally with a speed equal 2g − − √ m/sec to take the time ( =1/ 2g − − √ = τ 2 )< τ B . The function t(ν ) suffers a jump discontinuity at ν= ν c , we offer some resolution.

Measuring stress levels in loaded structures is crucial to assess and monitor their health, and to predict the length of their remaining structural life. However, measuring stress non-destructively has proved quite challenging. Many ultrasonic methods are able to accurately predict in-plane stresses in a controlled laboratory environment, but struggle to be robust outside, in a real world setting. That is because they rely either on knowing beforehand the material constants (which are difficult to acquire) or they require significant calibration for each specimen. Here we present a simple ultrasonic method to evaluate the in-plane stress in situ directly, without knowing any material constants. This method only requires measuring the speed of two angled shear waves. It is based on a formula which is exact for incompressible solids, such as soft gels or tissues, and is approximately true for compressible "hard" solids, such as steel and other metals. We validate the formula against virtual experiments using Finite Element simulations, and find it displays excellent accuracy, with a small error of the order of 1%.

The goal of this paper is to study the electrostatic field due to an arbitrary charge distribution on a dielectric layer in a dielectric-loaded rectangular waveguide. In order to obtain this electrostatic field, the potential due to a point charge on the dielectric layer is solved in advance. The high computational complexity of this problem requires the use of different numerical integration techniques (e.g. Filon, Gauss-Kronrod, Lobatto, ...) and interpolation methods. Using the principle of superposition, the potential due to an arbitrary charge distribution on a dielectric layer is obtained by adding the individual contribution of each point charge. Finally, a numerical differentiation of the potential is carried out to obtain the electrostatic field in the waveguide. The results of this electrostatic problem are going to be extended to model the multipactor effect, which is a problem of great interest in the space industry.

This paper analyses the surface wave mode propagating along a simplified planar Goubau line consisting of a perfectly conducting circular wire on top of a dielectric substrate of finite thickness but infinite width. An approximate equation for the propagation constant is derived and solved through numerical integration. The dependence of the propagation constant on various system parameters is calculated and the results agree well with full numerical simulations. In addition, the spatial distribution of the longitudinal electric field is reported and excellent agreement with the numerical simulation and previous studies is found. Moreover, validation against experimental phase velocity measurements is also reported. Finally, insights gained from the model are considered for a Goubau line with a rectangular conductor. These results present the first step towards an analytic model of the planar Goubau line.

In this dissertation, we analytically study the apex field enhancement factor (FEF), γ a , by constructing a method which consists in minimizing an error function defined as to measure the deviation of the potential at the boundary, yielding approximate axial multipole coefficients of a general axial-symmetric conducting emitter shape, on which the apex FEF can be calculated by summing its respective Legendre series. Such method is analytically applied for a conducting hemisphere on a flat plate, confirming the known result of γ a =3 . Also, it is applied on a hemi-ellipsoid on a plate where the values of the apex FEF are compared with the ones extracted from the analytical expression. Then, the method is applied for the hemisphere on a cylindrical post (HCP) model. In this case, to analytically estimate the apex FEF from first principles is a problem of considerable complexity. Despite the slow convergence of the apex FEF, useful analytical conclusions are drawn and explored, such as, it is confirmed that all even multipole contributions of the HCP model are zero, which in turn leads to restrictions on the charge density distribution: it will be shown the surface charge density must be an odd function with respect to height in an equivalent system. Also, expressions found for the apex FEF depend explicitly on the aspect ratio, that is, the ratio of height by base radius. Using the dominant multipole contribution, the dipole, at sufficient large distances, it is shown that, for two interacting emitters, as their separation distance increases, the fractional change in apex FEF, δ , decreases following a power law with exponent −3 . The result is extended for conducting emitters having an arbitrary axially-symmetric shape, where it is also shown δ has a pre-factor depending on geometry, confirming the tendency observed in recent analytical and numerical results.

It is most common to construct the Hamiltonian function and Hamilton's canonical equations through a Legendre transformation of the Lagrangean function or through the central equation. These common perspectives, however, seem abstract and detached from classical analytical dynamics. A new and different approach is presented in which the Hamiltonian function is created as one investigates d'Alembert's equation of motion. This formulation directly ties the Hamiltonian function and Hamilton's canonical equations to the root of classical analytical dynamics more than any other approach.