J. M. Aguirregabiria

An example of surface charge distribution on conductors carrying steady currents

In order to constrain electrons to move along ohmic conductors carrying steady currents, there must be a surface charge density that is usually very difficult to calculate. An approximate analytic expression for this surface charge density on a conducting square ring is presented here where the only source of emf is a changing external magnetic field. The corresponding electric field is determined and it is checked that the energy balance for this system holds.

δ -function converging sequences

We discuss the usefulness and physical interpretation of a simple and general way of constructing sequences of functions that converge to the Dirac delta function. The main result, which seems to have been largely overlooked, includes most of the δ-function converging sequences found in textbooks, is easily extended, and can be used to introduce many useful generalized functions to physics students with little mathematical background. We show that some interesting delta-function identities are simple consequences of the one discussed here. An illustrative example in electrodynamics is also analyzed, with the surprising result that the formalism allows as a limit an uncharged massless particle which creates no electromagnetic field, but has a nonzero electromagnetic energy–momentum tensor.

Robust chaos with variable Lyapunov exponent in smooth one-dimensional maps

Abstract We present several new easy ways of generating smooth one-dimensional maps displaying robust chaos, i.e., chaos for whole intervals of the parameter. Unlike what happens with previous methods, the Lyapunov exponent of the maps constructed here varies widely with the parameter. We show that the condition of negative Schwarzian derivative, which was used in previous works, is not a necessary condition for robust chaos. Finally we show that the maps constructed in previous works have always the Lyapunov exponent ln 2 because they are conjugated to each other and to the tent map by means of smooth homeomorphisms. In the methods presented here, the maps have variable Lyapunov coefficients because they are conjugated through non-smooth homeomorphisms similar to Minkowski’s question mark function.

Linear momentum density in quasistatic electromagnetic systems

We discuss a couple of simple quasistatic electromagnetic systems in which the density of electromagnetic linear momentum can be easily computed. The examples are also used to illustrate how the total electromagnetic linear momentum, which may also be calculated by using the vector potential, can be understood as a consequence of the violation of the action–reaction principle, because a non-null external force is required to maintain constant the mechanical linear momentum. We show how one can avoid the divergence in the interaction linear electromagnetic momentum of a system composed by an idealization often used in textbooks (an infinite straight current) and a point charge.

Magnetic braking revisited

The braking force acting on a conducting disk rotating under the influence of an external magnetic field of axial symmetry is calculated in a quasi-static approximation and the role played by the charge distributions induced in the disk is shown. The two cases of infinite and finite radius are considered to analyze the influence of edge effects and we obtain a general expression for the braking torque when the magnetic field has axial symmetry. The particular case of a uniform external magnetic field is used to show the working of a simplified model of a cylindrical battery. Analytical results are compared with those obtained by other authors.

A pure kinematical explanation of the gyromagnetic ratio g=2 of leptons and charged bosons

Abstract By analysing the structure of the spin operator, we give a pure kinematical explanation of the origin of the gyromagnetic ratio of elementary particles.

Falling elastic bars and springs

We analyze the initial motion of an elastic bar that is suddenly released after being hung from one end. The analytical solutions uncover some unexpected properties, which can be checked with a digital camera or camcorder in an alternative setup in which a spring is substituted for the bar. The model and the experiments are useful for understanding the similarities and differences between the elastic properties of bars and springs. Students can use the simple experiments to improve their understanding of elastic waves.

Surface charges and energy flow in a ring rotating in a magnetic field

An ohmic ring that rotates with constant angular velocity in an external uniform magnetic field is considered as a simple model for a current generator. Under the assumption that all quantities vary slowly in time, the lowest‐order approximation to the surface charge density is found. The flux of the Poynting vector through the loop surface is also computed. Unlike in the examples that are given in textbooks, this flux is not always incoming: It has the outgoing direction around the loop parts where the electrons are moving against electrostatic forces.

A Lewis-Tolman-like paradox

A non-relativistic Lewis-Tolman-like paradox is proposed. It is checked by direct calculation that the paradox disappears if linear and angular momenta are attached to the static electromagnetic field. The storage of linear momentum in the electromagnetic field during the assembling process is also analysed. Finally a naive model of the electron suggested by this system is proposed.

Velocity fields inside a conducting sphere near a slowly moving charge

Explicit expressions for first‐order electric and magnetic fields created inside a conducting sphere by a nearby slowly moving charge are given. They are found to be independent of the sphere radius. On the contrary, outer first‐order fields, which are also computed, depend on it. The energy dissipation by Joule effect is calculated and shown to agree with the external first‐order work done on the charge to maintain its uniform motion.