We present the numerical analysis of the capacitance dependence of a conducting sphere on the distance to another sphere and its relative size. Among other results, we find that the greatest deviations from the model of a secluded sphere are observed for the case of the presence of infinite conducting plane near it. We provide a simple physical explanation of this fact. For example, our numerical calculations show that the value of relative error in the determination of the sphere capacitance is equal to 5% if the distance from the sphere center to the infinite plate is ten times more than its radius. The consideration of this problem will be useful for advanced undergraduates, who study the methods for solving electrostatic problems.
We present a quantitative analysis of some ‘infinite’ models, occurring in the course of electricity and magnetism. First, we estimate the deviations from the models of an ideal parallel-plate capacitor and solenoid arising due to the bending field effect. Next, we consider the models of an infinite wire and plane of uniform charge. We show that in the first case the real electric field differs from the idealized one because of the finite length of the wire. In the second case this difference is caused also by the non-uniform surface charge density. The issues outlined in this article will be useful for advanced undergraduates, studying solutions to electrostatic and magnetostatic problems.
In this paper we present an overview of methods of resistance finding for the conductor with variable cross-section or (and) length. First, we consider the cases of the finite size resistors and an infinite homogeneous weakly conducting medium with two electrodes. Next, we continue the Romano and Price analysis about the truncated cone problem by means of comprehensive numerical calculations done for the trapezoid plates. We conclude that in general case of a conductor with variable cross-section the homogeneous electric field approximation gives only the lower limit estimation for the resistance value. This fact can be explained on the basis of the minimum electric power principle. The issues outlined in this article will be useful for advanced undergraduates, who study methods for solving electrostatic problems.Keywords: problem solving, electrical resistance, trapezoid plate problem.
Abstract We discuss the influence of several factors on the deviations from energy spectrum of an infinite square quantum well (QW) for real microscopic systems that can be approximately modelled using particle in a box. We introduce the “blurring” potential in the form of the modified Woods-Saxon potential and solve the corresponding Schrödinger equation. It is found that the increase of the degree of blurring δ of the QW leads to the increase of number of the energy levels inside it and to increase of deviations from the quadratic dependence ε (n) (ε is the particle energy, n is the energy level number) typical for the infinite square QW, especially, for the energy levels close to the QW “tops”. It is most surprising that for relatively “large” values of δ the difference between the levels energies of such well and the appropriate (with the same n) levels energies of the square QW with the same depth changes sign (from positive to negative) as number n increases. We also conclude that the asymmetry of the QW and non-equality m i n ≠ m o u t (where m i n and m o u t are the particle effective mass inside and outside the QW) play a significant role for the relatively “shallow” well near the QW top.