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The slowly charging capacitor is the standard example used to illustrate that the displacement current density is needed in Ampere's law if we want to correctly determine the magnetic field between capacitor plates. However, in any quasi-static situation the magnetic field can also be determined using the Biot-Savart law including only the real current densities. In this work, we will numerically calculate the magnetic field due to the surface currents on the capacitor plates and add it to the magnetic field due to the charging wire and show how they combine to create the correct magnetic field thoughout all space. For regions to the left or right of the capacitor, we find the surprising result that the surface currents replicate the magnetic field that would have been created by the missing section of the charging wire between the plates. For points between the capacitor plates, the magnetic field due to the surface currents mostly cancels the magnetic field from the near-infinite length charging wire, resulting in the well-known reduced field in that interior region. We will also illustrate the impact of finite capacitor plates on these results and briefly comment on how textbook and/or classroom discussions could be improved by carefully discussing these details.

We illustrate the use of Maxwell's term in Ampere's law and also the conservation laws of electromagnetic theory. First, we outline the conservation laws for energy and linear momentum which often appear in textbooks of electromagnetism, and then we show how these conservation laws can be extended to the conservation law for angular momentum and to the relativistic conservation law involving the center of energy. Next we treat the traditional example of charging a parallel-plate capacitor but explore a number of energy-conservation aspects and variations which are not usually found in textbooks. Finally, we discuss the charging of spherical conductors where we can illustrate the use of Maxwell's term and can provide non-trivial examples of three of the four conservation laws.

The use of Faraday's Law of induction for calculating the induced currents in an extended conducting body is discussed. In a general case with arbitrary geometry, the solution to the problem of a moving metal object in the presence of a magnetic field is difficult and implies solving Maxwell's equations in a time-dependent situation. In many cases, including cases with good conductors (but not superconductors) Ampere's Law can be neglected and a simpler solution based solely in Faraday's law can be obtained. The integral form of Faraday's Law along any loop in the conducting body is equivalent to a Kirkhhoff's voltage law of a circuit. Therefore, a numerical solution can be obtained by solving a linear system of equations corresponding to a discrete number of loops in the body.

We show that the concept of electric current was elaborated only after the discovery by Oersted in 1820 of a connection between electricity and magnetism, and thanks to the subsequent work of Ampere. In his study of the interaction between a compass and an electric circuit, Ampere set up a crucial experiment when he put a compass above his Voltaic pile, and another one above the connecting wire. Indeed, this experiment supported his creation of a new physical quantity, independent of the nature of physical phenomena, identical in the pile and in the wire, and only characterized by its direction and its intensity. To the experimental definition of this physical quantity—the electric current—by the oriented deviation of a magnetic needle, Ampere added in his manuscripts the substance of the two present theoretical definitions of the intensity of the current, namely, the ratio of charge to time q/t, and the ratio of electromotive force to the conducting wire's resistance E/R.

In 1822 Ampere created a new kind of motor when he succeeded in spinning a cylindrical magnet around its axis by connecting it to a battery generating a steady current. Nowadays, it is easy to present such a motor in the classroom utilizing a neodymium magnet, a D battery, a steel nail, and a short piece of copper wire. Although it is very simple to observe the rotation, the explanation of this effect is still under dispute. This work presents the history of this motor including the controversy between Ampere and Faraday, as well as the modern explanation based on the field concept. We emphasize the positive outcomes to be gained in the classroom by presenting this device to students.

The integral form of Ampere’s law applied to an infinitely long solenoid gives a good first approximation to the magnetic field of finite air-cored solenoids with length L to width w ratios (aspect ratios) significantly greater than 1. It is less well known that this approximation requires much larger aspect ratios to be applicable to solenoids whose relative core permeability μr is much greater than that of the surrounding medium. An exact expression is derived for the magnetic field and inductance of a linear permeable core spheroidal solenoid, and an inequality is obtained for the validity of the infinite solenoid approximation. It is demonstrated that the magnetic field and inductance of a spheroidal solenoid with a linear core approximate well with those of a cylindrical solenoid of the same length and width.

This paper describes the design, implementation, and evaluation of an instructional sequence intended to improve student understanding of Ampere’s law among engineering students taking introductory physics. Participating students performed better on qualitative and quantitative test questions about Ampere’s law than students in a similar class that devoted the same amount of time to the topic but followed traditional instruction. The results suggest that the strategy employed in the sequence can help students to apply and explain the relation between electrical currents and magnetic fields.

We update Ampere’s theory using vector notation and derive his expression for the force between two current elements. We assume that the two elements are in different current loops and integrate over one to obtain the force on a differential element in the second. This procedure allows us to define the magnetic field in a natural manner and to derive the Lorentz force for a current segment. We equate the magnetic moments of current and permanent magnet dipoles and show that Biot and Savart could have performed their experiment using a small current loop, thus establishing the Biot-Savart law as a consequence of Ampere’s theory.

A more effective experimental method is used to analyze the effect of magnetic damping on a nonferromagnetic air track. Due to the continuous interaction between the horizontal air track and the moving magnets fixed on the air-track glider, the experiment can be easily done with low-cost photoelectric detecting techniques. A simple model is proposed based on Faraday’s law, the Lorentz force law, and Ampere’s law. Analytic expressions for the position, the velocity, and the magnetic damping force of the moving magnets as a function of time are obtained. Systematic measurements were performed and the results are in good agreement with the model. The method provides a simple teaching platform for introductory physics demonstrations and undergraduate courses in experimental physics.

Because mathematicians and physicists think differently about mathematics, they have different goals for their courses and teach different ways of thinking about the material. As a consequence, there are a number of capabilities that physics majors need in order to be successful that might not be addressed by any traditional course. The result is that the total cognitive load is too high for many students at the transition from the calculus and introductory physics sequences to upper-division courses for physics majors. We illustrate typical student difficulties in the context of an Ampere’s law problem.

Francois Arago brought the news of Oersted’s discovery of the effect of a current-carrying wire on a magnet to Paris on September 4, 1820. Andre-Marie Ampere and the team of Jean-Baptiste Biot and Felix Savart quickly set to work to establish a quantitative law for this effect. Their approaches were quite different. This paper describes the experiments of Biot and Savart and their results. It also briefly discusses the approach of Andre-Marie Ampere, who coined the name “electrodynamics” and whose fundamental formula gave the force between two infinitesimal current elements. Ampere’s formula fell into disuse after the advent of Maxwell’s field approach. Biot and Savart’s experimental law, in the modern form of the differential magnetic field due to a current element, became the standard starting point for calculating the magnetic field due to steady currents.

Experiments designed to illustrate the principles embodied in Ampere’s Law and Faraday’s Law often depend on the knowledge of the number of turns of wire on various types of laboratory coils. The lack of direct measurement of this parameter can be overcome with the use of inexpensive (

There has recently been a revival of an old controversy over whether Ampere’s original law of force between current elements is to be preferred over the more familiar and universally used Biot–Savart–Lorentz law. Although it is agreed that the two laws give identical results when used to calculate the force between two circuits, it has been claimed that if Ampere’s law is applied to the action of a circuit upon part of itself it predicts internal longitudinal forces that are excluded by Biot–Savart–Lorentz. The existence of longitudinal forces has been inferred from observations of the buckling of the rails in a railgun, attributed to compressive stress, and the fracture of long wires subjected to pulsed currents, attributed to ‘‘Ampere tension.’’ An experiment has been performed that should provide an unambiguous demonstration of this longitudinal force, if it exists. The apparatus consisted of a rigid coaxial circuit in which an unsymmetrical section of the center conductor was free to move in the axial direction under the influence of the longitudinal force. Pulsed currents of up to 100 kA were passed through the circuit but no evidence of a longitudinal force was found. It is shown that this result is consistent with Ampere’s force law, and that claims that the law predicts longitudinal forces are based on the incomplete application of the law.

From the perspective of Ampere’s circuital law, either displacement current or conduction current can be viewed as the source for the magnetic field inside a circular capacitor that is slowly being charged. The Biot–Savart law is more selective. How it can be used with conduction current alone is shown. Also considered is the ‘‘leaky capacitor. Here it is shown that an isolated charged capacitor which discharges slowly in a homogeneous Ohmic dielectric produces no magnetic field anywhere. Alternatively, a field is produced if the conducting material is confined to a limited region. This field is calculated for a circular capacitor when only the material in the gap is conducting.

We reproduce here Fig. 3 from that paper in its correct originally submitted form, which also appeared in the proof sent to the author. Subsequently, in their desire for a more uniform typographical style, the figure was altered by the American Institute of Physics staff at Woodbury, New York, without consultation with either the author or the editor. In so doing the letters indicating the figure sub‐units were pasted back in the wrong order. The Woodbury staff has apologized to the author for this regrettable error. That apology is hereby transmitted to the readers of this Journal.

In 1832, after Michael Faraday had announced his discovery of electromagnetic induction, Andre‐Marie Ampere claimed that he had actually discovered the induction of one current by another in 1822. In fact, he had, but did not really publish the fact at that time. This article explores the reasons for Ampere’s failure to lay claim to a discovery that would have guaranteed him scientific immortality.

This article describes the use of inexpensive Hall‐effect integrated circuits to build magnetic field transducers and the use of these transducers in a sophomore‐level experiment that directly illustrates Ampere’s law in its integral form. This experiment has the pedagogical advantages of permitting direct measurement of the vector components of the magnetic field and of illustrating the precise operational meaning of a line integral around a closed path.

This article gives a combined form of the laws of Ampere and Biot–Savart which allows us to calculate the infinitesimal magnetic circulation around a closed loop ∑δB⋅δl in terms of the current flow i through an infinitesimal current path δs, and the change in the solid angle subtending the closed loop δΩ as the apex of the solid angle traverses δs:∑δB⋅δl=iδΩ×10−7. After giving its derivation from the law of Biot–Savart, the combined form is applied to a variety of magnetostatic problems. The illustrative examples, none of which requires the use of calculus for its solution, include the usual introductory course examples such as determining the magnetic field inside a toroidal coil, and at the center and along the axis of a circular current loop. In addition, the potentialities of the combined law are demonstrated by determining the field anywhere in the plane and along the axis of a rectangular current loop. Appendices include discussions of solid angles and relativistic effects.

Within six years of the time Ampere learned of Oersted’s observations, he had completely elucidated the principles by which the magnetic interactions between steady currents can be calculated. In constructing his demostration, he used Oersted’s experiment only to infer that, in general, two electrical circuits will exert forces on each other; and he made no use of previous knowledge of magnetics. Eschewing such restrictions, we give a simpler demonstration of a simpler but equivalent result. Our method is one that Ampere could have chosen, and that (unlike his) can be understood by any junior–senior student in physics or engineering.

In Ampere’s circuital law, two separate terms are required to describe the contributions to the magnetic field of the charged‐particle current and the ’’displacement current.’’ In this paper, it is shown how both terms arise by applying the Biot–Savart law to a calculation of the displacement current resulting from the motion of a single charged particle. The conclusions drawn from this calculation are applied to the case of a steady straight current filament.

The force laws of Ampere and Biot–Savart in magnetostatics are compared using the geometrical model of a closed curve to represent a current loop. The two laws give identical results when forces between separate current loops are considered and also for the force exerted by a current loop on a rectilinear part of itself. According to both laws, these self‐forces diverge wherever the curvature of the curve representing the current loop is not equal to zero. Differences in the predictions of the two laws are shown to appear only as differences in diverging forces when evaluating forces of a current loop on a part of itself and to be entirely due to the oversimplified and unrealistic geometrical model used for the current loop.

Ampere’s circuital law for magnetostatics and two‐dimensional analogs of Gauss’ law for electrostatics and magnetostatics are obtained by constructing appropriate complex functions and applying the residue theorem. The approach is motivated by the formal similarities between the residue theorem and Ampere’s circuital law for magnetostatics.

One usually derives Ampere's law by taking the curl of the magnetic field given by the law of Biot and Savartfor a volume distribution of currents J. In the derivation, one drops a term that contains ∇ · J, in order to treat stationary processes. It is shown that if that term is kept one gets, for ∇ × H, a term which is the usual conduction current J plus a second term similar to Maxwell's displacement current, present even outside the current distribution J. That extra term is δD0/δt, where D0 = e0E0 and E0 is the instantaneous Coulomb field produced by the charges in question.

The equations of electromagnetism, whether written in the electrostatic, electromagnetic, or symmetric system, whether in rationalized or nonrationalized form, express an invariant set of physical relationships. The same set of letter symbols is employed in each, system of equations; these symbols represent related, but different, quantities in the various systems. The relationships among corresponding symbols are given and applied to precise statements about the relation between the oersted and the ampere per meter, the abampere and the ampere, etc.

The paper discusses the erroneous interpretation of the double layer as a source of a presumed nonconservative field, and traces back that very old mistake to works of Ampere, Gauss, and Maxwell. That mistake is repeated by all authors up to the present time. It is shown that the magnetic double layer creates a strictly conservative field, which is not completely equivalent to the field of a filamentary current loop, since that field is nonconservative. It is further shown that the solid angle is a single-valued function with the step discontinuity, and by no means a multiple-valued function.

This paper shows that a conducting spiral behaves as a unipolar machine. As a first approximation, the current in the spiral can be viewed as a superposition of a system of circular currents, which provide a magnetic field perpendicular to the plane of the spiral, and a radial current on which the Ampere forces produce a self-torque. An essential feature of unipolar machines is stressed, i.e., the relevant electromagnetic interactions take place between perpendicular currents and not between parallel ones, as happens in the other machines. A simplified series unipolar machine for laboratory and classroom instruction has been built and is discussed.

Some general features of several elementary methods of determining the magnetic field of a current configuration are examined. Ampere's law, the circuital form of Maxwell's equation and a derived relation between the magnetic field and the vector curl of the current, are each applied to the simple problem of computing the magnetic field due to a thin current dipole in a conducting medium. The significance of closed current paths and the conditions under which the return currents in the medium must be included in the computation are discussed for the dc case.The conclusions concerning dc currents are then considered in relation to ac problems and the use of the vector potential in antenna problems. These considerations indicate why only antenna currents enter into the computation of vector potential and the nature of the “thick antenna” problem.

Maxwell's formulation of electromagnetism is not the only formulation, though it is the most familiar. In particular, the whole subject of electrodynamics can be expressed as an extension of Coulomb's equation, with the magnetic field entirely eliminated. The paper reviews the history of this attack, beginning with the work of Coulomb, Ampere, and Gauss, and concluding with recent developments.

If one drops the Faraday induction term from Maxwell’s equations, they become exactly Galilei invariant. This suggests that if Maxwell had worked between Ampere and Faraday, he could have developed this Galilei‐invariant electromagnetic theory so that Faraday’s discovery would have confronted physicists with the dilemma: give up the Galileian relativity principle for electromagnetism (ether hypothesis), or modify it (special relativity). This suggests a new pedagogical approach to electromagnetic theory, in which the displacement current and the Galileian relativity principle are introduced before the induction term is discussed.

The need for a strict definition of fundamental units of measurement is pointed out. The fact that the ampere is defined by an equation in terms of other previously defined quantities requires that it be defined in terms of those quantities. This would give the ampere the unit of newton12, and would make the permeability unitless. The instructional value of this analysis is pointed out by showing the ease with which the undergraduate student may balance, dimensionally, equations involving electrical quantities.

It is shown that charge conservation is not necessary in deriving the displacement current in Ampere's law.

The balance described in this paper was employed, in preliminary experiments, to determine factors governing the force between parallel electric currents. It was then employed to measure the ampere in absolute units. A current range of 4.0 to 6.0 amperes was used in the work and the results obtained as described in the paper indicate an average error of 0.3 percent.

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