Alex A. Kaufman, Richard O. Hansen, Robert L.K. Kleinberg

Abstract
Publisher Summary

This chapter focuses on the magnetic field in a nonmagnetic medium. Numerous experiments performed at the beginning of the19th century demonstrated that constant currents interact with each other that mean mechanical forces act at every element of the circuit. Certainly, this is one of the amazing phenomena of the nature and would have been very difficult to predict. In fact, it is almost impossible to expect that the motion of electrons inside of wire may cause a force on moving charges somewhere else, for instance, in another wire with current, and for this reason the phenomenon of this interaction was discovered by chance. By analogy with the attraction field caused by masses, it is proper to assume that constant (time-invariant) currents create a field and because of the existence of this field and of the existence of this field, other current elements experience the action of the force F. Such a field is called the magnetic field, and it can be introduced from Ampere’s law.

1.1 Interaction of constant currents and Ampere's law

Numerous experiments performed at the beginning of the19th century demonstrated that constant currents interact with each other; that is mechanical forces act at every element of the circuit. Certainly, this is one of the amazing phenomena of the nature and would have been very difficult to predict. In fact, it is almost impossible to expect that the motion of electrons inside of wire may cause a force on moving charges somewhere else, for instance, in another wire with current, and for this reason the phenomenon of this interaction was discovered by chance. It turns out that this force of interaction between currents in two circuits depends on the magnitude of these currents, the direction of charge movement, the shape and dimensions of circuits, as well as the their mutual position with respect to each other. The list of factors clearly shows that the mathematical formulation of the interaction of currents should be much more complicated task than that for masses or electric charges. In spite of this fact, Ampere was able to find a relatively simple expression for the force of the interaction of so-called elementary currents:

     (1.1)

where I1 and I2 are magnitudes of the currents in the linear elements dl(p) and dl(q), respectively, and their direction coincides with that of the current density; Lqp the distance between these elements and is directed from the point q to the point p, which can be located at the center of these elements; finally μ0 is a constant equal to

and is often called the magnetic permeability of free space. Certainly, this is confusing definition, since free space does not have any magnetic properties. We will use the S.I. system of units where the distance is measured in meters and force in newtons. Of course, with a change of the system of units the value of μ0 varies too. In applying Ampere's law (Equation (1.1)), it is essential that the separation between current elements must be much greater than their length; that is

Correspondingly, points: p and q can be located anywhere inside their elements. It is easy to see some similarity of Ampere's law and Newton's law of attraction; they describe a force between either elementary currents or elementary masses. Let us illustrate Equation (1.1) by three examples shown in Fig. 1.1. Suppose that elements dl(p) and dl(q) are in parallel with each other. Then, as follows from definition of the cross product, the force dF(p) is directed toward the element dl(q), and two current elements attract each other (Fig. 1.1(a)). If two current elements have opposite directions, the force dF(p) tries to increase the distance between elements, and therefore they repeal each other (Fig. 1.1(b)). If the elements dl(p) and dl(q) are perpendicular to each other, as is shown in Fig. 1.1(c), then in accordance with Equation (1.1) the magnitude of the force acting at the element dl(p) equals

and it is parallel to the element dl(q). At the same time, the force dF(q) at the point q is equal to zero, that is Newton's third law becomes invalid. This contradiction results from the fact that Equation (1.1) describes an interaction between current elements instead of closed current circuits. In other words, this equation is written for unrealistic case, since we cannot create a constant current in an open circuit, but as all experiments show, Equation (1.1) gives the correct result for closed current lines. For instance, applying the principle of superposition, the force of interaction between two arbitrary and closed currents (Fig. 1.1(d)) is defined as

     (1.2)

The internal integral in Equation (1.2) characterizes the force acting at some point of the current line L1, for instance, point p1 and caused by all elements of the current line L2. Thus, the force F represents a sum (integral) of forces applied at different points of the same circuit and, as is well known, its action causes in general a deformation, translation and a rotation of the current line L1. It is obvious that in the case of closed circuits the interaction between them obeys Newton's third law.

The relationship between the force F and currents (Equation (1.2)) is called Ampere's law for closed circuits with constant currents, and it is impossible to overestimate its importance, since it is the theoretical foundation of many devices measuring magnetic field as well as electromotors, transforming electric energy into mechanical energy, and it has numerous applications in physics and technology. Finally, it is proper to notice the following:

Fig. 1.2 (a) Illustration of Equation (1.4). (b) Field due to the surface currents.

1.2.1 General form of Biot–Savart law

Now we generalize Equation (1.4) assuming that along with linear currents there are also volume and surface currents. First let us represent the product I dl...