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The macroscopic electric charge on a body is determined from the quantity of electricity carried by particles constituting the material. Although some electric phenomena were familiar before discoveries of these particles, such an origin of electricity came to our knowledge after numerous investigations of the structure of matter. Unlike the mass that represents mechanical properties, two kinds of electric charges different in sign were discovered in nature, signified by attractive and repulsive interactions between charged bodies. While electric charges can be combined as in algebraic addition, carrier particles tend to form neutral species in equilibrium states of matter, corresponding to zero of the charge in macroscopic scale.

In early physics, static electricity was studied as a subject independent from magnetism; it was after Oersted’s experiment that the relation between an electric current and the magnetic field was recognized. Today, static and dynamic electricity are viewed as clearly exclusive phenomena; however, many early findings on static phenomena significantly contributed to establishing present day knowledge of electromagnetism.

Spherical and cylindrical capacitors are important devices in practical applications as well as for simple analysis of a static electric field. In these capacitors, field-lines can be visualized precisely as radial, thereby the model of a point and line charge can be established.

One of the basic properties of the electric field is the force on a small electric charge brought in from outside the field or on a hypothetical charge of 1 C that can be assumed as if present inside the field. On the other hand, the field as a whole is depicted by distributed lines originating from electric charges.

In electrostatics a basic problem is to find the potential function for a given charge distribution that can be solved with the Laplace-Poisson equations under specified boundary conditions. Although the cause for distributed charges is unspecified in the given system, here we are only concerned about the electrostatic problem. For many applications we are interested in the potential at a distant point from distributed charges in a small region. The corresponding field is often dominated by deviations from spherical symmetry, a deviation that can be conveniently viewed with polar coordinates with respect to the center of distribution. Such a charge distribution is often described with respect to a unique direction for deformation that arises from an internal origin.

Humans have utilized magnetized materials such as iron, magnetite, hematite, and other minerals for centuries. In particular, a thin iron bar was utilized as a compass for navigation. To explain the terrestrial magnetism, Gilbert (ca. 1600) first considered the planet Earth as a giant magnet. Since that time, magnetism became a subject for serious scientific studies, bringing about discoveries of many other magnetic materials. Due to their attractive and repulsive nature, each magnet was characterized by two distinct “poles,” named north and south poles (N- and Spoles), according to the way they interacted. These poles were recognized to exist always in pairs, and they were understood to be inseparable, unlike electric charges. Further, because of their delocalization on magnetized surfaces, polar charges were not easily measurable. Nevertheless, Coulomb (1785) defined magnetic poles according to the force between them in an idealized case, analogous to the static force between electric charges.

The magnetic field is a property of the space that surrounds an electric current and a magnet. In early physics powdered iron was used to visualize its geometric character, and the distributed field intensities were studied with a compass needle. However, such observations were hardly subject to systematic investigations, until a coiled wire became available for studying a uniform magnetic field at a measurable intensity. Indicated by a ripple pattern of iron particles, the field-lines revealed the distribution in space, which were recognized as significant for describing the nature of a magnetic field. Faraday (1832) made a number of sketches of field-lines emphasizing the significance of a distributed pattern, and he discussed the field with the concept of flux of field-lines. In fact, in the previous chapters we used his idea for electric fields for which electric charges on conductors were responsible. For a magnetic field, in contrast, we cannot argue further using electric analogies because magnetic charges are absent in nature. Nevertheless, with the concept of flux Faraday described the induction effect in magnetic fields, which lead him to establish the fundamental laws of electromagnetism. Owing to Faraday’s discovery, the flux of magnetic field-lines, as determined by induced voltages, allows a quantitative description of the magnetic field with no responsible magnetic charges.

Natural magnets played the central role in magnetic studies in early days, leading to the Ampère law of a magnetic field, which was discovered by using iron magnets. Characterized as N- and S-poles, magnetic charges were postulated by analogy to electric charges. However, it was only after Heisenberg’s quantum theory (1928) that the origin of magnetism was attributed to correlated electronic spin angular momenta in materials.

In a uniform solenoid of the cross-sectional area S, number of turns N, and length l , the number of field-lines is given by the flux Φ = N BS , where B = μ_o H and H = N I / l if a current I is flowing through it.

Static charges and steady currents are responsible for electrostatic and steady magnetic fields, respectively. A static field arises from a charge Q at rest, whereas, for a steady field, a constant current I = d Q /d t is responsible; both represent states of the charge carriers relative to the observer. However, charge and current are not exclusive, and both electric and magnetic fields can be observed as related with time-dependent Q and I , as discussed for magnetic induction.