To define the surface charge density, mount a Perpendicular to that surface, the charge density is a one-dimensional The one-dimensional singularity in charge density is representedīy the surface charge density. In general, l is a function of position along the 1.3.4, the line charge per unit length l (the lineĬharge density) is defined as the limit where the cross-sectionalĪrea of the volume goes to zero, goes to infinity, but theĮlement having cross-section da used to define line charge In terms of the filamentary volume shown inįig. It is the mathematical abstraction representing a thinĬharge filament. In the limit where the volume 4 R 3 /3 goes to zero, while q =Ī line charge density represents a two-dimensional singularity inĬharge density. The point charge can be pictured as a small charge-filled region, Point charge, (b) line charge, (c) surface charge.Ī point charge is the limit of an infinite charge density These singular distributions are defined in terms of integrals.įigure 1.3.3 Singular charge distributions: (a) Like the temporal impulse function of circuit theory, Point, line, and surface distributions of, as illustrated inįig. Three spatial analogues of the temporal impulse function. Only one "dimension." In three-dimensional field theory, there are The one independent variable, namely time, circuit theory is Increase in proportion to r 2, but the enclosed charge remains constant.Ĭircuit theory are impulse and step functions. Figure 1.3.2 illustrates this dependence, as wellĪs the exterior field decay. Increases like the square of the radius, the enclosed charge increasesĮven more rapidly. Increases with the square of the radius because even though the Inside the spherical charged region, the radial electric field With the volume and surface integrals evaluated in (5) and Thus, for the spherical surface at the arbitrary radius r, To evaluate the left-hand side of (1), note that A similar argument shows that E also is zero.Īnd it follows that for a spherical volume having arbitrary radius r, However, the rotation leaves the source of thatįield, the charge distribution, unaltered. Rotation of the system about theĪxis shown results in a component of E in some new direction At a given point, the components ofĪppear as shown in Fig. Indeed, suppose that in addition to this r component the field (b) Axis of rotation forĭemonstration that the components of E transverse to the radial Shows that the only possible component of E is radial.įigure 1.3.2 (a) Spherically symmetric chargeĭistribution, showing radial dependence of charge density andĪssociated radial electric field intensity. An argument based on the spherical symmetry In the spherical coordinate system of Fig. AnĮxample is the distribution of charge density Not directly useful unless there is a great deal of symmetry. Given the charge and current distributions, the integral lawsįully determine the electric and magnetic fields. The following example illustrates the mechanics of carrying outĮxample 1.3.1. Out of any regionĬontaining net charge, there must be a net displacement flux. The quantity o E is called the electric displacement flux density and, Summation over this same closed surface of the differentialĬontributions of flux o E d a. The net charge enclosed by the surface S. To express Maxwell's equations in SI units. = 8.854 x 10 -12 farad/meter, is an empirical constant needed With the surface normal defined as directed outward, the volume is Volume V that is enclosed by a surface S is related to the net The first of Maxwell's equations toīe considered, Gauss' law, describes how the electric field This chapter are concerned with the reaction of the moving charges 1.1 expresses the effect ofĮlectromagnetic fields on a moving charge. Gauss' Integral Law of Electric Field Intensity
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