Editing Content :: Admin Scraper

Chapter

Content type

Position

Value

Physics discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field. Further, the electric field due to the comb is not uniform. This non-uniformity of the field makes a dipole to experience a net force on it. In this situation, it is easily seen that the paper should move in the direction of the comb! 1.12 C ONTINUOUS C HARGE D ISTRIBUTION We have so far dealt with charge configurations involving discrete charges q1, q2, ...n q . One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus. For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions. For example, on the surface of a charged conductor, it is impractical to specify thechargedistributionintermsofthelocationsofthemicroscopiccharged constituents. It is more feasible to consider an area element S (Fig. 1.21) on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge Q on that element. We then define a surface charge density at the area element by Q σ = (1.23) S We can do this at different points on the conductor and thus arrive at a continuous function , called the surface charge density. The surface charge density so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*.represents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element S which, as said before, is large microscopically but small macroscopically. The 2 units for are C/m . FIGURE 1.21 Similarconsiderationsapplyforalinechargedistributionandavolume Definition of linear, charge distribution. The linear charge density of a wire is defined by surface and volume charge densities. λ = Q (1.24) In each case, the l element ( l, S, V) where l is a small line element of wire on the macroscopic scale that, chosen is small on however, includes a large number of microscopic charged constituents, the macroscopic and Q is the charge contained in that line element. The units for are scale but contains C/m. The volumechargedensity (sometimes simply called charge density) a very large number is defined in a similar manner: of microscopic constituents. Q ρ = (1.25) V where Q is the charge included in the macroscopically small volume element V that includes a large number of microscopic charged constituents. The units for are C/m . The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics. When we refer to 28 * At the microscopic level, charge distribution is discontinuous, because they are discrete charges separated by intervening space where there is no charge. Reprint 2026-27