Thus the functionρcannot be continuous in the way a mathematicianmight prefer. When we let our regionVishrink down in the course ofdemonstrating the differential form of Gauss’s law, we know as physiciststhat we musn’t let it shrink too far. That is awkward perhaps, but the factis that we make out very well with the continuum model in large-scaleFigure 2.29.(a) How electric charge density, electricpotential, and electric field are related. Theintegral relations involve the line integral and thevolume integral. The differential relations involvethe gradient, the divergence, and div·grad(equivalently∇2), the Laplacian operator. Thecharge densityρis in coulomb/meter3, thepotentialφis in volts, the fieldEis in volt/meter,and all lengths are in meters. (b) The samerelations in Gaussian units. The charge densityρis in esu/cm3, the potentialφis in statvolts, thefieldEis in statvolt/meter, and all lengths are incentimeters.E=−∇fErff= −E• dsE=r2rrduElectricpotentialf=rrdur= −∇2f14πChargedensityElectric field(b)r=14π∇• EE=−∇fEr= 0∇• Er= −0∇2ff= −E• dsrfE=1r2rrdu4π0Electricpotentialf=1rrdu4π0ChargedensityElectric field(a)
90The electric potentialelectrical systems. In the atomic world we have the elementary particles,and vacuum. Inside the particles, even if Coulomb’s law turns out tohave some kind of meaning, much else is going on. The vacuum, so faras electrostatics is concerned, is ruled by Laplace’s equation. Still, wecannot be sure that, even in the vacuum, passage to a limit of zero sizehasphysicalmeaning.CdsFC1BC2ai(a)(b)(c)Figure 2.30.For the subdivided loop, the sum of all thecirculationsiaround the sections is equal tothe circulationaround the original curveC.2.14 The curl of a vector functionNote: Study of this section and the remainder of Chapter 2 can be post-poned until Chapter 6 is reached. Until then our only application of thecurl will be the demonstration that an electrostatic field is characterizedbycurlE=0, as explained in Section 2.17. The reason we are intro-ducing the curl now is that the derivation so closely parallels the abovederivation of the divergence.We developed the concept of divergence, a local property of a vectorfield, by starting from the surface integral over a large closed surface.In the same spirit, let us consider the line integral of some vector fieldF(x,y,z), taken around a closed path, some curveCthat comes backto join itself. The curveCcan be visualized as the boundary of somesurfaceSthat spans it. A good name for the magnitude of such a closed-path line integral iscirculation; we shall use(capital gamma) as itssymbol:=CF·ds.(2.77)In the integrand,dsis the element of path, an infinitesimal vector locallytangent toC(Fig. 2.30(a)). There are two senses in whichCcould betraversed; we have to pick one to make the direction ofdsunambiguous.