Thus the function ρ cannot be continuous in the way

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Thus the function ρ cannot be continuous in the way a mathematician might prefer. When we let our region V i shrink down in the course of demonstrating the differential form of Gauss’s law, we know as physicists that we musn’t let it shrink too far. That is awkward perhaps, but the fact is that we make out very well with the continuum model in large-scale Figure 2.29. (a) How electric charge density, electric potential, and electric field are related. The integral relations involve the line integral and the volume integral. The differential relations involve the gradient, the divergence, and div · grad (equivalently 2 ), the Laplacian operator. The charge density ρ is in coulomb/meter 3 , the potential φ is in volts, the field E is in volt/meter, and all lengths are in meters. (b) The same relations in Gaussian units. The charge density ρ is in esu/cm 3 , the potential φ is in statvolts, the field E is in statvolt/meter, and all lengths are in centimeters. E = −∇ f E r f f = E d s E = r 2 r r d u E lec tri c p o t en tia l f = r r d u r = 2 f 1 4 π C h arg e d en sity E lec tri c fi el d ( b ) r = 1 4 π E E = −∇ f E r = 0 E r = 0 2 f f = E d s r f E = 1 r 2 r r d u 4 π 0 E lec tri c p o t en tia l f = 1 r r d u 4 π 0 C h arg e d en sity E lec tri c fi el d (a)
90 The electric potential electrical systems. In the atomic world we have the elementary particles, and vacuum. Inside the particles, even if Coulomb’s law turns out to have some kind of meaning, much else is going on. The vacuum, so far as electrostatics is concerned, is ruled by Laplace’s equation. Still, we cannot be sure that, even in the vacuum, passage to a limit of zero size has physical meaning. C d s F C 1 B C 2 a i (a) ( b ) ( c ) Figure 2.30. For the subdivided loop, the sum of all the circulations i around the sections is equal to the circulation around the original curve C . 2.14 The curl of a vector function Note: 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 the curl will be the demonstration that an electrostatic field is characterized by curl E = 0 , as explained in Section 2.17. The reason we are intro- ducing the curl now is that the derivation so closely parallels the above derivation of the divergence. We developed the concept of divergence, a local property of a vector field, by starting from the surface integral over a large closed surface. In the same spirit, let us consider the line integral of some vector field F ( x , y , z ) , taken around a closed path, some curve C that comes back to join itself. The curve C can be visualized as the boundary of some surface S that spans it. A good name for the magnitude of such a closed- path line integral is circulation ; we shall use (capital gamma) as its symbol: = C F · d s . (2.77) In the integrand, d s is the element of path, an infinitesimal vector locally tangent to C (Fig. 2.30(a)). There are two senses in which C could be traversed; we have to pick one to make the direction of d s unambiguous.

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