# R e v or c j 1 210 this is the potential difference

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r E [ V or C J 1 - ] (2.10) This is the potential difference between the two points, which depends only on the values of the potential at the two end points of the path. In particular for a closed path of integration, such as the one displayed in Figure 2.3, one gets Figure 2.3: Closed path of integration from field a P to field point a b P P = 0 d d d cl cl cl = - - = = - b b a a V V V V V V l l r r r E [ V or C J 1 - ] (2.11) Applying Stoke’s theorem to the LHS shows that ( ) ( ) 0 d d d cl = × = × - V V a a a a r r r E where a is the area of any surface bounded by the closed path. For any a , these arbitrary surface integrals would agree with the null result on the RHS only if the integrands vanish, that is 0 E r r × = × - V [ 2 m V - ] (2.12) This is the differential form of (2.11). We refer to this as the zero second order spatial derivative of the electrostatic scalar potential. Either form constitutes a foundation of the Faraday-Lenz’s law . Equation (2.12) can be verified by taking the curl of (2.5) and then applying equation (1.68). 2.4 Calculation of the electric field and electric scalar potential The integrals of equations (2.1) and (2.8b), = 2 0 ˆ d 4 1 R q R E πε r (2.13) = = R R R q V d 4 1 0 πε (2.14) ( ) 0 , 0 , 0 O a b P P = a b r r r r = path Closed
4 can be used to directly evaluate the electric field E r and electric scalar potential V at P due to a known electric scalar charge distribution. As 0 = V at = R , this is used as the reference limit of integration. When the electric field E is known and equation (2.11) is rewritten as - = = - b a b a b V V V l r r d d a E (2.15) it serves as an indirect method of evaluating the electric scalar potential difference between any two points, and the electric scalar potential at any point when the other point is set at = R . 2.5 Five density types for electric scalar charge arrays 2.5.1 Electric scalar charge arrays and their cross-sections Continuous charge distribution in space is of three kinds: on a line of length l , on a surface of area a , or in a 3-dimensional space of volume v . Figure 2.4 shows our models of spatial distributions of an elemental electric scalar charge q d spread on a line of length l d , over a surface of area a d and within a space of volume v d . These distributions have the respective Figure 2.4: Line l d λ , surface a d σ , volume v d ρ electric scalar charge elements on longitudinal line l d and transverse point P , line l d , surface a d of sizes 1 , l d , a d . arbitrary cross ( × )-sections of a point × P , a line of length × l and a surface of area × a , as well as spatial line λ , surface σ and volume ρ scalar charge densities, which can be signified and associated by line a along spread , d d d l l λ λ λ = q (2.16a) surface a over spread , d d d d d d d d l l l r r σ λ σ σ σ = × = = × l l a q (2.16b) volume a in spread , d d d d d d d d l l r r ρ λ ρ ρ ρ = = = × a l a v q (2.16c) Here the longitudinal line , the tangential surface and the volume type of spaces, over which a q d