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Unformatted text preview: 1 PHYS809 Class 4 Notes Electrostatic energy The electrostatic energy of a pair of charges q 1 and q 2 at positions r 1 and r 2 is ( ) ( ) 1 2 2 1 2 1 1 2 1 2 1 2 1 , 4 q q U q q = =- =-- r r r r r r where ( ) 1 r is the potential due to charge q 1 . The electrostatic energy of a set of discrete charge is equal to the sum of the electrostatic energy of each pair ( ) , , 1 1 1 . 2 4 2 i j i j i j i j i j i j q q U q = =-- r r r r The factor one half is necessary due to the double counting of pairs in the sums. For a continuous distribution of charge inside a volume V , the total electrostatic energy is ( ) ( ) ( ) ( ) 3 3 3 1 1 1 . 2 4 2 V V V U d d d = = - r r r r r r r r r This can also be written in terms of the electric field by using the Poisson equation for the electrostatic potential ( ) 2 2 1 2 2 2 . 2 2 V V V S V U dV dV dV d E dV = = - = - - = - + S If we let V be all space and all charge lies inside a finite volume so that the potential is zero at infinity, then 2 1 . 2 all space U E dV = We see that the electrostatic energy density is 2 2. E 2 Multipole expansion of the potential Consider a localized charge distribution, i.e. one for which there exists a spherical surface that bounds all Consider a localized charge distribution, i....
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