4.3 The Charge Density of a Point Electric Dipole The text used Poisson's equation to show that the charge density of a
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4.3 The Charge Density of a Point Electric Dipole The text used Poisson's equation to show that the charge density of a point electric dipole with moment p located at the point ro is pp(r) = -p. Vår - ro). (a) Derive the given formula for pp(r) using a limiting process analogous to the one used in the text to find its electrostatic potential. (b) Show that the formula given above for pp(r) is correct by demonstrating that it produces the expected dipole potential when inserted into the Coulomb integral for y(r).
+g/s r-sb-ro —r-r- 0-9/s The charge density of this system is - ro - sb) - (r-ro]. For the point electric dipole, we are interested in the limit as s + 0. Therefore, we expand the argument of the delta function of the positive charge to get PD (r) = lim ? [(r - ro) – sb. 78(r - ro) +...-5(r - ro)]. All the higher terms in the expansion are proportional to s, s?, etc. and thus go to zero in the limit. Therefore, with p= qb, we get the advertised result, pd(r) = -p 98(r-ro).
(b) The suggested charged density is correct because the electrostatic potential it produces (1) The suggested charged density to co) e corda av procent - parte. Saw Pala EP (r) = 13.PD(r) Ir-r' ,V'8(r' - ro) r-r' 471 EO - rol PV P 4T60 r - rol