1. The problem statement, all variables and given/known data Show that the force caused by external electric field on a charge distribution is given by: [tex]F=qE(0)+(p\cdot \nabla)E(0)+...[/tex] (it's enough to show the first two terms), where E(0) is the electric field in the origin which we choose to develop the expression around it. p is the electric dipole. 2. Relevant equations 3. The attempt at a solution So, [tex]F=\int d^3 r E(r)\rho(r)[/tex], from integration by parts I get: [tex]F= r \rho(r) E(r) -\int r[\nabla(\rho(r)) E(r) +div(E(r))\rho(r)]d^3 r[/tex] Now I develop E(r) around E(0) by taylor expansion: [tex]E(r)=E(0)+div(E(0))r+...[/tex], so div(E(r))=div(E(0))+... Now the first term in F is zero by boundary conditions, Now I know that [tex]p=\int d^3 r \rho(r) r[/tex], but if I plug div(E(r))=div(E(0))+... to the third term I get something else than the second term in the question, the first term I do get, because: [tex]\int r \nabla(rho) d^3 r=r\rho(r) - \int \rho(r) d^3r[/tex] and plugging it back and inserting E(r)=E(0)+div(E(0))r... I get that the third term is: div(E(0))p, and the second term is -2div(E(0))p+-qE(0), so in summary I get: qE(0)+p.div(E(0))+... But it's not the same as what I need to show here, but I don't see where did I got this wrong, can someone enlight me?