Question

2. Dipole-dipole force The charge distribution where two equal but opposite charges are separated by a fixed distance a dipole and is very common and important in nature. Consider two dipoles, each consisting of charges ±q separated by a distance δ. The axes of two dipoles are parallel and their midpoints are separated by a distance r. One is inverted compared to the other. See the figure tq +9 (a) Calculate the force exerted on each dipole by the other b) Find an approximate expression for t he force that each dipole exerts on the other if. The magnitude of the force should be in the form FCr" Note that n is not necessarily an integer. C is some combination of given quantities and numerical factors that you will have to find. The answer is not zero, but should approach zero as r -> oo (why?) Hint: When lel « 1, the two largest terms in the Taylor erpansion of (1 +e)k are

Answer 1

F^bar = q^2/4 pi epsilon_0 [- 1/r^2 - 1/r^2 + 2r/(r^2 + delta^2)^3/2] i^^+ q^2/4 pi epsilon_0 [delta/(r^2 + delta^2)^3/2 - delta/(r^2 + delta^2)^3/2] j^^F^bar = q^2/4 pi epsilon_0 [2r/(r^2 + delta^2)^3/2 - 2/r^2] i^^If r > > delta, delta/r < < 1 (r^2 + delta^2)^-3/2 = r^-3 (1 + delta^2/r^2)^-3/2 = r^3 (1 - 3 delta^2/2 r^2) F^bar = q^2/4 pi epsilon_0 [2/r^2 (1 - 3 delta^2/2 r^2) - 2/r^2] i^^ Vertical-bar F^bar Vertical-bar = q^2/4 pi epsilon_0 (3 delta^2/r^4), F = Cr^n C = 3 q^2 delta^2/4 pi epsilon_0, n = 4