Physics · Ch 1 — Electrostatics
Electrostatic potential at a point due to an electric dipole
Electrostatic potential at a point due to an electric dipole
Consider a dipole with charges +q and -q separated by a small distance 2a, and a field point P located a distance r from the dipole's midpoint O, where the line OP makes an angle theta with the dipole axis. Applying the point-charge potential formula to each charge separately and adding them as scalars (V = kq/r1 - kq/r2, where r1 and r2 are the slightly different distances from P to +q and -q respectively) and simplifying for the case r much greater than a gives the general result V(r, theta) is approximately equal to (1/4 pi epsilon0) x (p cos theta)/r^2, where p is the dipole moment magnitude. Two special cases follow immediately: along the axial line, theta = 0 (on the +q side) gives the maximum positive potential V_axial is approximately kp/r^2, while theta = 180 degrees (on the -q side) gives the maximum-magnitude negative potential V_axial is approximately -kp/r^2; on the equatorial line (the perpendicular bisector, where theta = 90 degrees), cos(theta) = 0 and the potential is exactly zero everywhere -- a positive test charge brought in from i …
What this figure shows. A dipole with -q and +q separated by 2a is drawn with its midpoint at O, and a general field point P is marked at distance r from O, with the angle theta between the line OP and the dipole axis explicitly labelled. Two shorter dashed lines are drawn from P to each individual charge, showing the slightly different distances r1 (to +q) and r2 (to -q) that must be combined -- by the ordinary scalar addition of potentials -- to arrive at the dipole's net pote …