Physics · Ch 1 — Electrostatics
Torque experienced by an electric dipole in the uniform electric field
Torque experienced by an electric dipole in the uniform electric field
Consider a dipole of moment p placed inside a uniform external electric field E, whose field lines are parallel and equally spaced everywhere. The +q charge experiences a force +qE and the -q charge an equal and opposite force -qE; because the field is uniform, these two forces are exactly equal in magnitude and opposite in direction, so the net translational force on the dipole is zero -- a dipole in a perfectly uniform field does not get pushed anywhere. However, because the two equal-and-opposite forces act at two different points (the two ends of the dipole, separated by 2a), they constitute a couple, which produces a net torque even though the net force is zero. Working through the geometry gives the torque magnitude tau = pE sin(theta), where theta is the angle between the dipole moment p and the field E, or in vector notation tau = p x E. This torque acts to rotate the dipole toward alignment with the field: it vanishes at theta = 0 degrees (p parallel to E, a stable equilibrium orientation) and also at theta = 180 degrees (p anti-parallel to E, an unstable equilibrium -- any small perturbation grows away from this orientation rather than being restored), and is maximum, equal to pE, when p is perpendicular to E (theta = 90 degrees). If instead the external field is non-uniform, the forces on the two charges no longer have exactly equal magnitude (since the field strength differs at their two slightly different locations), so in addition to the t …
What this figure shows. A dipole with +q and -q at either end is drawn tilted at an angle theta to a set of parallel, evenly-spaced field lines representing a uniform external field E. Equal-magnitude force vectors qE are drawn on each charge, both parallel to E, but because they act at different points (offset by the dipole's length) they form a couple whose net effect is a rotational torque tending to twist the dipole so that its moment vector p swings …
What this figure shows. A dipole is placed in a region where the field lines are drawn closer together (denser, stronger field) on one side and more spread apart (weaker field) on the other, so that the force qE pulling on the charge nearer the strong-field side is drawn noticeably longer than the oppositely-directed force on the charge nearer the weak-field side. Because the two forces no longer have equal magnitude, they do not cancel to zero net force the way they do in a uniform field; the dipole experiences both a torque and a net translational force pulling it toward the region of stronger field, which is the physical mechanism used later to explain why a charg …
Worked out. In a uniform field E, the +q charge experiences a force +qE and the -q charge experiences an equal and opposite force -qE; because the field is uniform these two forces are equal in magnitude and opposite in direction, so they exert zero net translational force on the dipole as a whole -- but since they act at the two different ends of the dipole rather than at the same point, they form a couple (a torque-producing pair). Taking moments about the dipole's midpoint and working through the geometry of the angle theta between p and E, the magnitude of the net torque comes out to tau = pE sin(theta), or in vector form tau = p x E, directed so as to rotate the dipole toward alignment with the field. The torque is zero when theta = 0 (p already aligned with E, a stable equilibrium) and also zero at theta = 180 degrees (p exactly anti-aligned with E, an unstable equilibrium, since any small disturbance grows rather than being restor …