# Dipole in an electric field

## Homework Statement

I have a dipole in the electric field of a parallel plate capacitor. The plates are circles with radii r and they are at plus or minus d on the z axis. The coordinate system thus has the origin directly below/above the centres of the plates with x and y axes. I have to deduce the variation of force on the dipole as it moves along the y axis from y=-(3/2)r to y=(3/2)r for situations in which the dipole is parallel to each of the three axes.

U=-p.E
F=-∇U

## The Attempt at a Solution

I'm a little bemused by the movement from -(3/2)r to (3/2)r. Is the factor of 3/2 significant or is it chosen so that the force will safely be zero that far out from the plates? Anyway, the force would always be zero but for edge effects, so I guess they should be considered.
I should probably let the upper plate be at a higher potential than the top. If the dipole moment is parallel to the x axis, then I believe we would never have any net force, only a torque. Parallel to the z axis, we would have zero force, then increasing to some net force towards the origin as we get to the edges of the field, then dropping to zero as we enter the field, then increasing again at the other edge acting towards the origin, then back to zero again. I'm not too sure about those two anyway, but at least have an idea. Then when parallel to the y axis, I guess it would be similar to when parallel to the z axis. Anyway I'm not too sure so if somebody could have a check that would be nice :)

rude man
Homework Helper
Gold Member
Since no one else has responded:

Your cop-out answer is to point out that the only stable orientation for the dipole is when its moment is co-directional with the E field. This precludes dipole moment orientation along either the x or y axes, and also precludes dipole moment orientation in opposition to the E field. On either the x or y axis the E field is everywhere directed along the z axis.

The answer for force on the dipole for the stable case is obvious, assuming the dipole's center never leaves the y axis.

This really is a copout since even the case of co-directional dipole moment and E field represents force metastability. The case of anti-directional dipole moment and E field represents torque metastability. Dipole moment directions other than along the z axis represent unconditional torque instability.

If the dipole is to be considered somehow fixed as it moves along the y axis irrespective of moment orientation I have unfortunately nothing to contribute. I'm afraid that's what they want.