〈 Berger | Dillon 〉

What's going on? • The needle has been given an electrostatic charge by rubbing it on silk • Water droplets are placed by the needle w/ velocity v≠ 0 • H₂O has a slight separation of charge (i.e a dipole moment) so it feels the needle's charge + Coulomb's law --> orbit

Are you sure this is correct? The force due to induced polarization is ~ 1/r^5 for a point charge, and ~ 1/r^3 for a line charge, neither of which admits a stable orbit.

Just to be pedantic (excuse me) the normal Coulomb force admits a stable circular orbit because the attractive force is ~ 1/r^2, while the outward "centrifugal force" is mv^2/r = L^2/(m r^3).

So if r is perturbed slightly from its stable equilibrium the orbit is either pushed back out (by the larger centrifugal force) or back in (by the larger Coulomb force).

For a dielectric sphere attracted to a point charge, the force is F = d(p E)/dr, where p ~ R^3 E is the induced dipole moment (R is the radius of the sphere). So F ~ 1/r^5, and this does not admit a stable orbit. For a dielectric sphere attracted to a line charge F ~ 1/r^3

F ~ 1/r^3 does not admit a truly stable orbit, although maybe you could get a few periods of marginal stability.