# Is it possible for two polarizable bodies to induce dipoles in each other in the absence of an external electric field?

If there exist two initially neutral bodies (say atoms) some distance apart, with no external electric field applied, can they induce dipoles within each other?

• Objects can and do induce temporary dipoles in each other. This is the origin of the London dispersion force. Are you asking if there is a state where permanent dipoles exist that has a lower energy then the unpolarised state? – John Rennie May 19 '15 at 9:53

Indeed they can. This can even be seen in a classical set of equations if the interaction between the two polarizable objects is strong enough. Take a simple 1d problem with two polarizable point particles in a line. For low polarizations, we assume a linear relation:

$$p_1 = \alpha E(r_1) = \alpha (E_2 + E_{ext}) = \alpha (k p_2 + E_{ext})$$

$$p_2 = \alpha (k p_1 + E_{ext})$$

where the $p$s are dipole moments, $\alpha$ is the polarizability, $E$s are fields (i.e., $E_1$ is the field at $p_2$'s position due to the polarization at $p_1$), and k relates the polarization of $p_1$ to the field it produces at $p_2$ (scales like the inverse distance cubed). This is solved by

$$p_1 = p_2 = \alpha \frac{\alpha k +1}{1-\alpha^2 k^2} E_{ext}$$

It's clear that something strange happens as $\alpha k$ approaches 1: polarization diverges. This is because we didn't include the fact that polarization has a finite limit. Let's call the maximum polarizability $p_m$. We can include this in our model as such:

$$p_2 = \alpha (k p_1 + E_{ext})(1-\frac{p_2^2}{p_m^2})$$

and vice versa for $p_1$. Solving this is tedious, but doable. What you find is that for $\alpha k < 1$, there is a single real solution. However, as $\alpha k$ increases, this solution becomes unstable and two new solutions emerge that form a hysteresis loop in $E_{ext}$. In other words, the two objects together become "ferro-electric" with a remanent dipole moment $\pm \frac{\alpha k -1}{\alpha k} p_m$ that persists in the absence of an external field.