What is the equilibrium angle - dipoles?

[It's me]

Homework Statement

Two coplanar dipoles are oriented as shown below.

If θ is fixed, what is the equilibrium angle θ' ?

Homework Equations

The torque exerted by dipole P on dipole P' is given by $$\vec{N'}=\vec{P'}\times\vec{E}$$ where vector E is the electric field.

The Attempt at a Solution

I think $$\vec{E}(r, \theta)=\frac{P}{4\pi\epsilon_0r^3}(2\cos\theta\hat{r}+\sin\theta\hat{\theta})$$ where P is the magnitude of dipole P, and $$\vec{P'}=P'\cos{\theta'}\hat{r}+P'\sin{\theta'}\hat{\theta}$$ so $$\vec{N'}=\frac{PP'}{4\pi\epsilon_0r^3}(\cos{\theta'}\sin{\theta}-2\sin{\theta'}\cos{\theta})\hat{\phi}$$ and the equilibrium angle would be such that the torque is zero. However, that gives me $$\theta'=\tan^{-1}(\frac{\tan{\theta}}{2})$$, and I was expecting an answer more like θ'=180-θ.

Am I doing something wrong?

 

[TSny]

Your answer looks correct to me. Note that the torque is zero when $\vec{P}'$ is parallel to $\vec{E}$. So, you just need to find the direction of $\vec{E}$.

Why were you expecting θ'=180-θ? To gain confidence that your answer might be correct, check some limiting cases such as θ = 0 and θ = 90^o.

 

[It's me]

I was expecting that answer because of the fact that dipoles tend to counter-align with each other

 

[TSny]

If you allow both θ and θ' to adjust themselves to give minimum potential energy for the system of two dipoles, the dipoles will align parallel to each other and parallel to the line connecting them. That is, you find that the lowest energy is θ = θ' = 0 or θ = θ' = 180^o.

If you fix θ at 90^o, then the other dipole will orient itself anti-parallel to the first dipole to achieve minimum energy (i.e., θ' = 90^o). But the overall energy will not be as low as the θ = θ' = 0 case.

For a system of many dipoles in a lattice, you get interesting patterns.
See http://www.evsc.net/projects/dipole-spin-system and https://vimeo.com/album/16185