How Does Dipole Interaction Energy Vary with Orientation?

[physconomic]

Homework Statement

We consider two dipoles p1 and p2 and take r to be the position vector of p2
measured from p1.

Find Uint (interaction energy)

Draw graphs showing how Uint depends upon the relative orientation of the dipoles
in the following cases:
(i) p1 is parallel to r,
(ii) p1 is perpendicular to r.

Relevant Equations

U = -p.E
UInt = 1/(4*pi*epsilon0*r^3)*[p1.p2-3(p1.r^)(p2.r^)]

Draw graphs showing how interaction energy depends upon the relative orientation of two dipoles
if
(i) p1 is parallel to r,
(ii) p1 is perpendicular to r.

I've done the first part and found the interaction energy as
UInt = 1/(4*pi*epsilon0*r^3)*[p1.p2-3(p1.r^)(p2.r^)]
which I know is correct.

I know for the perpendicular case the dot product of p1 and r would be 0 - but I'm not sure what the graph would look like.

 

[BvU]

Hello @physconomic , !

Apparently your dipoles are electric dipoles ?

which I know is correct

How so ?

For the first part, I would expect to see a graph for $U(\theta)$, or at least the expression; I see no $\theta$ ?

UInt = 1/(4*pi*epsilon0*r^3)*[p1.p2-3(p1.r^)(p2.r^)]

is hard to read. Perhaps you want to learn some $\LaTeX$ ?

$$U_\text{int}= {1\over 4\pi\varepsilon_0 r_{ij}^3}\Biggl[ \vec p_i\cdot\vec p_j - 3{(\vec p_i\cdot\vec r_{ij})(\vec r_{ij}\cdot\vec p_j) \over r_{ij}^2}\Biggr ]$$ is a lot easier on the eyes.

For the second part:
Well, what did you find for $U$ ?

 

[BvU]

@physconomic wasn't seen since Wednesday .. ?