Two point dipoles of dipole moment $$\vec{p_1}$$ and $$\vec{p_2}$$ are at a distance x from each other and $$\vec{p_1} \| \vec{p_2}$$. The force between the dipoles is :

The interaction potential energy ($$U$$) between two point dipoles $$\vec{p}_1$$ and $$\vec{p}_2$$ separated by a distance $$r$$ is given by:

$$U = \frac{1}{4\pi\epsilon_0 r^3} \left[ \vec{p}_1 \cdot \vec{p}_2 - 3(\vec{p}_1 \cdot \hat{r})(\vec{p}_2 \cdot \hat{r}) \right]$$

In the axial configuration where both dipoles are collinear and point along the line joining them (i.e., $$\vec{p}_1 \parallel \vec{p}_2 \parallel \vec{r}$$):

$$\vec{p}_1 \cdot \vec{p}_2 = p_1 p_2$$, $$\vec{p}_1 \cdot \hat{r} = p_1$$, $$\vec{p}_2 \cdot \hat{r} = p_2$$

$$U = \frac{1}{4\pi\epsilon_0 x^3} \left[ p_1 p_2 - 3(p_1)(p_2) \right]$$

$$U = -\frac{2 p_1 p_2}{4\pi\epsilon_0 x^3}$$

$$F = -\frac{dU}{dx}$$

$$F = -\frac{6 p_1 p_2}{4\pi\epsilon_0 x^4}$$ (the $$-ve$$ sign indicates attraction)