  For a smooth bounded domain $G\subset\mathbb{R}^3$ we consider maps
$n\colon\mathbb R^3\setminus G\to\mathbb S^2$ minimizing the energy
$E(n)=\int_{\mathbb R^3\setminus G}|\nabla n|^2 +F_s(n_{\lfloor\partial G})$
among $\mathbb S^2$-valued map such that $n(x)\approx n_0$ as $|x|\to\infty$.
This is a model for a particle $G$ immersed in nematic liquid crystal. The
surface energy $F_s$ describes the anchoring properties of the particle, and
can be quite general. We prove that such minimizing map $n$ has an asymptotic
expansion in powers of $1/r$. Further, we show that the leading order $1/r$
term is uniquely determined by the far-field condition $n_0$ for almost all
$n_0\in\mathbb S^2$, by relating it to the gradient of the minimal energy with
respect to $n_0$. We derive various consequences of this relation in physically
motivated situations: when the orientation of the particle $G$ is stable
relative to a prescribed far-field alignment $n_0$; and when the particle $G$
has some rotational symmetries. In particular, these corollaries justify some
approximations that can be found in the physics literature to describe nematic
suspensions via a so-called electrostatics analogy.
