= Solution
For <separation of variables>, write $B=F(r)\Theta(\theta)$. The exterior <Laplace equation> separates into
$$
\frac{(r^2F')'}F=-\frac{(\sin\theta\,\Theta')'}{\sin\theta\,\Theta}=\lambda.
$$
With $\mu=\cos\theta$, angular regularity at the poles selects $\lambda=\ell(\ell+1)$ and the <Legendre polynomial> $P_\ell(\mu)$. The radial equation is $(r^2F')'-\ell(\ell+1)F=0$, an <Euler-Cauchy equation> with solutions $r^\ell,r^{-\ell-1}$. Decay at infinity excludes the first branch for every $\ell$, including its constant $\ell=0$ case. Therefore
$$
\boxed{B_{\rm out}(r,\theta)=\sum_{\ell=0}^\infty A_\ell r^{-\ell-1}P_\ell(\cos\theta),\qquad r>R.}
$$
The expansion uses axisymmetric <spherical harmonics>. The written scalar boundary-value problem permits the $\ell=0$ term; any additional physical exclusion of that sector must be stated separately.
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