= Solution
Balance upward surface-tension <force> $2\pi r\gamma\cos\theta$ against the weight $\rho g\pi r^2h$ of the liquid column. The <capillary action> is
$$
h=\frac{2\gamma\cos\theta}{\rho g r}
=\frac{4\gamma\cos\theta}{\rho g D}.
$$
With perfect wetting, $\cos\theta=1$, $D=10^{-3}\ \mathrm m$, $\rho=1000\ \mathrm{kg\,m^{-3}}$ and $g=9.81\ \mathrm{m\,s^{-2}}$, \b[$h=2.94\times10^{-2}\ \mathrm m=29.4\ \mathrm{mm}$]. The diameter, not the radius, is $1\ \mathrm{mm}$. This neglects the small meniscus-volume correction, gas-density buoyancy and contact-angle hysteresis.
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