  The dynamical state of star-forming molecular clouds cannot be understood
without determining the structure and strength of their magnetic fields.
Measurements of polarized far-infrared radiation from thermally aligned dust
grains are used to map the orientation of the field and estimate its strength,
but the accuracy of the results has remained in doubt. In order to assess the
reliability of this method, we apply it to simulated far-infrared polarization
maps derived from three-dimensional simulations of supersonic
magnetohydrodynamical turbulence, and compare the estimated values to the known
magnetic field strengths in the simulations. We investigate the effects of
limited telescope resolution and self-gravity on the structure of the maps.
Limited observational resolution affects the field structure such that small
scale variations can be completely suppressed, thus giving the impression of a
very homogeneous field. The Chandrasekhar-Fermi method of estimating the mean
magnetic field in a turbulent medium is tested, and we suggest an extension to
measure the rms field. Both methods yield results within a factor of 2 for
field strengths typical of molecular clouds, with the modified version
returning more reliable estimates for slightly weaker fields. However, neither
method alone works well for very weak fields, missing them by a factor of up to
150. Taking the geometric mean of both methods estimates even the weakest
fields accurately within a factor of 2.5. Limited telescope resolution leads to
a systematic overestimation of the field strengths for all methods. We discuss
the effects responsible for this overestimation and show how to extract
information on the underlying (turbulent) power spectrum.
