[abridged] The role of magnetic fields in the process of star formation is a matter of continuous debate. Clear observational proof of the general influence of magnetic fields on the early phase of cloud collapse is still pending. First results on Bok globules with simple structures indicate dominant magnetic fields across large spatial scales (Bertrang+2014). The aim of this study is to test the magnetic field influence across Bok globules with more complex density structures. We apply near-infrared polarimetry to trace the magnetic field structure on scales of 10^4-10^5au in selected Bok globules. The combination of these measurements with archival data in the optical and sub-mm wavelength range allows us to characterize the magnetic field on scales of 10^3-10^6au. We present polarimetric data in the near-infrared wavelength range for the three Bok globules CB34, CB56, and [OMK2002]18, combined with archival polarimetric data in the optical wavelength range for CB34 and CB56, and in the sub-millimeter wavelength range for CB34 and [OMK2002]18. We find a strong polarization signal (P>2%) in the near-infrared and strongly aligned polarization segments on large scales (10^4-10^6au) for all three globules. This indicates dominant magnetic fields across Bok globules with complex density structures. To reconcile our findings in globules, the lowest mass clouds known, and the results on intermediate (e.g., Taurus) and more massive (e.g., Orion) clouds, we postulate a mass dependent role of magnetic fields, whereby magnetic fields appear to be dominant on low and high mass but rather sub-dominant on intermediate mass clouds.
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