The detailed magnetic field structure of the starless dense core CB81 ( L1774 , Pipe 42 ) in the Pipe Nebula was determined based on near-infrared polarimetric observations of background stars to measure dichroically polarized light produced by magnetically aligned dust grains in the core . The magnetic fields pervading CB81 were mapped using 147 stars and axisymmetrically distorted hourglass-like fields were identified . On the basis of simple 2D and 3D magnetic field modeling , the magnetic inclination angles in the plane-of-sky and line-of-sight directions were determined to be 4 ^ { \circ } \pm 8 ^ { \circ } and 20 ^ { \circ } \pm 20 ^ { \circ } , respectively . The total magnetic field strength of CB81 was found to be 7.2 \pm 2.3 \mu { G } . Taking into account the effects of thermal/turbulent pressure and magnetic fields , the critical mass of CB81 was calculated to be M _ { cr } = 4.03 \pm 0.40 M _ { \odot } , which is close to the observed core mass of M _ { core } = 3.37 \pm 0.51 M _ { \odot } . We thus conclude that CB81 is in a condition close to the critical state . In addition , a spatial offset of 92 ^ { \prime \prime } was found between the center of magnetic field geometry and the dust extinction distribution ; this offset structure could not have been produced by self-gravity . The data also indicate a linear relationship between polarization and extinction up to A _ { V } \sim 30 mag going toward the core center . This result confirms that near-infrared polarization can accurately trace the overall magnetic field structure of the core .