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Since its introduction over a decade ago, the spun birefringent single-mode optical fiber is increasingly being considered for current-sensing applications. In this paper, we study the behavior of such a fiber when it is bent into a sensing coil. We show that bending weakens the polarization-holding capability of the spun birefringent fiber and causes the average polarization state to slowly oscillate along the fiber, potentially degrading the performance of the current sensor. The amplitude of this oscillation increases with tighter bending. Both the amplitude and the period of this oscillation, which are important parameters for designing current-sensing coils, are calculated by applying an appropriate perturbation theory. An experiment confirming the theory is also reported.