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The curved structure of a coaxial magnetically insulated transmission line (MITL) causes it to generate a nonuniform azimuthal magnetic field, which can disturb the electron flow and lead to a higher current loss when compared with a straight MITL. In this paper, the current loss mechanism and nonuniformity of the azimuthal magnetic field inherent to curved MITLs are investigated using theoretical analysis and particle-in-cell simulations. Results show that the critical current for magnetic insulation of the curved MITL increases with nonuniformity of the azimuthal magnetic field. The magnitude of the azimuthal magnetic field of the curved MITL can be described by a cosine distribution, lower in the outer region and higher in the inner region of the curved MITL. Nonuniformity of the azimuthal magnetic field decreases as electron current increases. Finally, an empirical expression relating to the structural parameters of curved MITLs is obtained for nonuniformity of the azimuthal magnetic field, which can be used to estimate the critical current for magnetic insulation in curved MITLs. This work extends the understanding of magnetic insulation in curved coaxial transmission lines. [ABSTRACT FROM AUTHOR]