  In nonequilibrium systems, the relative fluctuation of a current has a
universal trade-off relation with the entropy production, called the
thermodynamic uncertainty relation (TUR). For systems with broken time reversal
symmetry, its violation has been reported in specific models or in the linear
response regime. Here, we derive a modified version of the TUR analytically in
the overdamped limit for general Langevin dynamics with a magnetic Lorentz
force causing time reversal broken. Remarkably, this modified version is simply
given by the conventional TUR scaled by the ratio of the reduced effective
temperature of the overdamped motion to the reservoir temperature, permitting a
violation of the conventional TUR. Without the Lorentz force, this ratio
becomes unity and the conventional TUR is restored. We verify our results both
analytically and numerically in a specific solvable system.
