  The Galactic centre supermassive black hole (SMBH), in sharp contrast with
its complex environment, is characterized by only three classical parameters --
mass, spin, and electric charge. Its charge is poorly constrained. It is,
however, usually assumed to be zero because of neutralization due to the
presence of plasma. We revisit the question of the SMBH charge and put
realistic limits on its value, timescales of charging and discharging, and
observable consequences of the potential, small charge associated with the
Galactic centre black hole. The electric charge due to classical arguments
based on the mass difference between protons and electrons is $\lesssim
10^9\,{\rm C}$ and is of a transient nature on the viscous time-scale. However,
the rotation of a black hole in magnetic field generates electric field due to
the twisting of magnetic field lines. This electric field can be associated
with induced charge, for which we estimate an upper limit of $\lesssim
10^{15}\,{\rm C}$. Moreover, this charge is most likely positive due to an
expected alignment between the magnetic field and the black-hole spin. Even a
small charge of this order significantly shifts the position of the innermost
stable circular orbit (ISCO) of charged particles. In addition, we propose a
novel observational test based on the presence of the bremsstrahlung surface
brightness decrease, which is more sensitive for smaller unshielded electric
charges than the black-hole shadow size. Based on this test, the current upper
observational limit on the charge of Sgr A* is $\lesssim 3\times 10^{8}\,{\rm
C}$.
