  We investigate time evolution of $S$-wave charmonium populations under a
time-dependent homogeneous magnetic field and evaluate survival probabilities
of the low-lying charmonia to the goal of estimating the magnetic field
strength at heavy-ion collisions. Our approach implements mixing between
different spin eigenstates and transitions to radially excited states. We show
that the survival probabilities can change even by an extremely short magnetic
field. Furthermore, we find that the survival probabilities depend on the
initial spin states. We propose the sum of the survival probabilities over spin
partners as an observable insensitive to the initial states. We also find that
the sum can be approximately given as a function of $\sigma B_0^2$ with a
duration time $\sigma$ and the maximum strength of the magnetic field $B_0$.
