  The magnetic dipole transitions between the vector mesons $B_c^*$ and their
relevant pseudoscalar mesons $B_c$ ($B_c$, $B_c^*$, $B_c(2S)$, $B_c^*(2S)$,
$B_c(3S)$ and $B_c^*(3S)$ etc, the binding states of $(c\bar{b})$ system) of
the $B_c$ family are interesting. To see the `hyperfine' splitting due to
spin-spin interaction is an important topic for understanding the spin-spin
interaction and the spectrum of the the $(c\bar{b})$ binding system. The
knowledge about the magnetic dipole transitions is also very useful for
identifying the vector boson $B_c^*$ mesons experimentally, whose masses are
just slightly above the masses of their relevant pseudoscalar mesons $B_c$
accordingly. Considering the possibility to observe the vector mesons via the
transitions at $Z^0$ factory and the potentially usages of the theoretical
estimate on the transitions, we fucus our efforts on calculating the magnetic
dipole transitions, i.e. precisely to calculate the rates for the transitions
such as decays $B_c^*\to B_c\gamma$ and $B_c^*\to B_c e^+e^-$, and particularly
work in the Behte-Salpeter framework. In the estimate, as a typical example, we
carefully investigate the dependance of the rate $\Gamma(B_c^*\to B_c\gamma)$
on the mass difference $\Delta M=M_{B_c^*}-M_{B_c}$ as well.
