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The magnetic dipole transitions between the vector mesons $B_c^*$ and theirrelevant 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) ofthe $B_c$ family are interesting. To see the `hyperfine' splitting due tospin-spin interaction is an important topic for understanding the spin-spininteraction and the spectrum of the the $(c\bar{b})$ binding system. Theknowledge about the magnetic dipole transitions is also very useful foridentifying the vector boson $B_c^*$ mesons experimentally, whose masses arejust slightly above the masses of their relevant pseudoscalar mesons $B_c$accordingly. Considering the possibility to observe the vector mesons via thetransitions at $Z^0$ factory and the potentially usages of the theoreticalestimate on the transitions, we fucus our efforts on calculating the magneticdipole transitions, i.e. precisely to calculate the rates for the transitionssuch as decays $B_c^*\to B_c\gamma$ and $B_c^*\to B_c e^+e^-$, and particularlywork in the Behte-Salpeter framework. In the estimate, as a typical example, wecarefully 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.

The magnetic dipole transitions in the $(c\bar{b})$ binding system