Josephson Current and Dissipation of the $\mathit{c}$-Axis Transport in Magnetic Fields in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\mathit{\delta }}$

We have measured the $c$-axis resistivity, magnetoresistance, current-voltage characteristics, and the maximum Josephson current $I_{\mathrm{c}}$ in magnetic fields $H$ parallel to the $c$ axis for a very thin mesa of a ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ crystal, which is equivalent to 20 tunnel Josephson junctions in series. It is found that $I_{\mathrm{c}}$ decreases proportionally to $H^{-1\mathrm{}}$, which agrees with a recent theory and Josephson plasma resonance experiments. This provides a basis for the straightforward application of the Ambegaokar-Halperin theory to the dissipation mechanism of the $c$-axis transport.