Fragmentation of spiral arms can drive the formation of giant clumps and induce intense star formation in disc galaxies. Based on the spiral-arm instability analysis of our Paper I, we present linear perturbation theory of dynamical instability of self-gravitating spiral arms of magnetised gas, focusing on the effect of toroidal magnetic fields. Spiral arms can be destabilised by the toroidal fields which cancel Coriolis force, i.e. magneto-Jeans instability. Our analysis can be applied to multi-component systems that consist of gas and stars. To test our analysis, we perform ideal magneto-hydrodynamics simulations of isolated disc galaxies and examine the simulation results. We find that our analysis can characterise dynamical instability leading arms to fragment and form clumps if magnetic fields are nearly toroidal. We propose that dimensionless growth rate of the most unstable perturbation, which is computed from our analysis, can be used to predict fragmentation of spiral arms within an orbital time-scale. Our analysis is applicable as long as magnetic fields are nearly toroidal. Using our analytic model, we estimate a typical mass of clumps forming from spiral-arm fragmentation to be consistent with observed giant clumps $sim10^{7-8}~{rm M_odot}$. Furthermore, we find that, although the magnetic destabilisation can cause low-density spiral arms to fragment, the estimated mass of resultant clumps is almost independent from strength of magnetic fields since marginal instability occurs at long wavelengths which compensate the low densities of magnetically destabilised arms.
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