The existence of various types of (fast) magnetoacoustic modes in different frequency regimes in a magnetized dusty plasma consisting of electrons, ions and dust particles is investigated. The analysis is carried out using an effective two-fluid MHD-like model which allows for the non-frozen motion of the component fluids. For frequencies much smaller than the dust particle gyro-frequency, we obtain a magnetoacoustic mode that is a generalization of the usual compressional fast hydromagnetic wave in an electron-ion plasma. In the higher-frequency regimes, we show the existence of two new types of modes called 'Dust-magnetoacoustic waves'. Both modes are accompanied by compressional magnetic field and plasma number density perturbations, and are the electromagnetic generalizations of the 'dust-acoustic waves in an unmagnetized dusty plasma with thermal electrons and ions. For a two-component plasma, all three modes degenerate into the same fast magneto-acoustic wave found in the usual electron-ion plasmas. We also obtain another novel type of magneto-acoustic mode called a 'dust-ion-magneto-acoustic wave', which is an electromagnetic generalization of the dust—ion-acoustic wave. The dispersion relations as well as the frequency regimes for the existence of the various modes are explicitly obtained. An alternative derivation of the relevant governing equations using an approach similar to that employed in so-called 'electron magnetohydrodynamics' (EMHD) is also presented.
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