  A theoretical exploration and an analytical model for the
electro-magneto-hydrodynamics (EMHD) of leaky dielectric liquid droplets,
suspended in an immiscible confined fluid domain has been presented. The
analytical solution for the system, under small deformation approximation, in
creeping flow regime, has been put forward. Study of the droplet deformation
suggests that its temporal evolution is exponential, and dependents on the
electric and magnetic field interaction. Further, the direction of the applied
magnetic field with respect to the electric field decides whether the
contribution of magnetic forces opposes or aids the interfacial net electrical
force due to the electric field. Validation of the proposed model at the
asymptotic limits of vanishing magnetic field show that the model accurately
reduces to the case of transient electrohydrodynamic model. We also propose a
magnetic discriminating function to quantify the steady-state droplet
deformation in the presence of interacting electric and magnetic fields. The
change of droplets from spherical shape to prolate, and oblate spheroids,
correspond to magnetic discriminating function >0 and <0 regimes, respectively.
It is shown that with the aid of low magnitude magnetic field, a substantial
augmentation in the deformation parameter, and the associated EMHD circulation
within and around the droplet is achieved. The analysis also reveals the
deformation lag and specific critical parameters that aid or suppressed this
lag behaviour; discussed in terms of relevant non-dimensional parameters.
