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Compact stars can have either hadronic matter or can have exotic states ofmatter like strange quark matter or color superconducting matter. Stars alsocan have a quark core surrounded by hadronic matter, known as hybrid stars(HS). The HS is likely to have a mixed phase in between the hadron and quarkphase. Observational results suggest huge surface magnetic field in certainneutron stars (NS) called magnetars. Here we study the effect of strongmagnetic field on the respective EOS of matter under extreme conditions. Wefurther study the hadron-quark phase transition in the interiors of NS givingrise to hybrid stars (HS) in presence of strong magnetic field. The hadronicmatter EOS is described based on relativistic mean field theory and we includethe effect of strong magnetic fields leading to Landau quantization of thecharged particles. For the quark phase we use the simple MIT bag model. Weassume density dependent bag pressure and magnetic field. The magnetic fieldstrength increases going from the surface to the center of the star. Weconstruct the intermediate mixed phase using Glendenning conjecture. Themagnetic field softens the EOS of both the matter phases. The effect ofmagnetic field is insignificant unless the field strength is above $10^{14}$G.A varying magnetic field, with surface field strength of $10^{14}$G and thecentral field strength of the order of $10^{17}$G has significant effect onboth the stiffness and the mixed phase regime of the EOS. We finally study themass-radius relationship for such type of mixed HS, calculating their maximummass, and compare them with the recent observation of pulsar PSR J1614-2230,which is about 2 solar mass. The observations puts a severe constraint on theEOS of matter at extreme conditions. The maximum mass with our EOS can reachthe limit set by the observation.

Mixed phase in a compact star with strong magnetic field