  Rotating bodies in General Relativity produce frame dragging, also known as
the {\it gravitomagnetic effect} in analogy with classical electromagnetism. In
this work, we study the effect of magnetic field on the gravitomagnetic effect
in neutron stars with poloidal geometry, which is produced as a result of its
rotation. We show that the magnetic field has a non-negligible impact on frame
dragging. The maximum effect of the magnetic field appears along the polar
direction, where the frame-dragging frequency decreases with increase in
magnetic field, and along the equatorial direction, where its magnitude
increases. For intermediate angles, the effect of the magnetic field decreases,
and goes through a minimum for a particular angular value at which magnetic
field has no effect on gravitomagnetism. Beyond that particular angle
gravitomagnetic effect increases with increasing magnetic field. We try to
identify this `null region' for the case of magnetized neutron stars, both
inside and outside, as a function of the magnetic field, and suggest a thought
experiment to find the null region of a particular pulsar using the frame
dragging effect.
