  The three-dimensional XY model is investigated in the presence of a uniform
magnetic field applied in the $X$-direction. The nearest neighbour intraplanar
interaction is considered ferromagnetic, and the interplanar nearest neighbour
interaction is chosen to be antiferromagnetic. Starting from a high-temperature
initial random spin configuration, the equilibrium phase of the system at any
finite temperature was achieved by cooling the system using the Monte Carlo
single spin-flip Metropolis algorithm with a random updating rule. The
components of total magnetisation and the sublattice magnetisations were
calculated. The variance of the antiferromagnetic order parameter and the
susceptibility have been calculated. In a specific range of relative strengths
of interactions (antiferromagnetic/ferromagnetic) and the applied magnetic
field, the system shows the equilibrium phase transitions at different
temperatures. The phase diagrams (in the field-temperature plane) were obtained
for different values of the relative interaction strengths. The ordered region
bounded by the phase boundary was found to increase as the ratio of relative
interaction strength increased. Furthermore, the maximum value of the
susceptibility ($\chi^m_{ay}$) was found to increase with the system size
($L$). For $\chi^m_{ay} \sim L^{{{\gamma} \over {\nu}}}$, the exponent
$\gamma/\nu$ has been estimated to be 2.10$\pm0.11$ .
