  We study the creation of vortices by triggering the rotating Bose-Einstein
condensates in a toroidal trap with trap parameters such as laser beam waist
and Gaussian potential depth. By numerically solving the time-dependent
Gross-Pitaevskii equation in two dimensions, we observe a change in vortex
structure and a considerable increase in the number of vortices when the waist
of the irradiated laser beam is in consonance with the area of the condensate
as we vary the Gaussian potential depth. By computing the root mean square
radius of the condensate, we confirm the variation in the number of vortices
generated as a function of the ratio between the root-mean-square radius of the
condensate and the laser beam waist. In particular, the number of hidden
vortices reaches the maximum value when the above ratio is close to the value
0.7. We find the variation in the number of vortices is rapid for deeper
Gaussian potentials, and we conclude that the larger beam waist and deeper
Gaussian potentials generate more vortices. Further, we calculate the number of
vortices using the Feynman rule with Thomas Fermi approximation and compare
them with the numerical results. We also observe that the critical rotation
frequency decreases with an increase in depth of Gaussian potential.
