We investigate the space-time evolution of the magnetization induced in a 2D ferromagnetic island by a short pulse of a rotating electric field. The field generates elastic twists that act on the magnetization via the Barnett effect: magnetization by rotation. Analytical studies are conducted within classical electrodynamics of continuous media and continuous elastic theory, while numerical studies are performed using discretized Landau-Lifshitz spin dynamics on the atomic lattice. The effect is studied for typical parameters of magnetic oxides at various field frequencies and amplitudes, and for various strengths of magnetic anisotropy, exchange, and damping. The possibility of reversing the island magnetization with an electric-field pulse is demonstrated.