The generation of parallel electric fields by a shear Alfv¿n wave standing along closed magnetic field lines in Earth's magnetosphere is considered under the condition that the electron bounce period is smaller than the wave period. The effects of the inhomogeneity of the magnetic field, plasma density, and electron temperature are analyzed in terms of the magnitude of the parallel potential, its location, and the distribution function of the current-carrying electrons. The theory demonstrates a possible mechanism for keV electron energization in the polar magnetosphere.