We consider a gas of hydrogen atoms illuminated by a broadband unpolarized radiation with zero anisotropy. In the absence of external fields, atomic J-levels are thus isotropically populated. While this condition persists in the presence of a magnetic field, we show instead that electric fields can induce the alignment of these levels. We also show that this electric alignment cannot occur for a two-term model of hydrogen (e.g. if only the Lyα transition is excited), or if the level populations are distributed according to Boltzmann's law.