  Aims. On the basis the theoretical model proposed by Bekenstein for
{\alpha}'s variation, we analyze the equations that describe the energy
exchange between matter and both the electromagnetic and the scalar fields.
Methods. We determine how the energy flow of the material is modified by the
presence of a scalar field. We estimate the total magnetic energy of matter
from the "sum rules techniques". We compare the results with data obtained from
the thermal evolution of the Earth and other planets. Results. We obtain
stringent upper limits to the variations in {\alpha} that are comparable with
those obtained from atomic clock frequency variations. Conclusions. Our
constraints imply that the fundamental length scale of Bekenstein's theory "lB"
cannot be larger than Planck's length "lP".
