  We examined the effects of the weak magnetic field on the dissociation of
heavy quarkonia immersed in a thermal medium of quarks and gluons. For that
purpose, we have revisited the general structure of gluon self-energy tensor in
the presence of a weak magnetic field in thermal medium and obtained the
relevant structure functions. The structure functions give the real and
imaginary parts of the resummed gluon propagator, which further give the
respective dielectric permittivities. The real and imaginary parts of the
dielectric permittivity will be used to evaluate the real and imaginary parts
of the complex heavy quark potential. The real-part of the potential is found
to be more screened, whereas the magnitude of the imaginary-part of the
potential gets increased on increasing the value of both temperature and
magnetic field. In addition to this, we have observed that the real-part gets
slightly more screened while the imaginary part gets increased in the presence
of a weak magnetic field as compared to their counterparts in the absence of a
magnetic field. The increase in the screening of the real-part of the potential
leads to the decrease of binding energies of $J/\Psi$ and $\Upsilon$, whereas
the increase in the magnitude of the imaginary part leads to the increase of
thermal width with the temperature and magnetic field both. Also the binding
energy and thermal width in the presence of weak magnetic field become smaller
and larger, respectively, as compared to that in the pure thermal case. We have
finally obtained the dissociation temperatures for $J/\Psi$ and $\Upsilon$,
which become slightly lower in the presence of weak magnetic field. This
observation leads to the slightly early dissociation of quarkonia because of
the presence of a weak magnetic field.
