  We have investigated the effects of strong magnetic field on the properties
of quarkonia immersed in a thermal medium of quarks and gluons and studied its
quasi-free dissociation due to the Landau-damping. Thermalizing the Schwinger
propagator in the lowest Landau levels for quarks and the Feynman propagator
for gluons in real-time formalism, we have calculated the resummed retarded and
symmetric propagators, which in turn give the real and imaginary components of
dielectric permittivity, respectively. The magnetic field affects the
large-distance interaction more than the short-distance interaction, as a
result, the real part of potential becomes more attractive and the magnitude of
imaginary part too becomes larger, compared to the thermal medium in absence of
strong magnetic field. As a consequence the average size of $J/\psi$'s and
$\psi^\prime$'s are increased but $\chi_c$'s get shrunk. Similarly the magnetic
field affects the binding of $J/\psi$'s and $\chi_c$'s discriminately, i.e. it
decreases the binding of $J/\psi$ and increases for $\chi_c$. However, the
further increase in magnetic field results in the decrease of binding energies.
On contrary the magnetic field increases the width of the resonances, unless
the temperature is sufficiently high. We have finally studied how the presence
of magnetic field affects the dissolution of quarkonia in a thermal medium due
to the Landau damping, where the dissociation temperatures are found to
increase compared to the thermal medium in absence of magnetic field. However,
further increase of magnetic field decreases the dissociation temperatures. For
example, $J/\psi$'s and $\chi_c$'s are dissociated at higher temperatures at 2
$T_c$ and 1.1 $T_c$ at a magnetic field $eB \approx 6~{\rm{and}}~4~m_\pi^2$,
respectively, compared to the values 1.60 $T_c$ and 0.8 $T_c$ in the absence of
magnetic field, respectively.
