  We investigate the luminosity and cooling of highly magnetized white dwarfs
with electron-degenerate cores and non-degenerate surface layers where cooling
occurs by diffusion of photons. We find the temperature and density profiles in
the surface layers or envelope of white dwarfs by solving the magnetostatic
equilibrium and photon diffusion equations in a Newtonian framework. We also
obtain the properties of white dwarfs at the core-envelope interface, when the
core is assumed to be practically isothermal. With the increase in magnetic
field, the interface temperature increases whereas the interface radius
decreases. For a given age of the white dwarf and for fixed interface radius or
interface temperature, we find that the luminosity decreases significantly from
about $10^{-6}\, L_{\odot}$ to $10^{-9}\, L_{\odot}$ as the magnetic field
strength increases from about $10^9$ to $10^{12}\,$G at the interface and hence
the envelope. This is remarkable because it argues that magnetized white dwarfs
are fainter and can be practically hidden in an observed Hertzsprung--Russell
diagram. We also find the cooling rates corresponding to these luminosities.
Interestingly, the decrease in temperature with time, for the fields under
consideration, is not found to be appreciable.
