  We propose a new theory of thermal Casimir effect, holding for the
experimentally important case of metallic surfaces with a roughness having a
spatial scale smaller than the skin depth. The theory is based on a simple
phenomenological model for a rough conductor, that explicitly takes account of
the fact that ohmic conduction in the immediate vicinity of the surface of a
conductor is much impeded by surface roughness, if the amplitude of roughness
is smaller than the skin depth. As a result of the new model, we find that
surface roughness strongly influences the magnitude of the thermal correction
to the Casimir force, independently of the plates separation. Our model, while
consistent with recent accurate measurements of the Casimir force in the
submicron range, leads to a new prediction for the not yet observed thermal
correction to the Casimir force at large plates separation. Besides the thermal
Casimir problem, our model is relevant for the correct theoretical
interpretation of current experiments probing other proximity effects between
conductors, like radiative heat transfer and quantum friction.
