  We have studied the thermal properties of atom chips consisting o high
thermal conductivity Aluminum Nitride (AlN) substrates on which gold microwires
are directly deposited. We have measured the heating of wires of several widths
and with different thermal couplings to the copper mount holding the chip. The
results are in good agreement with a theoretical model where the copper mount
is treated as a heat sink and the thermal interface resistance between the wire
and the substrate is vanishing. We give analytical formulas describing the
different transient heating regimes and the steady state. We identify criteria
to optimize the design of a chip as well as the maximal currents $I_c$ that can
be fed in the wires. For a 600 $\mu$m thick-chip glued on a copper block with
Epotek H77, we find $I_c=16$ A for a 3 $\mu$m high, 200 $\mu$m wide-wire.
