  A type I superconductor expels a magnetic field from its interior to a
surface layer of thickness $\lambda_L$, the London penetration depth.
$\lambda_L$ is a function of temperature, becoming smaller as the temperature
decreases. Here we analyze the process of cooling (or heating) a type I
superconductor in a magnetic field, with the system remaining always in the
superconducting state. The conventional theory predicts that Joule heat is
generated in this process, the amount of which depends on the rate at which the
temperature changes. Assuming the final state of the superconductor is
independent of history, as the conventional theory assumes, we show that this
process violates the first and second laws of thermodynamics. We conclude that
the conventional theory of superconductivity is internally inconsistent.
Instead, we suggest that the alternative theory of hole superconductivity may
be able to resolve this problem.
