The component of magnetic force along the slope has magnitude . The rate at which the sliding bar does work against this magnetic braking is
(1)
By Ohm's law, , so
(2)
This exactly equals the Joule heating in the resistor in the negligible-self-inductance approximation. The normal force does no work because it is perpendicular to the velocity.
The gravitational power is instead . By the work-energy theorem,
(3)
It equals the Joule heating only at constant speed. For a concrete counterexample, during the initial acceleration the kinetic energy increases, so some gravitational power remains mechanical.

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