Joule's law of heating states that an accelerated electron loses its energy, which is then converted into heat energy, by colliding with vibrating atom i.e ions in their lattice site. but we know atom consist of electrons and a nucleus. Where does it collide? How does energy get transferred?
Answer
That's a very hard question to answer with the appropriate level of detail! Very broadly speaking in an ideal metal all atoms are forming a perfectly regular crystal lattice. Conduction band electrons can move freely around these atoms, which makes it easy to pass a current trough the metal.
In a (theoretical) metal with perfect crystal lattice the electrons wouldn't be losing any energy and the metal would behave similar to a superconductor. However, in reality metals are never ideal, they have so called defects. A defect is a place in the lattice where an atom is missing or where it's sitting in the wrong position, or there could be an atom of a different element replacing one of the metal's own.
When electrons pass such a defect, they encounter a discontinuity and get deflected from their ideal path. This can only happen if the momentum of the electrons change and because of momentum conservation that has to change the momentum of the atoms around the defect. The momentum change also transfers kinetic energy from the electrons to the metal lattice, which is the heat that is generated when a current flows trough a conductor.
In reality these processes have to be described with quantum mechanics and that's so complicated that we are still researching many of these phenomena (although simple resistive Joule heating is understood fairly well).
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