In the NCERT class-12 book on current electricity (see here) , it is derived in page-98 that the average drift velocity electrons would be given as accompanied by the following text:
$$ v_d = \frac{-eE}{m} \tau $$
This last result is surprising. It tells us that the electrons move with an average velocity which is independent of time, although electrons are accelerated. This is the phenomenon of drift and the velocity $v_d$ in is called the drift velocity
Till here it makes sense to me but things got confusing page-105 where the following arguments were given to motivate the resistor must dissipate heat:
Consider a conductor with end points $A$ and$ B$, in which a current $I$ is flowing from $A$ to $B$. The electric potential at $A$ and $B$ are denoted by $V(A)$ and $V(B)$ respectively. Since current is flowing from $A$ to$ B$, $V(A) > V (B)$ and the potential difference across $AB$ is $V = V(A) – V(B) > 0$. In a time interval $∆t$, an amount of charge $∆Q = I ∆t$ travels from $A$ to $B$. The potential energy of the charge at$ A$, by definition, was $Q V(A)$ and similarly at$ B$, it is $Q V(B)$. Thus, change in its potential energy $∆U_{pot}$ is $$∆U_{pot} = \text{Final potential energy} – \text{Initial potential energy}$$ $$= ∆Q[(V (B) – V (A)] = –∆Q V = –I V∆t $$
In case charges were moving freely through the conductor under the action of electric field, their kinetic energy would increase as they move. We have, however, seen earlier that on the average, charge carriers do not move with acceleration but with a steady drift velocity. This is because of the collisions with ions and atoms during transit. During collisions, the energy gained by the charges thus is shared with the atoms. The atoms vibrate more vigorously, i.e., the conductor heats up. Thus, in an actual conductor, an amount of energy dissipated as heat in the conductor during the time interval $\Delta t$ is , $\Delta W = IV \Delta T$ and $P=IV$
I can't understand how it was theoretically concluded that the energy must go into heating. Usually in collision related phenomena, heating occurs when we have inelastic collisions. In this case, I don't understand how two electrons can hit each other and become meld. (Similar to how if we hit two clay balls together, then they under go inelastic collision where they become one) r
As of right now, the quoted text gives me an impression that we can't predict where the energy goes theoretically from the Drude model and we have to introduce the experimental fact that the conductor gets hot to make everything consistent.
Hopefully someone can giving a clearer account of the above.
