# How does an increase in drift velocity increase the temperature of a conductor?

I have some intuition on how temperature affect drift velocity, but what about the other way around? In a fixed circuit, drift velocity is proportional to current, and current to temperature.

In the Drude model, electrons are assumed to be moving around even with no voltage, bouncing around and colliding with atoms. I'd assume that temperature is correlated with the amount of collisions and the strength of the collisions - but I don't see how changing some general tendency of movement, the drift velocity, would change either of those, on average.

So how exactly do the atoms gain more kinetic energy due to an increase in drift velocity?

Drude model:

$$m\vec{a} = e \vec{E}- \frac{m}{T} \vec{v}$$

When a = 0

$$e \vec{E} = \frac{m}{T} \vec{v}$$

$$\vec{v} = \frac{T}{m}(e\vec{E})$$

The term $$- \frac{m}{T} \vec{v}$$ represents a resistive force, this resistive force does NEGATIVE work on the electrons. The electric field is doing positive work against the resistive force, to move the electrons. Aka the resistive force is "stealing" energy from the field.

If $$e\vec{E}$$ is bigger, the term $$\frac{m}{T} \vec{v}$$ is also bigger. Which corresponds to more work being transfered to the particles making up the resistive force.

$$\vec{v} \propto e\vec{E}$$

So increasing V, increases the amount of energy transfered to the lattice of the wire.

Intuitively, ofcourse a higher drift velocity means more collisions with the lattice.