Why is the drift velocity directly proportional to the electric field? If I double the electric field, that should double the acceleration of electrons inside the conductor in the general direction of the electric field. But why does that double the drift velocity, and in turn, double the current? Normally when we have a fixed path length the velocity ends up getting multiplied by $\sqrt{2}$ when we double the acceleration. Thanks!
 A: Recall that Ohm’s law relates the three physical objects namely the electric field $\vec{E}$, the conductivity $\sigma$ and the current density $\vec{J}$:
$ \vec{J} = \sigma \vec{E} $.
Assuming that the electrons will flow in a direction perpendicular to some surface in the direction of $ \vec{J}$ we can find the current using the following equation:
$I = J A$ , where $J = |\vec{J}|$ and $A$ the surface area.
Another useful thing to note is that the current of some electrons with electron density $n$ and  some velocity which we define to be the drift velocity $v_d$ then we can also write:
$I = n A v_d e$ where $e$ is the charge of an electron.
This equation could best be derived using for example a cylinder with cross section surface area $A$ and electrons moving with a speed $v_d$ a time $t$ in the direction perpendicular to the surface. Then by definition of the current we have that $I = Q/t = \frac{A n v_d t e}{t} = A n  v_d e$. 
Now we take these equations together. 
$ I = n A e v_d = J A = \sigma E A $ and solving for the drift velocity we get: 
$ v_d = \sigma E / (n e) $.
This shows that the drift velocity is indeed proportional to the applied electric field. How to check this? Do an experiment, I just did a basic derivation of the drift velocity that consist of a lot of assumptions. 
As a final note mark that a drift velocity is the average velocity of the electrons. Please also note that since the acceleration is constant we have that $v_d = a \tau$ with $a$ the acceleration and $\tau$ the mean free time. This is always the case, could you give an example where the velocity changes quadratically with $a$ ? Hope it helps, I am not an expert in this subject and I also had to look it up. 
