There is actually a student-friendly microscopic model how to derive the **real** Ohm's law

$$\vec{j} = \sigma \vec{E}$$

After its derivation you can put it into the more common shape using the answer by Nesp.

The idea goes as following:

In material you have certain amount of almost "free" electrons.  Those electrons behave like particle in the gas, they are crashing between themselves and from atom cores and bouncing back and forth and there is actually no net movement.

However, if you put some potential on the ends of material, you actually put electric field in the material, which strength is

$$E = \frac{V}{l}$$

All electrons start to accelerate in the direction of the positive potential and you can easily obtain this acceleration using

$$\vec{a} = \frac{\vec{F}}{m} = \frac{q\vec{E}}{m}$$

So you actually get net movement of electrons.  And now comes the beauty of Ohms law.  You should ask yourself: **if electrons accelerate, how come current (which is proportional to average speed of electrons, so-called drift speed) isn't becoming larger and larger with the time**?

The reason is that electrons keep crashing into atom cores and between themselves, so you can define some typical time between two crashes $\tau$.