Step 1: State Ohm's law.
For a conductor at constant temperature, \(V = IR\), i.e. the current is proportional to the applied voltage. We now derive this from the microscopic motion of electrons.
Step 2: Start from current density.
Define current density \(J = I/A\). In terms of drift velocity, \(J = ne\,v_d\), where \(n\) is free-electron density and \(e\) the electronic charge.
Step 3: Insert the drift velocity.
An electron gains drift velocity \(v_d = \dfrac{eE\tau}{m}\) under field \(E\). Hence
\(J = ne\cdot\dfrac{eE\tau}{m} = \dfrac{ne^2\tau}{m}\,E = \sigma E\),
where \(\sigma = \dfrac{ne^2\tau}{m}\) is the conductivity. This is the microscopic form of Ohm's law, \(J = \sigma E\).
Step 4: Move to the macroscopic form.
Put \(J = I/A\) and \(E = V/L\):
\(\dfrac{I}{A} = \sigma\dfrac{V}{L}\ \Rightarrow\ I = \dfrac{\sigma A}{L}V\).
Step 5: Read off resistance.
Therefore \(V = I\cdot\dfrac{L}{\sigma A} = IR\) with \(R = \dfrac{L}{\sigma A} = \dfrac{mL}{ne^2\tau A}\), a constant for the conductor. Hence \(V \propto I\).
\[\boxed{J = \sigma E \ \Rightarrow\ V = IR}\]