Points to Ponder

Current is a scalar, even though we draw arrows to show its direction. It does not follow vector addition rules. This is clear from its definition: the current $I$ through a surface is the scalar (dot) product of two vectors, current density $\mathbf{j}$ and area element $\Delta\mathbf{S}$, i.e., $I = \mathbf{j} \cdot \Delta\mathbf{S}$.

The equation $V = IR$ defines resistance and applies to any device, whether it obeys Ohm's law or not. Ohm's law specifically states that the $I$–$V$ graph is a straight line, meaning $R$ is constant and independent of $V$. Another form of Ohm's law is $\mathbf{E} = \rho \mathbf{j}$, which holds only when resistivity $\rho$ does not depend on the magnitude or direction of the applied electric field.

Even materials that normally obey Ohm's law, like pure silver or doped germanium, show deviations if the applied electric field becomes too strong. Ohm's law is valid only within a certain range of field strengths.

The motion of conduction electrons in an electric field $\mathbf{E}$ has two parts: random thermal motion (which averages to zero and contributes nothing to drift velocity) and the directed motion due to $\mathbf{E}$. The drift velocity $\mathbf{v}_d$ arises only from the applied field. …