Q.Obtain the macroscopic form of Ohm's law from its microscopic form and discuss its limitations.

From the microscopic to the macroscopic form. Starting from the microscopic relation $J=\sigma E$, consider a uniform wire of length l and cross-sectional area A carrying a uniform field, so the potential difference along it is $V=El$ (i.e. $E=V/l$), and the current density is $J=I/A$. Substituting both into $J=\sigma E$ gives $I/A=\sigma V/l$, which rearranges to $V=I\left(\dfrac{l}{\sigma A}\right)$. The bracketed quantity, which depends only on the conductor's own geometry and material, is DEFINED as its resistance, R -- this converts the equation into the familiar, everyday, macroscopic statement of Ohm's law:

Equivalently, $R=V/I$: resistance is simply the ratio of the potential difference across a conductor to the current flowing through it, with SI unit the ohm ($\Omega$).

Ohmic versus non-ohmic behaviour. Because R stays constant (independent of V or I) for many everyday conductors, Ohm's law predicts that a graph of current I against voltage V for such a material is a straight line through the origin, with slope $1/R$. Materials that genuinely produce this straight-line I-V graph are called ohmic. Not every material or device obeys this simple rule, however -- devices such as the diode (met formally in Unit 9) instead show a curved, non-linear I-V graph with no single constant resistance value at all; these are called non-ohmic devices, and their more complex behaviour is exactly why Ohm's law, despite its name, is a special-case empirical relation rather than a fundamental law of physics on the same footing as, say, Kirchhoff's rules.

Limitations of the macroscopic form. The macroscopic law $V=IR$ is a simplification that assumes R is a fixed, well-defined constant for the conductor in question -- valid for ohmic conductors under steady conditions, but it silently breaks down for any material whose resistance itself changes with the current, voltage, or temperature (as happens, for instance, in a filament lamp that heats up as more current flows, or in any genuinely non-ohmic device). The microscopic form $\vec J=\sigma\vec E$ is more fundamental precisely because it is built directly from the material's underlying electron density, charge and mean free time, and remains valid (in principle, with $\sigma$ evaluated locally) even in situations where the simple macroscopic $V=IR$ picture, with one single constant R, no longer applies cleanly.