Q.A galvanometer of resistance 100 Ω which can measure a maximum current of 1 mA is converted into an ohmmeter by connecting a battery of emf 1 V and a fixed resistance of 900 Ω in series with the galvanometer. When an external resistance is measured the current reading is 0.1 mA. Calculate the value of the resistance.

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. …