Ohm's Law

For ohmic conductors, voltage is proportional to current at constant temperature. R is the resistance. Ohm's law is not a universal law — it holds only for materials where R is independent of V and I.

From microscopic to macroscopic

For a conductor of length $L$ and cross-section $A$ with uniform field $E$:

Potential difference: $V = EL$

Current density: $J = \sigma E$, so $I = JA = \sigma EA$

Eliminating $E = V/L$:

$$I = \sigma \cdot \frac{V}{L} \cdot A = \frac{\sigma A}{L} V$$

Defining resistance $R = L/\sigma A = \rho L/A$:

$$\boxed{V = IR}$$

When Ohm's law holds

Ohm's law requires $\sigma$ (or $\rho$) to be independent of $E$ — i.e., the material's response is linear. This holds for:

• Metallic conductors at constant temperature
• Electrolyte solutions at low fields

It fails for:

• Semiconductors (non-linear $I$-$V$)
• Diodes, transistors
• Conductors at very high fields or varying temperature

$V$-$I$ characteristics

For ohmic devices: straight line through origin with slope $1/R$. Non-ohmic devices show curves.