Ohm's law in Vector Form

Step 1 - Start with resistance relation:

Resistance of a conductor of length l, area A, and resistivity ρ:

Step 2 - Substitute into scalar Ohm's law V = IR:

Step 3 - Identify physical quantities:

• V/l = E (electric field)
• I/A = j (current density)
• \[\frac {m}{ne^2τ}\] = ρ (resistivity)

Step 4 - Rewrite the equation:

Step 5 - Use ρ = 1/σ to express in terms of conductivity:

This is Ohm's Law in Vector Form.

• For an isotropic conductor, current density \[\vec j\]​ is directly proportional to the applied electric field \[\vec E\].
• Both vectors point in the same direction at every point inside the conductor.
• σ\sigmaσ (conductivity) is the proportionality constant — a scalar for isotropic materials.

Real-Life Analogy: Think of water flowing through a wide, uniform pipe. The rate of water flow per unit cross-sectional area (like current density) is proportional to the pressure gradient pushing it (like electric field) — and both act in the same direction along the pipe.