Ohm's Law & Resistance
Current density proportional to field; V = Ri.
Lesson at a glanceShow the key points
- j = σE (32.6); ρ = 1/σ (32.7); σ independent of E for metals.
- V = Ri (32.8), R = ρl/A (32.9); ohm, Ω m, mho m⁻¹.
- Not valid for semiconductors, electrolytes, insulators.
- Colour code: digit, digit, multiplier, tolerance (gold 5 %, silver 10 %, none 20 %).
Combine the drift picture with i = nAevd and one of the most used rules in physics falls out: current density is proportional to the field.
Ohm’s law
Putting vd = (eτ/2m)E into j = nevd:
The content of Ohm’s law is that σ does not depend on the field: double E and j doubles. This holds for metals over a wide range of fields.
Resistance
Take a conductor of length l and uniform cross-section A with potential difference V across its ends. Inside, E = V/l and j = i/A, so i/A = σV/l, i.e. V = (ρl/A) i:
- R is the resistance, in ohms (Ω); 1/R is the conductance.
- Resistivity is in Ω m; conductivity in (Ω m)⁻¹, also written mho m⁻¹.
- A component made to have a chosen resistance is a resistor, drawn as a zig-zag.
Find the resistance of a copper wire 20 m long with a cross-section of 1.0 mm² (ρ = 1.72 × 10⁻⁸ Ω m).
The colour code
Carbon resistors are cylinders with wire leads, labelled by four coloured bands. Read from the end the bands are bunched toward: the first two are digits, the third is the power of ten, the fourth the tolerance (gold ±5 %, silver ±10 %; no fourth band means ±20 %).
A resistor is banded red, violet, orange, gold. What is its value?
- 1Ohm's law: j = σE (32.6), ρ = 1/σ (32.7); σ independent of E.
- 2V = Ri (32.8) with R = ρl/A (32.9); unit ohm; conductance 1/R.
- 3Not valid for semiconductors, electrolytes, insulators…
- 4Colour code: digit, digit, multiplier, tolerance.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.