Resistance and Ohm's law

Put a potential difference V across any two-terminal component, measure the current I through it, and define R ≡ V/I. That number always exists, and its unit is the volt per ampere, named the ohm: 1 Ω = 1 V A⁻¹. Nothing has been assumed about the component. Ohm's law is a second and much stronger statement — that for this component, over a stated range of conditions, R comes out the same whatever V you apply. A 100 Ω metal-film resistor draws 15 mA at 1.5 V, 40 mA at 4.0 V and 90 mA at 9.0 V, and the ratio never budges. That constancy is the law, and it is a fact about the material, not a rearrangement of algebra. Writing V = IR uses the law; writing R = V/I only names a ratio. Keeping them apart is what lets you say, without contradiction, that a filament lamp has a resistance of 25 Ω at 6.0 V and does not obey Ohm's law.

Plot I up the page against V across it and both statements become lines you can point at. For an ohmic component the graph is a straight line through the origin whose gradient is I/V = 1/R; a 150 Ω resistor gives 6.67 mA per volt. For anything else the graph bends, and there are then two resistances at every point. The chord from the origin to the operating point has gradient I/V, so its reciprocal is R = V/I — the number you want when you need the current the device actually draws. The tangent has gradient dI/dV, and its reciprocal r_d = dV/dI is the slope or dynamic resistance — the number a small change sees. A 6 V filament lamp drawing 240 mA has R = 6.0 V ÷ 0.240 A = 25.0 Ω, while raising the supply by 1.0 V near that point adds only 16 mA, so r_d = 1.0 V ÷ 0.016 A = 62.5 Ω. Two honest answers to two different questions. On an ohmic device chord and tangent coincide, and there is only one.

The constancy is not obvious, so it is worth seeing where it comes from. A field E inside the metal pushes every conduction electron, but collisions with the lattice interrupt the push after a mean free time τ, so instead of accelerating without limit the electrons settle at a steady drift speed proportional to the field, v_d = e E τ / m, directed against E because their charge is negative. The current density follows, and it points along E: J = n e v_d = (n e² τ / m) E. That is Ohm's law written for the material rather than for the object — J = σE, with conductivity σ = n e² τ / m. The whole argument rests on τ not depending on E, and it does not, because the electrons' random motion dwarfs the drift the field adds. In copper carrying 5 × 10⁶ A m⁻² the drift speed is about 3.7 × 10⁻⁴ m s⁻¹, while the speed that sets the collision rate is of order 10⁶ m s⁻¹, larger by some nine orders of magnitude. The field nudges a motion it cannot restructure, so σ does not notice it, and V/I does not move.

A passive component never shows a negative chord resistance V/I, because current always enters at its higher-potential terminal; a bent curve can hand you a negative slope dV/dI, as a gas discharge does, but never a negative V/I. Adopt the passive sign convention — draw the current arrow into the terminal you marked + — and V and I are then positive together, so R = V/I is positive by construction. Walking along the component in the direction of the current, the potential changes by ΔV = −I R; walking against the current it changes by +I R. The convention also survives a wrong guess. Assign a current left to right through a 220 Ω resistor, solve, and find I = −27 mA: that is 6.0 V ÷ 220 Ω with the arrow drawn the wrong way round. The magnitude is right, R is still +220 Ω because V and I flipped together, and only the product I R changes sign with your choice of direction.

The reciprocal G = 1/R is the conductance, measured in siemens, with 1 S = 1 Ω⁻¹ = 1 A V⁻¹, and Ohm's law reads I = G V. It is the same physics turned round, and the choice between the two is about which quantity you control. When the current through a component is what you set, R is natural, because it turns that current into a potential difference. When the voltage across it is what you set, G is natural, because it turns that voltage straight into a current: a 125 Ω resistor has G = 1 ÷ 125 Ω = 8.00 mS, so across 5.0 V it draws 5.0 V × 8.00 × 10⁻³ S = 40 mA with no division left to do. Conductance also behaves better at the extremes. An open switch has R = ∞ and G = 0; an ideal superconducting link has R = 0 and G = ∞. Whichever of the pair stays finite is the one to compute with.

Three common components leave Ohm's law in three different directions. A filament lamp heats as it conducts, so its resistance climbs with current and its I–V curve bends towards the V axis. An NTC thermistor does the opposite — heating lowers its resistance, the curve bends towards the I axis, and the device runs away unless something limits the current. A semiconductor diode is nowhere near linear: I = Iₛ (e^(V/V_T) − 1), with V_T = kT/e ≈ 25.9 mV at 300 K, so the current climbs by roughly a factor of ten for every 60 mV. None of this stops you working. Fix an operating point, quote R = V/I there for the steady current, and quote r_d = dV/dI there for anything that wiggles. On a diode the gap between the two is a ratio you can write down in advance: R/r_d = (V/I) ÷ (V_T/I) = V/V_T, which at a forward drop of 0.65 V is about 25. One number can no longer answer both questions — that is all that non-ohmic means.