Distinguish a resistance ratio from a nonlinear response

Define topology, recipient and initial conditions

Ohmic behavior is a physical model: voltage is proportional to current under specified conditions, with constant resistance. Writing R=V/I at one point does not establish that proportionality across a range. The supplied invented device instead follows V=10I+βI³. It is passive in the displayed nonnegative-current range, but is not claimed to be a particular lamp or semiconductor.

Derive the quantitative response

At nonzero current, the chord ratio is V/I=10+βI². For a finite increment h, expansion gives [V(I+h)−V(I)]/h=10+β(3I²+3Ih+h²). This finite-step response generally differs from V/I and depends on the step. At I=0 the ratio V/I is undefined, although the nearby finite-step response can still be calculated. No derivative is required for this comparison.

Check evidence and the limits of the model

Electrical transfer rate is P=IV even when the device is non-ohmic; I²R only becomes a useful expression when R is defined consistently at that operating point. Current is charge crossing a section per elapsed time. In a simple uniform conductor, I=nqAv_d relates number density, carrier charge, area and mean drift speed; the electron drift direction is opposite conventional current. Microscopic scattering transfers energy, but the supplied cubic law does not determine a temperature or microscopic mechanism. A changed response could reflect temperature, geometry, material or measurement effects.

A separate microscopic current packet

For an ideal uniform carrier population, use magnitudes n=5×10²⁸ m⁻³, |q|=1.6×10⁻¹⁹ C, area10⁻⁶ m² and drift speed10⁻⁴ m/s. Then I=n|q|Av_d=.8 A. Predict/check: doubling the area at unchanged density and drift doubles current to1.6 A; keeping current fixed instead would halve drift speed. For electrons, their mean motion is opposite the conventional current arrow. This packet does not identify the mechanism behind the separate cubic voltage law. Primary current/drift verification.

Worked original case

At I=1 A and β=20 V/A³, voltage is 30 V and power 30 W. Chord resistance is 30 Ω. For h=.1 A, the finite-step response is 10+20(3+.3+.01)=76.2 Ω; multiplying by .1 A gives a 7.62 V increment.

Investigation

Compare I=0,.5,1 A at fixed β. Retain the undefined zero-current chord state instead of inventing a zero resistance. Change h on paper and show why a finite-step response is not automatically the same as a chord ratio.

Changed-case practice

Try the changed case before feedback

For V=10I+10I³, find the chord ratio V/I at I=2 A in ohms.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and evidence boundary

Original numerical cases and prose checked 30 October 2026 against the relevant primary explanation. All records are synthetic and all diagrams mathematical. No practical wiring, battery testing, capacitor handling, mains measurement or equipment safety rating is requested. Public feedback does not verify independent mastery; retain first predictions, contrasting recipes and criterion-based critique in the separate course evidence workspace.