Integrate material resistance without confusing graph slopes

Separate material from object

Resistivity describes a specified material state; resistance also depends on geometry. For a uniform segment, R = rho L/A. For uniform area but varying rho(x), a small series element contributes dR = rho(x) dx/A, so total R is its length integral. In steady state, the same current passes each section even though local field and voltage gradient can vary.

Carry the spatial integral

With rho(x) = rho0(1 + alpha x/L), integration gives R = rho0 L(1 + alpha/2)/A. The potential drop from 0 to x is I rho0[x + alpha x²/(2L)]/A. Differentiating this drop recovers the local field I rho(x)/A; integrating it over the full resistor recovers the applied voltage.

Use the axes to interpret a slope

For a fixed ideal ohmic resistance, I = V/R. Thus a graph with I vertically and V horizontally has slope 1/R; a V-versus-I graph has slope R. In a changed supplied non-ohmic law V = (2 ohm)I + (1 ohm/A²)I³, the secant V/I and differential dV/dI differ: at I = 1 A they are 3 ohm and 5 ohm. Neither can be silently treated as a single constant resistance over every current.

Qualify the temperature idealization

The constant-resistance model holds material conditions fixed. Ordinary metals often increase resistivity as temperature rises, and Joule heating can alter the operating state. Ohmic response at a fixed temperature does not establish temperature-independent resistivity under all conditions. We interpret the framework’s ideal constant-parameter statement as a model restriction, not a universal material law. For a supplied rho(T) = rho_ref[1 + 0.004(T − Tref)] with temperatures in kelvin differences, a 50 K rise increases resistance by 20% at unchanged geometry; the ideal unchanged-R prediction then fails.

Worked original case

For rho0 = 2 micro-ohm m, alpha = 1, A = 1 mm² and L = 1 m, R = 3 ohm. At 10 V, I = 10/3 A. The I-versus-V slope is 1/3 A/V; local field grows from 20/3 to 40/3 V/m while its integrated drop remains 10 V.

Investigation

Compare uniform and linearly varying resistivity with the same rho0. Derive the cumulative voltage curve, then differentiate it. Predict both graph slopes. Finally use the supplied temperature and nonlinear packets to identify where a fixed-parameter ohmic model is no longer appropriate.

Changed-case practice

Sources and evidence boundary

Original teaching checked 31 October 2026 against the current official framework, Unit 11 and OpenStax resistivity and resistance. All diagrams and numerical records are virtual, supplied or generated. No physical wiring, battery connection, short circuit, meter operation or electrical handling is requested. Public feedback and retained self-reports do not certify independent reasoning, actual laboratory participation or institutional credit.