Integrate resistance rather than averaging a cross section
A long, gently tapered conductor can be represented by thin resistive slices when section-average current is a justified approximation. Here length is 1 m, entrance area A0=1 mm² and radius grows as r0(1+αx/L). Resistivity is fixed at 2×10⁻⁸ Ω m. This is a stipulated one-dimensional element, not an exact three-dimensional boundary solution.
Learning outcome: Compare α=0,1,2 at the same current. Derive the integral before reading the ledger, and reconcile its power with the plotted spatial density. State why this model does not resolve lateral fields near the taper ends.
Starting recap
Review charge continuity, definite integrals, first-order differential equations, initial conditions and signed source work. Distinguish displayed time from physical time.
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With α=1 the final radius is twice the initial radius and R=0.01 Ω. At 2 A the voltage drop is 0.02 V and conversion is 0.04 W. Entrance power per length is 0.08 W/m, exit power per length 0.02 W/m; their simple endpoint average is not the integral.
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For α=1 and current 2 A, what is the total resistive power?
Experiment
Compare α=0,1,2 at the same current. Derive the integral before reading the ledger, and reconcile its power with the plotted spatial density. State why this model does not resolve lateral fields near the taper ends.
Integrate a slowly tapered resistive element
Worked teaching model · synthetic values · One-dimensional slender-element approximation: length 1 m, entrance area 1 mm², radius grows as r0(1+alpha x/L), resistivity 2e−8 ohm m. Steady prescribed current, uniform section averages and fixed temperature. No exact three-dimensional field, heating evolution or material identification.
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Distributed resistive conversion
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: one-dimensional element model (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Distributed resistive conversion
one-dimensional element model
| position along element (m) | power per length (W/m) |
|---|---|
| 0 | 0.08 |
| 0.0125 | 0.07803688 |
| 0.025 | 0.07614515 |
| 0.0375 | 0.07432138 |
| 0.05 | 0.07256236 |
| 0.0625 | 0.07086505 |
| 0.075 | 0.06922661 |
| 0.0875 | 0.06764434 |
| 0.1 | 0.0661157 |
| 0.1125 | 0.0646383 |
| 0.125 | 0.06320988 |
| 0.1375 | 0.06182828 |
| 0.15 | 0.06049149 |
| 0.1625 | 0.0591976 |
| 0.175 | 0.05794477 |
| 0.1875 | 0.0567313 |
| 0.2 | 0.05555556 |
| 0.2125 | 0.05441598 |
| 0.225 | 0.05331112 |
| 0.2375 | 0.05223957 |
| 0.25 | 0.0512 |
| 0.2625 | 0.05019116 |
| 0.275 | 0.04921184 |
| 0.2875 | 0.04826091 |
| 0.3 | 0.04733728 |
| 0.3125 | 0.04643991 |
| 0.325 | 0.04556782 |
| 0.3375 | 0.04472006 |
| 0.35 | 0.04389575 |
| 0.3625 | 0.04309402 |
| 0.375 | 0.04231405 |
| 0.3875 | 0.04155507 |
| 0.4 | 0.04081633 |
| 0.4125 | 0.04009711 |
| 0.425 | 0.03939674 |
| 0.4375 | 0.03871456 |
| 0.45 | 0.03804994 |
| 0.4625 | 0.03740229 |
| 0.475 | 0.03677104 |
| 0.4875 | 0.03615564 |
| 0.5 | 0.03555556 |
| 0.5125 | 0.03497029 |
| 0.525 | 0.03439936 |
| 0.5375 | 0.03384229 |
| 0.55 | 0.03329865 |
| 0.5625 | 0.032768 |
| 0.575 | 0.03224994 |
| 0.5875 | 0.03174406 |
| 0.6 | 0.03125 |
| 0.6125 | 0.03076738 |
| 0.625 | 0.03029586 |
| 0.6375 | 0.02983509 |
| 0.65 | 0.02938476 |
| 0.6625 | 0.02894454 |
| 0.675 | 0.02851415 |
| 0.6875 | 0.02809328 |
| 0.7 | 0.02768166 |
| 0.7125 | 0.02727902 |
| 0.725 | 0.02688511 |
| 0.7375 | 0.02649966 |
| 0.75 | 0.02612245 |
| 0.7625 | 0.02575323 |
| 0.775 | 0.02539179 |
| 0.7875 | 0.0250379 |
| 0.8 | 0.02469136 |
| 0.8125 | 0.02435196 |
| 0.825 | 0.02401952 |
| 0.8375 | 0.02369383 |
| 0.85 | 0.02337473 |
| 0.8625 | 0.02306202 |
| 0.875 | 0.02275556 |
| 0.8875 | 0.02245516 |
| 0.9 | 0.02216066 |
| 0.9125 | 0.02187193 |
| 0.925 | 0.0215888 |
| 0.9375 | 0.02131113 |
| 0.95 | 0.02103879 |
| 0.9625 | 0.02077163 |
| 0.975 | 0.02050953 |
| 0.9875 | 0.02025236 |
| 1 | 0.02 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Integrated resistance | 0.01 | ohm |
| Terminal voltage | 0.02 | V |
| Resistive power | 0.04 | W |
| Entrance voltage gradient magnitude | 0.04 | V/m |
| Exit voltage gradient magnitude | 0.01 | V/m |
| Entrance power per length | 0.08 | W/m |
| Exit power per length | 0.02 | W/m |
The same current passes every section while current density and local conversion vary.
With A(x)=A0(1+alpha x/L)², dR=rho dx/A(x). Integrating gives R=rho L/[A0(1+alpha)], including the alpha=0 limit. The voltage drop and integrated power are I R and I² R. Replacing the varying area with an arithmetic average generally changes the result. This slender-section model does not resolve end fields, lateral current redistribution or temperature-dependent resistivity.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The same current passes every section while current density and local conversion vary.
With A(x)=A0(1+alpha x/L)², dR=rho dx/A(x). Integrating gives R=rho L/[A0(1+alpha)], including the alpha=0 limit. The voltage drop and integrated power are I R and I² R. Replacing the varying area with an arithmetic average generally changes the result. This slender-section model does not resolve end fields, lateral current redistribution or temperature-dependent resistivity.
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