Conduction models

“Current flows” is a description of an output, not a model. Depending on scale, conduction may be represented by Ohm's law, a carrier-density calculation, a drift–diffusion equation, or a quantum transport calculation. Pick the smallest model that predicts the quantity you need and name what it deliberately ignores.

Think like a programmer

Keep material parameters separate from solver state. Resistivity, length, cross-sectional area, temperature, carrier density, mobility, and contact assumptions belong in an input record. Current, potential, and accumulated energy belong in the evolving or derived result.

Model checklist

Inputs: Voltage or electric field, conductor length and area, resistivity or carrier parameters, temperature, and contact assumptions.
State: Potential distribution, carrier densities when relevant, and time-integrated energy for transient models.
Rule: Use an explicitly chosen relationship such as Ohm's law or drift–diffusion.
Output: Resistance, current, current density, voltage drop, and power.
Check: Reversing an ideal applied voltage reverses current; zero voltage produces zero current.

Start with a resistor, then expose the assumptions

For a uniform ohmic conductor,

\[R=\rho\frac{L}{A},\qquad I=\frac{V}{R},\qquad P=IV\]

ρ is resistivity, L is length, and A is cross-sectional area. The formula predicts a linear relationship only while the material properties and temperature are treated as constant. It is an excellent first test fixture because changing one input has an expected direction: doubling length doubles resistance; doubling area halves it.

At a microscopic teaching level, the same current density can be written as

\[J=nqv_d\]

This connects a macroscopic current with a population of charge carriers. It does not mean signal propagation occurs at the slow drift speed. Circuit fields and boundary changes establish on different timescales, which a resistor-only model hides.

const resistanceOhms = resistivityOhmMetres * lengthMetres / areaSquareMetres;
const currentAmps = appliedVolts / resistanceOhms;
expect(currentAt(0)).toBe(0);

The final assertion is a limiting-case check. If it fails, inspect the sign convention and any source offset before blaming the material model.

Decide whether the model needs space or time

Use a lumped resistor when geometry can be represented by L and A and you only need steady-state terminal values. Move to a spatial potential or carrier model when local heating, nonuniform material, junctions, or transient charge storage affects the question. A more detailed model needs more boundary data; complexity without measurements is not accuracy.

Try this experiment

Prediction: For an ideal uniform conductor held at fixed voltage, doubling length halves the current.

Choose ρ, A, and V. Calculate current for length L and 2L, then check the ratio. State the material change that could make the simple prediction fail in a real wire.

Where this model breaks

Ohm's law omits temperature rise, nonlinear contacts, ballistic transport, quantum effects, frequency dependence, magnetic effects, and spatially varying material. The carrier expression also assumes a meaningful average density and drift velocity.

Summary

Conduction is a modeling choice. Begin with an explicit input contract and a limiting-case test, then add carrier, spatial, or time dependence only when the question and available data require it.

Glossary

Resistivity: material property relating electric field and current density in an ohmic model.
Current density: current per cross-sectional area.
Drift velocity: average carrier velocity induced by an applied field.

Self-check

Which inputs control resistance in the uniform-resistor model?
What should happen to ideal current when voltage changes sign?
Why does drift velocity not by itself describe signal propagation?

Sources

OpenStax, University Physics Volume 3, Band Theory of Solids.
N. W. Ashcroft and N. D. Mermin, Solid State Physics, transport and band-theory chapters.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Conduction Models, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Atoms and Solids, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Conduction Models as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Conduction Models into a test

Select a scale-appropriate carrier model and make material and transport assumptions explicit.

Name the inputs and units that the atoms and solids model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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