Resistance and the microscopic view

Electrical resistance describes how strongly a component opposes current for a given voltage difference. At the circuit level, an ideal resistor is represented by one number. At the material level, that number emerges from mobile charge carriers gaining ordered motion from an electric field and repeatedly scattering from the material.

These descriptions answer different questions. The circuit model predicts terminal voltage, current, and power. The microscopic model explains why material, temperature, length, and cross-sectional area matter. Ohm's law is a model of a device in a particular operating range, not a universal definition of every conductor.

Terminal relation for an ideal resistor

Choose a voltage polarity and a current direction. Under the passive sign convention, current enters the terminal labeled with the higher reference voltage. An ideal ohmic resistor then satisfies

\[V=IR\]

Here \(V\) is measured in volts, \(I\) in amperes, and \(R\) in ohms. Therefore

\[1\ \Omega=1\ \mathrm{V/A}\]

For a fixed positive resistance, voltage and current are proportional. Doubling the voltage doubles the current, while reversing the voltage reverses the current. At fixed voltage, doubling the resistance halves the current.

resistance, voltage = 1000.0, 5.0
assert resistance > 0
current = voltage / resistance
print(f"Current: {current:.6f} A")

The positivity check belongs to this simple passive-resistor model. Negative differential resistance can occur in specialized nonlinear devices, but such a device cannot be substituted blindly for a constant passive resistance.

From electric field to current density

Inside many ordinary conducting materials, a moderate electric field produces a current density approximately proportional to the field:

\[\mathbf J=\sigma\mathbf E\]

The conductivity \(\sigma\) measures how readily the material conducts. Resistivity is its reciprocal:

\[\rho=\frac{1}{\sigma}\]

In SI units, conductivity has units of siemens per metre and resistivity has units of ohm metres. Current density has units of amperes per square metre. The relation is local: it connects the field and current density at the same position in a homogeneous, isotropic ohmic material.

For a straight uniform wire of length \(L\) and cross-sectional area \(A\), suppose the field and current density are nearly uniform. The terminal voltage magnitude and current magnitude are then

\[V=EL,\qquad I=JA\]

Combining these relations with \(J=\sigma E\) gives

\[R=\rho\frac{L}{A}\]

The result explains the geometric trends. A longer wire provides a longer region over which the field transfers energy to the material, so resistance increases with length. A wider wire offers more parallel conducting paths, so resistance decreases with area. This formula assumes a uniform material, a constant cross-section, and contacts whose resistance is negligible or treated separately.

length, area, resistivity = 2.0, 1e-6, 1.68e-8
wire_resistance = resistivity * length / area
print(f"Wire resistance: {wire_resistance:.6f} ohm")

Why carriers do not accelerate forever

An electric field exerts force \(q\mathbf E\) on a carrier of charge \(q\). In empty space, that force would continually change the carrier's momentum. In a conductor, carriers interact with the lattice, impurities, defects, and vibrations. Those interactions interrupt the ordered motion and transfer energy to the material.

A simple Drude-type picture introduces a mean time \(\tau\) between momentum-randomizing collisions. For carrier mass \(m\), the average drift velocity is modeled as

\[\mathbf v_d=\frac{q\tau}{m}\mathbf E\]

Together with \(\mathbf J=nq\mathbf v_d\), where \(n\) is the number of mobile carriers per unit volume, this gives

\[\mathbf J=\frac{nq^2\tau}{m}\mathbf E\]

Thus the model predicts

\[\sigma=\frac{nq^2\tau}{m},\qquad \rho=\frac{m}{nq^2\tau}\]

The square \(q^2\) is important. Electrons have negative charge and drift opposite the field, but conventional current density points with the field in an ordinary metal. A negative carrier charge changes both the force direction and the charge-flow sign, leaving positive conductivity.

charge, mass, collision_time, density, field = (
    -1.602e-19,
    9.109e-31,
    2.5e-14,
    8.5e28,
    1.0,
)
drift_velocity = charge * field * collision_time / mass
current_density = density * charge * drift_velocity
print("Electron drift (m/s):", drift_velocity)
print("Conventional current density (A/m^2):", current_density)

This classical model is deliberately approximate. Modern solid-state physics uses quantum mechanics, energy bands, effective masses, and detailed scattering processes. The Drude model nevertheless captures why more carriers and longer relaxation times tend to increase conductivity.

Temperature dependence

Resistance is generally temperature dependent. Over a modest temperature interval, a metal is often approximated by

\[R(T)=R_0\left[1+\alpha(T-T_0)\right]\]

Here \(R_0\) is the resistance at \(T_0\), and \(\alpha\) is a material-dependent coefficient. For many metals near room temperature, \(\alpha\) is positive: lattice vibrations become stronger and carrier scattering increases. The coefficient is not constant over every temperature range, so the linear law must not be extrapolated indefinitely.

Semiconductors can show the opposite overall trend because warming can greatly increase the number of mobile carriers. Superconductors require a different model below their critical conditions.

Electrical power and thermal energy

Using the passive sign convention, the rate at which a resistor absorbs electrical energy is

\[P=VI=I^2R=\frac{V^2}{R}\]

These equal forms describe the same operating point, but comparisons require stating what is held fixed. At fixed voltage, increasing resistance reduces power according to \(P=V^2/R\). At fixed current, increasing resistance raises power according to \(P=I^2R\). There is no contradiction because an ideal voltage source and an ideal current source impose different constraints.

In a microscopic picture, collisions transfer energy from the organized electrical motion into lattice vibrations and other internal degrees of freedom. Circuit power is therefore an energy-transfer rate, not energy that disappears. For a time-varying operating point, the energy delivered during an interval is

\[\Delta E=\int_{t_0}^{t_1}V(t)I(t)\,dt\]

Worked comparison

Place \(12\ \mathrm V\) across an ideal \(3\ \Omega\) resistor. The current and power are

\[I=\frac{12\ \mathrm V}{3\ \Omega}=4\ \mathrm A, \qquad P=(12\ \mathrm V)(4\ \mathrm A)=48\ \mathrm W\]

If the resistance becomes \(6\ \Omega\) while the voltage remains fixed, the current becomes \(2\ \mathrm A\) and the power becomes \(24\ \mathrm W\). Both are halved. If instead an ideal source holds the current at \(4\ \mathrm A\), the six-ohm resistor would require \(24\ \mathrm V\) and absorb \(96\ \mathrm W\).

Verification and domain of validity

Check dimensions before using a result. In \(R=\rho L/A\), ohm metres times metres divided by square metres gives ohms. Reversing the assigned voltage polarity must reverse the calculated current without changing the positive resistance. Doubling length must double resistance; doubling wire diameter must reduce resistance by a factor of four because area depends on diameter squared.

An ideal resistor omits tolerance, contact resistance, self-heating, thermal lag, maximum voltage, maximum power, and frequency-dependent parasitic capacitance and inductance. Ohm's law can fail when temperature changes substantially, fields become large, geometry becomes microscopic, or the device is intrinsically nonlinear. A useful model states these omissions rather than silently extending \(V=IR\) beyond its evidence.

A worked Python notebook

Use a wire of cross-section one square millimetre and fixed illustrative resistivity. The question for this calculation is precise: how does resistance change when we change wire length, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Temperature and material composition are fixed and contact resistance is omitted. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.1 to 5 m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for wire length, expressed in m. The output is resistance, expressed in ohm. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 0.0168 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 2 m, the expression gives 0.0336 ohm. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("1.68e-8*q/1e-6", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted resistance:", reference_output, "ohm")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0.00168 ohm. The midpoint, at 2.55 m, gives 0.04284 ohm. The upper endpoint gives 0.084 ohm. Read these together with the assumptions: they describe different choices of wire length within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.1, 2.55, 5.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m -> {output_value:.6g} ohm")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows wire length; the vertical axis shows resistance. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.1, 5.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Wire length (m)")
ax.set_ylabel("Resistance (ohm)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does resistance change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 m is 0.0168 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 0.0168\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 2.0245
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "ohm")
print("Full expression:", full_prediction, "ohm")
print("Approximation difference:", full_prediction - linear_prediction, "ohm")

Here the changed input is 2.0245 m. The tangent estimate is 0.0340116 ohm, while the full expression gives 0.0340116 ohm. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 4.9
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of wire length inside the stated interval, predict whether resistance will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Temperature and material composition are fixed and contact resistance is omitted. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of ohm: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(5.0) - evaluate(0.1))
for points in [101, 201, 401]:
    grid = np.linspace(0.1, 5.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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