Current as charge flow

Electric current measures the rate at which charge crosses an oriented surface. It is not the amount of charge stored in a wire, and it is not the speed of one electron. A current value needs a boundary, a time interval, and a declared positive direction.

Conventional current follows the direction positive charge would move. In a metal, electrons usually drift in the opposite direction because their charge is negative.

Average and instantaneous current

If signed charge \(\Delta Q\) crosses during a positive time interval \(\Delta t\), the average current is

\[I_{\mathrm{avg}}=\frac{\Delta Q}{\Delta t}\]

The instantaneous current is the limiting rate

\[I(t)=\frac{dQ}{dt}\]

The SI unit is the ampere:

\[1\ \mathrm A=1\ \mathrm{C/s}\]

A negative current means the net conventional charge transfer is opposite the stored positive orientation. It does not mean the magnitude of flow is physically negative.

Current reconstructs transferred charge

Integrating a time-dependent current gives net transferred charge:

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

Positive and negative intervals cancel in the net result. A sinusoidal alternating current can transfer zero net charge over a complete cycle while still moving charge back and forth and delivering energy to a resistor.

For sampled current, a numerical quadrature rule estimates the integral. The sample times must be stored; multiplying a sum by one assumed interval is wrong for irregular sampling.

Conventional current and electron motion

Imagine electrons moving to the left through a wire. Negative charge moving left is equivalent to positive conventional current moving right. Circuit arrows normally show that conventional direction.

For one type of carrier with number density \(n\), signed charge \(q\), and average drift velocity \(\mathbf v_d\), the current density is

\[\mathbf J=nq\mathbf v_d\]

The sign of \(q\) automatically makes electron current density point opposite electron drift. For several carrier species, add their current-density vectors:

\[\mathbf J=\sum_s n_sq_s\mathbf v_{d,s}\]

Positive ions and electrons can move in opposite directions yet contribute current in the same conventional direction.

From current density to branch current

Current density has SI units of amperes per square meter. The current through an oriented surface \(S\) is

\[I=\int_S\mathbf J\cdot d\mathbf a\]

The vector area element \(d\mathbf a\) defines the positive direction. Reversing the surface normal reverses the reported current.

For uniform current density parallel to a wire of cross-sectional area \(A\),

\[I=JA=n|q|v_dA\]

The final magnitude form uses carrier charge magnitude and drift-speed magnitude. Direction is handled separately by the conventional-current orientation.

Drift speed is not signal speed

Conduction electrons have large random thermal or quantum velocities even when net current is zero. An applied electric field adds a small average drift. The random parts mostly cancel, leaving the drift contribution to current.

Electrical influence propagates through the electromagnetic field and circuit geometry at a substantial fraction of the speed of light in the medium, typically far faster than individual electron drift. A lamp can respond soon after a switch closes because charges and fields are already present throughout the circuit; an electron does not need to travel from the switch to the lamp first.

The lumped-circuit approximation treats propagation delay as negligible. Long cables and high-frequency signals require transmission-line or full field models.

A simple charge-flow example

Suppose net charge

\(\Delta Q=-6.0\ \mathrm C\)

crosses a right-oriented surface during \(\Delta t=2.0\ \mathrm s\). Then

\[I_{\mathrm{avg}} =\frac{-6.0\ \mathrm C}{2.0\ \mathrm s} =-3.0\ \mathrm A\]

The physical transfer is three amperes opposite the selected orientation. If the surface orientation is reversed without changing the event, the current becomes positive three amperes.

Local charge conservation

Charge density \(\rho\) and current density satisfy the continuity equation

\[\frac{\partial\rho}{\partial t} +\boldsymbol{\nabla}\cdot\mathbf J=0\]

Integrating over a fixed volume gives

\[\frac{dQ_{\mathrm{inside}}}{dt} =-\oint_{\partial V}\mathbf J\cdot d\mathbf a\]

Net outward current decreases charge stored inside. This is the physical foundation of node-current bookkeeping.

For a circuit node with every branch current defined positive into the node,

\[\frac{dQ_{\mathrm{node}}}{dt} =\sum_k I_k\]

If stored node charge is not changing, the current sum is zero. Kirchhoff's current law is therefore a steady or lumped approximation to charge conservation, not a rule that permits charge to disappear.

Direct and alternating current

Direct current keeps one direction, though its magnitude can vary with time. Alternating current reverses direction. A complete description includes time dependence, frequency, phase, and measurement bandwidth.

For a sinusoidal current

\[I(t)=I_0\cos(\omega t+\phi)\]

the average over a complete cycle is zero. Its root-mean-square value is

\[I_{\mathrm{rms}}=\frac{I_0}{\sqrt2}\]

The RMS value is useful for resistive power, but it does not replace instantaneous current when phase, charge accumulation, or transient behavior matters.

Measuring current

An ideal ammeter is placed in series so the branch current passes through it. It has zero resistance in the ideal model; a real meter has nonzero burden and can alter the circuit. Connecting an ammeter directly across a voltage source can create a dangerously large current.

Microscopic current measurements also have bandwidth and noise. An average over too long a window can hide reversals, while sparse sampling can alias a varying signal. Record the instrument range, sampling interval, and orientation.

Charge-event reduction

A current estimator from discrete transfer events can remain narrow and explicit.

events = [(0.1, 2e-6), (0.3, -0.5e-6), (0.8, 1e-6)]
start, end = 0.0, 0.5
assert end > start
net_charge = sum(charge for time, charge in events if start <= time < end)
current = net_charge / (end - start)
print(f"Average current in the positive surface direction: {current:.6g} A")

If the event records physical electron crossings, the carrier charge is negative. The orientation helper must use both motion direction and charge sign to return conventional current.

Verification and limitations

Transfer the same signed charge in half the time and confirm that average current doubles. Reverse the reference surface and confirm that current sign reverses. Integrate a known constant current and recover charge proportional to duration.

At a node, compare the sum of oriented branch currents with the measured rate of stored charge change. Use a capacitor transient as a nonzero-accumulation case rather than forcing the current sum to zero at every instant.

Check a sinusoid over an integer number of periods: net transferred charge and average current should approach zero, while RMS current remains nonzero. Refine the sample interval to expose quadrature or aliasing error.

At high frequencies, branch current can vary along a conductor because charge and fields are distributed. Radiation, displacement current, skin effect, and propagation delay then limit the simple lumped model. Within its domain, current is best understood as oriented charge flow tied directly to local conservation.

A worked Python notebook

A signed net charge of 1.5 microcoulombs crosses a surface during the chosen interval. The question for this calculation is precise: how does average current change when we change measurement duration, 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.

The fixed transferred charge is spread over different durations; this is an average current, not a resolved waveform. 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 2 s. 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 measurement duration, expressed in s. The output is average current, expressed in A. 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.

\[f(q) = \frac{1.5 \cdot 10^{-6}}{q}\]

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 0.5 s, the expression gives 3e-06 A. 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("1.5e-6/q", locals={"q": q})
reference_input = 0.5
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted average current:", reference_output, "A")

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 1.5e-05 A. The midpoint, at 1.05 s, gives 1.42857e-06 A. The upper endpoint gives 7.5e-07 A. Read these together with the assumptions: they describe different choices of measurement duration 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, 1.05, 2.0])
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} s -> {output_value:.6g} A")
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 measurement duration; the vertical axis shows average current. 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, 2.0, 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("Measurement duration (s)")
ax.set_ylabel("Average current (A)")
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 average current change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.5 s is -6e-06 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - \frac{1.5 \cdot 10^{-6}}{q^{2}}\]

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 = 0.5095
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, "A")
print("Full expression:", full_prediction, "A")
print("Approximation difference:", full_prediction - linear_prediction, "A")

Here the changed input is 0.5095 s. The tangent estimate is 2.943e-06 A, while the full expression gives 2.94406e-06 A. 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 = 1.9
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 measurement duration inside the stated interval, predict whether average current 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: The fixed transferred charge is spread over different durations; this is an average current, not a resolved waveform. 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 A: 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(2.0) - evaluate(0.1))
for points in [101, 201, 401]:
    grid = np.linspace(0.1, 2.0, 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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