Charge as discrete entities

Electric charge is a property of matter, not a substance painted onto an object. Charged particles create electric fields and respond to electric and magnetic fields. The sign of charge determines the direction of those interactions, while conservation determines which transfers and transformations are physically allowed.

At microscopic scales, ordinary matter carries charge in discrete units. At macroscopic scales, enormous numbers of carriers make a continuous charge density an excellent approximation.

Two signs and one neutral total

Charge comes with positive and negative signs. Like signs repel and unlike signs attract. A proton carries positive elementary charge, while an electron carries the same magnitude with negative sign:

\[q_p=+e,\qquad q_e=-e\]

The elementary charge has the exact SI value

\[e=1.602\,176\,634\times10^{-19}\,\mathrm{C}\]

The value is exact because the modern SI defines the coulomb through the fixed value of the elementary charge. The symbol \(\mathrm{C}\) means coulomb, the SI unit of electric charge.

An electrically neutral object has zero net charge, but it is not necessarily free of charge. Ordinary neutral matter contains large positive and negative contributions that cancel:

\[Q_{\mathrm{net}}=\sum_i q_i=0\]

The internal charges can still separate slightly, producing polarization and electric dipole effects.

Quantization of charge

For an isolated object built from ordinary protons and electrons, its net charge changes in integer multiples of the elementary charge:

\[Q=ne,\qquad n\in\mathbb Z\]

Here \(n\) is a signed integer. Adding one electron changes an object's charge by \(-e\); removing one electron changes it by \(+e\).

Quarks carry fractional multiples of \(e\), but individual quarks are not observed as free isolated particles under ordinary conditions. Composite particles such as protons and neutrons have integer total charge. The introductory integer rule is therefore accurate for freely observed ordinary charged objects, provided its domain is stated.

If a measured object has charge \(Q\), the estimated excess-electron count is

\[N_{\mathrm{excess}}=-\frac{Q}{e}\]

The minus sign matters: negative net charge corresponds to a positive number of excess electrons. A physical count must be an integer, though experimental uncertainty and a coarse measurement may make the calculated ratio appear slightly nonintegral.

Conservation of charge

The total electric charge of an isolated system is conserved. Charge can move from one object to another, but an internal transfer does not change the system total:

\[Q_{\mathrm{before}}=Q_{\mathrm{after}}\]

If object \(A\) transfers charge \(\Delta q\) to object \(B\), consistent updates are

\[Q_A'=Q_A-\Delta q, \qquad Q_B'=Q_B+\Delta q\]

Adding the updated charges cancels the transfer term. Charge has moved, but it has not been created or destroyed.

Pair creation also conserves net charge. Creating a negatively charged electron together with its positively charged antiparticle adds zero total charge. Conservation is more general than preserving the number of charged particles.

Local conservation and current

Charge conservation holds locally, not only as a final global check. Let \(\rho\) be charge per unit volume and \(\mathbf J\) be charge flow per unit area per unit time. They obey the continuity equation

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

A decrease of charge density in a small region must be explained by outward current. Integrating over a fixed volume gives

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

The outward area surface element \(d\mathbf a\) makes outward current reduce the enclosed charge. If a model's total charge changes, the boundary current or an explicitly modeled charge-producing reaction must explain it.

Charging usually moves electrons

In ordinary solids, protons remain bound inside nuclei while electrons are transferred or redistributed. Rubbing two materials can move electrons from one to the other. The objects acquire equal and opposite changes when the pair is treated as a closed system.

A conductor contains mobile charge carriers that rearrange readily. An insulator binds its charges more strongly, so excess charge tends to remain localized. These are material responses, not differences in the conservation law.

Grounding opens the chosen system boundary. Earth can accept or supply a very large number of electrons without a noticeable change in its electric potential. A grounded object can therefore change net charge even though the charge of the larger object-plus-Earth system remains conserved.

Charging by induction can redistribute charge without contact. A nearby charged object polarizes a conductor. If the conductor is then grounded and the grounding connection is removed in the correct order, it can retain a net charge. Every stage should identify whether charge merely moved inside the conductor or crossed the ground connection.

Polarization is not net charging

A neutral atom or molecule can have its positive and negative charge distributions slightly displaced by an external field. The total remains zero, but the separated charges form an induced dipole.

This explains why a charged object can attract a neutral object. The opposite sign is pulled slightly closer than the like sign is pushed away, and the electric interaction is stronger at the shorter distance. Neutrality alone does not imply zero electric force in a nonuniform field.

The dipole moment for separated point charges can be written

\[\mathbf p=q\mathbf d\]

Here \(q\) is the charge magnitude and \(\mathbf d\) points from the negative charge toward the positive charge. The dipole moment is a vector even though the total charge is zero.

Discrete carriers and continuous density

Counting individual electrons is useful for small devices and elementary transfer events. For macroscopic matter, the count is enormous, so charge is often represented by a density:

\[Q=\int_V\rho(\mathbf r)\,dV\]

Surface and line models similarly use surface density \(\sigma\) in coulombs per square meter and line density \(\lambda\) in coulombs per meter.

The continuum approximation does not deny quantization. It averages many discrete carriers over a region large compared with their spacing but small compared with the macroscopic geometry. A model should state that averaging scale.

Source charge and test charge

An electric field is defined as force per unit positive test charge in the limit that the test charge does not disturb the source configuration:

\[\mathbf E=\lim_{q_{\mathrm{test}}\to0} \frac{\mathbf F}{q_{\mathrm{test}}}\]

The ideal test charge is a conceptual probe. A real charged probe also creates a field and can move the sources. Treating sources as fixed is an electrostatic approximation, not a claim that Newton's third law stops applying.

A charge ledger in Python

Charge transfer should be atomic from the ledger's point of view: validate the event, apply opposite updates, and verify the total.

elementary_charge_c = 1.602176634e-19
# Store electron counts as integers to make the ledger exact.
electrons = {"sender": 12, "receiver": 3}
total_before = sum(electrons.values())
transfer = 2
if electrons["sender"] < transfer:
    raise ValueError("Sender lacks the requested electrons.")
electrons["sender"] -= transfer
electrons["receiver"] += transfer
assert sum(electrons.values()) == total_before
print("Electron counts:", electrons)
print("Transferred charge (C):", -transfer * elementary_charge_c)

For an open boundary, include a boundary account rather than disabling the check. Charge lost by the modeled interior is gained by the external account with the opposite sign.

Useful tests include transferring one elementary charge, moving an entity without changing its charge, reversing every charge sign, and polarizing a neutral pair without changing its total. The ledger checks conservation; separate field and force tests check whether signs are used consistently in interactions.

A worked Python notebook

Convert an integer electron transfer into signed electric charge. The question for this calculation is precise: how does transferred charge change when we change transferred electron count, 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.

Electron counts are integers; noninteger points on the plot show the proportional rule rather than allowed individual transfers. 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 to 100 count. 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 transferred electron count, expressed in count. The output is transferred charge, expressed in C. 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.

\[f(q) = - 1.602176634 \cdot 10^{-19} q\]

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 20 count, the expression gives -3.20435e-18 C. 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("-1.602176634e-19*q", locals={"q": q})
reference_input = 20.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "count")
print("Predicted transferred charge:", reference_output, "C")

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 C. The midpoint, at 50 count, gives -8.01088e-18 C. The upper endpoint gives -1.60218e-17 C. Read these together with the assumptions: they describe different choices of transferred electron count 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([0.0, 50.0, 100.0])
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} count -> {output_value:.6g} C")
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 transferred electron count; the vertical axis shows transferred charge. 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(0.0, 100.0, 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("Transferred electron count (count)")
ax.set_ylabel("Transferred charge (C)")
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 transferred charge change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 20 count is -1.60218e-19 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.

\[f'(q) = -1.602176634 \cdot 10^{-19}\]

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 = 20.5
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, "C")
print("Full expression:", full_prediction, "C")
print("Approximation difference:", full_prediction - linear_prediction, "C")

Here the changed input is 20.5 count. The tangent estimate is -3.28446e-18 C, while the full expression gives -3.28446e-18 C. 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 = 100.0
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 transferred electron count inside the stated interval, predict whether transferred charge 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: Electron counts are integers; noninteger points on the plot show the proportional rule rather than allowed individual transfers. 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 C: 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(100.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 100.0, 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

• OpenStax, University Physics Volume 2, Section 5.1, “Electric Charge”.
• OpenStax, University Physics Volume 2, Section 5.2, “Conductors, Insulators, and Charging by Induction”.
• Benjamin Crowell, Light and Matter, Chapter 21, “Electricity and Circuits”.
• Physics LibreTexts, “Electricity and Magnetism”.