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Capacitor models

A capacitor consists of two conductors separated by an insulating region. Moving charge from one conductor to the other creates equal and opposite conductor charges and a potential difference. Capacitance measures how much separated charge is stored per unit potential difference.

Capacitance is determined by geometry and material response in a linear electrostatic model. It is not a universal constant attached to the word "capacitor."

Definition and units

For a linear two-conductor capacitor,

\[C=\frac{Q}{\Delta V}\]

Here \(Q\) is the magnitude of charge on either conductor, whose charges are \(+Q\) and \(-Q\). The potential difference \(\Delta V\) is taken consistently between the same labeled conductors.

The SI unit of capacitance is the farad:

\[1\ \mathrm F=1\ \mathrm{C/V}\]

One farad is large for many ordinary components, so microfarads, nanofarads, and picofarads are common. Capacitance is positive under the magnitude convention even though the signed conductor charges and signed potential difference depend on labeling.

Parallel-plate derivation

Consider two large parallel conducting plates of facing area \(A\) separated by distance \(d\) and filled by a uniform linear dielectric of permittivity \(\epsilon\). Neglect edge fringing.

If the surface charge magnitude is \(\sigma=Q/A\), the fields of the two oppositely charged plates add between the plates and cancel outside:

\[E=\frac{\sigma}{\epsilon} =\frac{Q}{\epsilon A}\]

The potential difference magnitude is

\[\Delta V=Ed=\frac{Qd}{\epsilon A}\]

Therefore

\[C=\frac{\epsilon A}{d}\]

Larger area increases capacitance because more charge can be placed on the facing surfaces for the same field. Larger separation decreases capacitance because the same field acts across a greater distance and produces a larger voltage.

A numerical scale

For vacuum plates with area

\(A=1.00\times10^{-2}\ \mathrm{m^2}\)

and separation \(d=1.00\times10^{-3}\ \mathrm m\),

\[C =\frac{\epsilon_0A}{d} \approx8.85\times10^{-11}\ \mathrm F =88.5\ \mathrm{pF}\]

This calculation assumes the plate dimensions are much larger than the separation. If that ratio is not large, fringing fields increase the effective capacitance relative to the simplest uniform-field estimate.

Isolated and spherical capacitors

An isolated conducting sphere of radius \(R\) can be treated as a capacitor whose second conductor is effectively at infinity. Its potential is \(V=Q/(4\pi\epsilon_0R)\), so

\[C_{\mathrm{sphere}}=4\pi\epsilon_0R\]

A spherical capacitor made from concentric conductors of radii \(a\) and \(b\), with \(b>a\), has

\[C_{\mathrm{spherical}} =4\pi\epsilon \frac{ab}{b-a}\]

As the outer radius grows without bound, this expression approaches the isolated-sphere result. These examples show that capacitance depends on full conductor geometry, not only a nearest separation.

Cylindrical geometry

For long coaxial cylindrical conductors of length \(L\), inner radius \(a\), and outer radius \(b\), neglecting end effects,

\[C_{\mathrm{coax}} =\frac{2\pi\epsilon L}{\ln(b/a)}\]

The logarithm comes from integrating the inverse-distance radial field between the two conductors. A cable specified per unit length uses \(C/L\).

Dielectric materials

For a uniform linear dielectric,

\[\epsilon=\kappa\epsilon_0\]

where \(\kappa\) is the dimensionless relative permittivity. In a fully filled ideal capacitor, capacitance becomes \(\kappa\) times its vacuum value.

The physical response depends on the electrical connection:

A dielectric also has a breakdown field. Above that field, it can conduct or be damaged, so the linear insulating model fails even though the capacitance formula still returns a number.

Combining ideal capacitors

Capacitors connected in parallel share the same voltage, and their charges add:

\[C_{\mathrm{parallel}}=\sum_i C_i\]

The equivalent capacitance is larger than any individual positive capacitance.

Capacitors connected in series carry equal-magnitude charge in the ideal steady state, and their voltage differences add:

\[\frac{1}{C_{\mathrm{series}}} =\sum_i\frac{1}{C_i}\]

The series equivalent is smaller than the smallest individual capacitance. These rules assume ideal two-terminal components with no significant parasitic coupling to other conductors.

Choose the independent electrical state

The relation \(Q=C\Delta V\) means that charge and voltage cannot be independently assigned after capacitance is fixed. A model should accept one and derive the other.

epsilon, area, separation = 8.8541878128e-12, 0.02, 0.001
capacitance = epsilon * area / separation
fixed_voltage = 100.0
charge = capacitance * fixed_voltage
print(f"Capacitance: {capacitance:.6g} F; charge: {charge:.6g} C")
# Isolate the capacitor, then double the separation at fixed charge.
new_capacitance = epsilon * area / (2 * separation)
new_voltage = charge / new_capacitance
print(f"Isolated capacitor after widening: {new_voltage:.1f} V")

The field estimate in the final lines applies to the ideal parallel-plate geometry, not to arbitrary capacitors.

Scaling checks

For an ideal parallel-plate capacitor, doubling area at fixed separation doubles capacitance. Doubling separation at fixed area halves it. Multiplying permittivity by a factor multiplies capacitance by the same factor.

Check both electrical boundary conditions. At fixed voltage, doubling capacitance doubles charge. At fixed charge, doubling capacitance halves voltage. A system that accepts unchanged charge and unchanged voltage after changing capacitance is internally inconsistent.

For series and parallel combinations, test two equal capacitors. Parallel connection gives twice one capacitance; series connection gives half. Verify units and require positive finite capacitances.

Where ideal models need refinement

Finite plates have fringing fields. Conductors and wires add stray capacitance. Real dielectrics can leak, absorb charge, depend on frequency and temperature, exhibit loss, and become nonlinear at strong fields. Manufacturing tolerances make component values uncertain.

At frequencies where dimensions are no longer small compared with electromagnetic wavelength, a single lumped capacitance does not describe propagation and radiation. Very small structures can also require quantum and microscopic material models.

A trustworthy capacitor result states conductor geometry, dielectric, electrical boundary condition, neglected fringing or end effects, and the applicable frequency and field range. The compact formulas are powerful because their assumptions are visible.

The chosen state variable also affects a time-dependent model. Charge is continuous through an ideal capacitor unless the model includes an impulsive current, whereas an ideal prescribed voltage source can impose voltage directly. Recording which quantity is controlled prevents a solver from independently updating charge and voltage into an impossible pair.

A worked Python notebook

Change the gap of a vacuum parallel-plate capacitor with plate area 0.02 square metres. The question for this calculation is precise: how does capacitance change when we change plate separation, 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 gap must remain small compared with the plate dimensions for fringing to be neglected. 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 0.0005 to 0.005 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 plate separation, expressed in m. The output is capacitance, expressed in F. 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.

\[f(q) = \frac{1.77083756256 \cdot 10^{-13}}{q}\]

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 0.001 m, the expression gives 1.77084e-10 F. 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("8.8541878128e-12*0.02/q", locals={"q": q})
reference_input = 0.001
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted capacitance:", reference_output, "F")

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 3.54168e-10 F. The midpoint, at 0.00275 m, gives 6.43941e-11 F. The upper endpoint gives 3.54168e-11 F. Read these together with the assumptions: they describe different choices of plate separation 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([0.0005, 0.00275, 0.005])
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} F")
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 plate separation; the vertical axis shows capacitance. 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(0.0005, 0.005, 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("Plate separation (m)")
ax.set_ylabel("Capacitance (F)")
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 capacitance change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.001 m is -1.77084e-07 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.

\[f'(q) = - \frac{1.77083756256 \cdot 10^{-13}}{q^{2}}\]

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 = 0.0010225
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, "F")
print("Full expression:", full_prediction, "F")
print("Approximation difference:", full_prediction - linear_prediction, "F")

Here the changed input is 0.0010225 m. The tangent estimate is 1.73099e-10 F, while the full expression gives 1.73187e-10 F. 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 = 0.0045000000000000005
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 plate separation inside the stated interval, predict whether capacitance 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 gap must remain small compared with the plate dimensions for fringing to be neglected. 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 F: 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(0.005) - evaluate(0.0005))
for points in [101, 201, 401]:
    grid = np.linspace(0.0005, 0.005, 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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