Energy storage

Charging a capacitor separates positive and negative charge. Work done during that separation becomes energy stored in the electric field. The stored energy can be written using capacitance and voltage, capacitance and charge, or charge and voltage, but the quantities must describe the same electrical state.

Energy is derived from compatible state. It is not a fourth independent value that can be chosen separately from charge, voltage, and capacitance.

Work required to charge a capacitor

For an initially uncharged linear capacitor of capacitance \(C\), the voltage after charge \(q\) has accumulated is

\[V(q)=\frac{q}{C}\]

Moving an additional small charge \(dq\) requires work

\[dW=V(q)\,dq=\frac{q}{C}\,dq\]

Early charge is inexpensive to move because the voltage is small. Later charge is moved against a larger potential difference. Integrating from zero to final charge \(Q\) gives

\[U =\int_0^Q\frac{q}{C}\,dq =\frac{Q^2}{2C}\]

Using \(Q=CV\) produces the equivalent forms

\[U=\frac12CV^2 =\frac{Q^2}{2C} =\frac12QV\]

The factor one half appears because voltage rises linearly from zero to its final value. Multiplying final charge by final voltage without that factor overcounts the work.

Signs and nonnegative stored energy

Voltage and charge can be signed according to terminal labels, but for a passive linear capacitor they have consistent signs under \(Q=CV\) with positive capacitance. Therefore \(QV/2\) and the squared forms give nonnegative stored energy.

Swapping both terminal labels reverses \(Q\) and \(V\) but leaves their product and stored energy unchanged.

At fixed capacitance, doubling voltage magnitude quadruples energy. At fixed voltage, doubling capacitance doubles energy. At fixed charge, doubling capacitance halves energy. These trends differ because the external constraint differs.

Power entering a capacitor

With the passive sign convention, current enters the terminal labeled with positive voltage. For a constant linear capacitance,

\[i=C\frac{dV}{dt}\]

The instantaneous electrical power entering the capacitor is

\[p=Vi =CV\frac{dV}{dt} =\frac{d}{dt}\left(\frac12CV^2\right)\]

Positive power increases stored energy; negative power releases it. If capacitance itself changes with time, the simple derivative needs an additional term and mechanical or material energy transfer may occur.

Energy density between parallel plates

For an ideal parallel-plate capacitor with uniform linear dielectric permittivity \(\epsilon\), plate area \(A\), and separation \(d\),

\[C=\frac{\epsilon A}{d}, \qquad E=\frac{V}{d}\]

Substitution into the capacitor energy gives

\[U =\frac12\epsilon E^2(Ad)\]

Because \(Ad\) is the volume between the plates, the energy density is

\[u_E=\frac12\epsilon E^2\]

This result neglects fringing. In a nonlinear, dispersive, or lossy dielectric, the simple constant-permittivity energy density may not apply.

Charging through a resistor

Connect an initially uncharged capacitor through resistance \(R\) to an ideal constant-voltage source \(V_0\). The ideal charging solution is

\[V_C(t)=V_0\left(1-e^{-t/(RC)}\right), \qquad i(t)=\frac{V_0}{R}e^{-t/(RC)}\]

The product \(RC\) is the time constant. After many time constants, capacitor voltage approaches the source voltage and current approaches zero.

The capacitor's final energy is

\[U_C(\infty)=\frac12CV_0^2\]

The source supplies

\[W_{\mathrm{source}} =\int_0^\infty V_0i(t)\,dt =CV_0^2\]

The resistor dissipates

\[Q_R =\int_0^\infty i(t)^2R\,dt =\frac12CV_0^2\]

Half the source work is stored and half is dissipated in this ideal fixed-voltage charging process. The total dissipated energy is independent of the positive resistance value; resistance changes the rate, not this final ideal partition. The limit of zero resistance cannot be handled by simply inserting zero into the lumped equations because parasitic inductance, radiation, and finite source impedance then matter.

Discharging through a resistor

If a charged capacitor with initial voltage \(V_0\) discharges through a resistor with no source,

\[V_C(t)=V_0e^{-t/(RC)}\]

The stored energy decays as

\[U_C(t)=\frac12CV_0^2e^{-2t/(RC)}\]

All initial capacitor energy becomes resistor heating in the ideal model. Voltage has time constant \(RC\), while energy contains an exponential with twice the decay rate because it depends on voltage squared.

Changing capacitance and dielectrics

Suppose a dielectric is inserted and raises capacitance. If the capacitor is isolated, charge remains fixed:

\[U_Q=\frac{Q^2}{2C}\]

Energy falls as capacitance rises. The difference can appear as mechanical work pulling the dielectric inward or as other material energy transfer.

If an ideal voltage source remains connected, voltage stays fixed:

\[U_V=\frac12CV^2\]

Stored capacitor energy rises as capacitance rises because the source supplies additional charge and energy. The complete accounting includes work by the source and any mechanical work, not only the capacitor term.

A transient energy ledger

Sampled data require numerical integration of power. State the quadrature rule and use the same time intervals for every channel.

import numpy as np

R, C, supply = 1000.0, 1e-6, 5.0
times = np.linspace(0, 10 * R * C, 10001)
voltage = supply * (1 - np.exp(-times / (R * C)))
current = supply / R * np.exp(-times / (R * C))
source_work = np.trapezoid(supply * current, times)
resistor_heat = np.trapezoid(R * current**2, times)
stored = 0.5 * C * voltage[-1] ** 2
residual = source_work - resistor_heat - stored
print("Source work, heat, storage (J):", source_work, resistor_heat, stored)
print("Ledger residual (J):", residual)

For a discharging circuit without a source, the sign convention and initial-energy term must be adjusted explicitly. Do not force every interval to balance by overwriting one ledger entry with the residual.

Verification and safety limits

Check that all three ideal energy formulas agree for the same compatible values. Zero charge and zero voltage must give zero energy. Reversing terminal labels must leave energy unchanged. Doubling voltage at fixed capacitance must multiply energy by four.

For the RC model, compare integrated source work with stored-energy change plus resistor heating. Halve the numerical time step and confirm that the residual decreases. Also compare the analytic final partition with the numerical integral.

Real capacitors have voltage ratings, dielectric breakdown, leakage, equivalent series resistance, dielectric absorption, temperature dependence, and parasitic inductance. Stored energy can remain hazardous after power is disconnected. A model should compare electric field with the material breakdown limit and include discharge paths where safety matters.

At high frequency or large physical size, energy propagates through electromagnetic fields and a lumped capacitor is incomplete. The simple formulas remain valuable within their range because they connect charging work, circuit power, and spatial field energy in one consistent ledger.

For a passive ideal capacitor with positive capacitance, stored energy cannot be negative. That invariant is stronger than checking only that a program returned a finite number. A negative result points to an invalid capacitance, an algebraic sign mistake, or confusion between stored energy and signed energy transferred during an interval.

A worked Python notebook

Charge a one microfarad capacitor to several voltages. The question for this calculation is precise: how does stored energy change when we change capacitor voltage, 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.

Stored capacitor energy differs from total source work when the charging path contains resistance. 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 10 V. 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 capacitor voltage, expressed in V. The output is stored energy, expressed in J. 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.

\[f(q) = 5.0 \cdot 10^{-7} q^{2}\]

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 5 V, the expression gives 1.25e-05 J. 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("1e-6*q**2/2", locals={"q": q})
reference_input = 5.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "V")
print("Predicted stored energy:", reference_output, "J")

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 J. The midpoint, at 5 V, gives 1.25e-05 J. The upper endpoint gives 5e-05 J. Read these together with the assumptions: they describe different choices of capacitor voltage 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, 5.0, 10.0])
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} V -> {output_value:.6g} J")
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 capacitor voltage; the vertical axis shows stored energy. 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, 10.0, 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("Capacitor voltage (V)")
ax.set_ylabel("Stored energy (J)")
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 stored energy change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 5 V is 5e-06 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.

\[f'(q) = 1.0 \cdot 10^{-6} q\]

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 = 5.05
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, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 5.05 V. The tangent estimate is 1.275e-05 J, while the full expression gives 1.27512e-05 J. 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 = 10.0
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 capacitor voltage inside the stated interval, predict whether stored energy 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: Stored capacitor energy differs from total source work when the charging path contains resistance. 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 J: 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(10.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 10.0, 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

Share to: