Symbolic field calculations

A symbolic electric-field result is more than a compact formula. It exposes direction, symmetry, scaling, units, discontinuities, and limiting behavior. Those properties make analytic fields valuable references for numerical calculations, provided both descriptions use the same geometry, material, boundaries, and domain.

Computer algebra can differentiate and integrate expressions, but it does not decide the physical assumptions. A beautifully simplified expression can still describe the wrong source or silently exclude a singular point.

Integral and differential constraints

Gauss's law relates electric flux through a closed surface to enclosed charge:

\[\oint_{\partial V} \mathbf E\cdot d\mathbf a =\frac{Q_{\mathrm{enclosed}}}{\epsilon_0}\]

The vector area element \(d\mathbf a\) points outward from the closed surface. Only charge inside the surface contributes to the right side, although charges outside can contribute local field that enters and leaves with zero net flux.

The local differential form is

\[\boldsymbol{\nabla}\cdot\mathbf E =\frac{\rho}{\epsilon_0}\]

Here \(\rho\) is charge per unit volume. The equation states that electric charge is a source or sink of electric flux. It does not say that the field at one point depends only on the charge density at that point; boundary conditions and all sources determine the solution.

In electrostatics,

\[\boldsymbol{\nabla}\times\mathbf E=\mathbf 0\]

This condition makes the field conservative and permits a scalar potential with \(\mathbf E=-\boldsymbol{\nabla}V\).

Symmetry is the engine of Gauss-law solutions

Gauss's law is always true, but it determines the field easily only when symmetry makes the field magnitude constant over useful parts of a closed surface and fixes its direction relative to the surface normal.

For an infinite straight line with uniform line charge density \(\lambda\), cylindrical symmetry requires a radial field depending only on perpendicular distance \(s\). A coaxial Gaussian cylinder of radius \(s\) and length \(L\) gives

\[E(2\pi sL)=\frac{\lambda L}{\epsilon_0}\]

Therefore

\[\mathbf E(s) =\frac{\lambda}{2\pi\epsilon_0s}\, \hat{\mathbf s}\]

The unit vector \(\hat{\mathbf s}\) points away from the line. A negative line density reverses the field. The inverse first power of distance differs from the inverse-square point-charge field because the source extends infinitely in one dimension.

Infinite plane field

For an infinite plane with uniform surface charge density \(\sigma\), symmetry makes the field perpendicular to the plane and equal in magnitude on both sides. A thin cylindrical pillbox of face area \(A\) has flux through its two flat faces:

\[2EA=\frac{\sigma A}{\epsilon_0}\]

Thus

\[E=\frac{|\sigma|}{2\epsilon_0}\]

The magnitude does not decrease with distance because an infinite source contributes from ever more distant area as the observation point recedes. This is an idealization. A finite sheet approaches the plane result only close to its center compared with its lateral dimensions; far away, any bounded sheet behaves like a point charge with its total charge.

Spherical symmetry

For a spherically symmetric charge distribution, the field is radial and has constant magnitude on a centered Gaussian sphere. If \(Q_{\mathrm{enc}}(r)\) is the enclosed charge,

\[\mathbf E(r) =\frac{1}{4\pi\epsilon_0} \frac{Q_{\mathrm{enc}}(r)}{r^2}\, \hat{\mathbf r}\]

Outside a spherical distribution of total charge \(Q\), the field is exactly the same as a point charge at the center. Inside a thin spherical shell, \(Q_{\mathrm{enc}}=0\), so the field is zero.

For a solid sphere of radius \(R\) and uniform density \(\rho\), the enclosed charge at an interior radius is

\[Q_{\mathrm{enc}}(r) =\rho\frac{4\pi r^3}{3}\]

Substitution gives

\[\mathbf E(r) =\frac{\rho r}{3\epsilon_0}\, \hat{\mathbf r}, \qquad r<R\]

The interior field grows linearly from zero at the center. At the surface, it matches the exterior point-charge expression when the total charge and density are related consistently.

Coordinate formulas must match the geometry

The divergence operator changes form with coordinates. For a purely radial spherical field \(\mathbf E=E_r(r)\hat{\mathbf r}\),

\[\boldsymbol{\nabla}\cdot\mathbf E =\frac{1}{r^2} \frac{d}{dr}\left(r^2E_r\right)\]

Using only an ordinary Cartesian-looking derivative would omit the growth of spherical area. For a point-charge field, the expression is zero at every nonzero radius because the squared-radius factor cancels the inverse-square magnitude.

The source has not disappeared. At the origin the field is singular, and the complete distributional statement is

\[\boldsymbol{\nabla}\cdot \left(\frac{\hat{\mathbf r}}{r^2}\right) =4\pi\delta^{(3)}(\mathbf r)\]

The three-dimensional delta distribution represents the point source. A symbolic system that simplifies the divergence to zero has implicitly assumed \(r>0\). That domain restriction must be retained.

Limits reveal hidden mistakes

Every closed form should be tested at ordinary, symmetric, near-boundary, and far-field points. For a bounded source with total charge \(Q\), the leading far field is

\[\mathbf E(\mathbf r) \approx\frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}\hat{\mathbf r}, \qquad r\gg a\]

Here \(a\) is a source length scale. If total charge is zero, this leading term vanishes and dipole or higher multipole terms control the far field. Comparing a neutral source with a nonzero point-charge reference would be the wrong test.

At symmetry points, use absolute error because relative error against an exact zero is undefined. Away from zero, relative error can compare values over different scales.

Piecewise results are often physically necessary

Fields can have different expressions inside and outside a source. A symbolic result for a uniform sphere must retain the conditions \(r<R\) and \(r>R\). Merging the branches without their domains can create a formula that is correct nowhere.

Absolute values also matter. The distance from an infinite plane is nonnegative, while the field direction depends on which side of the plane contains the observation point. A scalar magnitude can use absolute value, but a vector result needs an appropriate normal direction and sign.

At a surface charge, the normal field component has a jump:

\[\left(\mathbf E_{\mathrm{above}} -\mathbf E_{\mathrm{below}}\right) \cdot\hat{\mathbf n} =\frac{\sigma}{\epsilon_0}\]

The unit normal \(\hat{\mathbf n}\) is chosen to point from the side labeled below toward the side labeled above. Evaluating exactly on the ideal sheet requires a convention; limits from the two sides are the physically informative quantities.

Symbolic-to-numerical verification

Choose tests that probe different failure modes. Exact symmetry checks signs and component assembly. An off-axis point checks the general geometry. A far-field ratio checks total charge and scaling. A boundary-side pair checks discontinuities.

import sympy as sp
import numpy as np

x, y, k, q = sp.symbols("x y k q", real=True)
potential = k * q / sp.sqrt(x * x + y * y)
field = [-sp.diff(potential, coordinate) for coordinate in (x, y)]
evaluate = sp.lambdify((x, y, k, q), field, "numpy")
probe = np.array([0.8, 0.6])
reference = evaluate(*probe, 8.9875517923e9, 1e-9)
vector_form = 8.9875517923 * probe / np.linalg.norm(probe) ** 3
assert np.allclose(reference, vector_form)
print("Symbolic field evaluated at probe:", reference)

Keep physical probe coordinates fixed during refinement. Moving probes with grid indices mixes spatial change with resolution change.

Dimensional and flux checks

The field must have units of newtons per coulomb or volts per meter. A line density must be multiplied by a length, a surface density by an area, and a volume density by a volume before the result represents charge.

Numerically integrate the analytic field's flux through a closed surface and compare it with enclosed charge divided by vacuum permittivity. Also calculate divergence away from sources and curl in the electrostatic region. These checks are independent only if they do not reuse the same mistaken derivative or source helper.

Symbolic algebra is most valuable when it exposes assumptions rather than hiding them. Record whether a source is infinite or finite, uniform or varying, conducting or prescribed, in vacuum or material, static or moving, and which singularities are excluded. A formula without that domain is not yet a reliable physical result.

A worked Python notebook

Differentiate the potential of a one nanocoulomb point charge along a positive radial axis. The question for this calculation is precise: how does point-charge potential change when we change axial distance, 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 derivative is the potential slope; the electric field is its negative. 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 0.5 to 3 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 axial distance, expressed in m. The output is point-charge potential, expressed in V. 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.

\[f(q) = \frac{8.9875517923}{q}\]

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 1 m, the expression gives 8.98755 V. 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("8.9875517923/q", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted point-charge potential:", reference_output, "V")

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 17.9751 V. The midpoint, at 1.75 m, gives 5.13574 V. The upper endpoint gives 2.99585 V. Read these together with the assumptions: they describe different choices of axial distance 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([0.5, 1.75, 3.0])
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} V")
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 axial distance; the vertical axis shows point-charge potential. 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(0.5, 3.0, 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("Axial distance (m)")
ax.set_ylabel("Point-charge potential (V)")
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 point-charge potential change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 m is -8.98755 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.

\[f'(q) = - \frac{8.9875517923}{q^{2}}\]

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 = 1.0125
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, "V")
print("Full expression:", full_prediction, "V")
print("Approximation difference:", full_prediction - linear_prediction, "V")

Here the changed input is 1.0125 m. The tangent estimate is 8.87521 V, while the full expression gives 8.87659 V. 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 = 2.5
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 axial distance inside the stated interval, predict whether point-charge potential 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 derivative is the potential slope; the electric field is its negative. 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 V: 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(3.0) - evaluate(0.5))
for points in [101, 201, 401]:
    grid = np.linspace(0.5, 3.0, 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 6.1, “Electric Flux”.
• OpenStax, University Physics Volume 2, Section 6.2, “Explaining Gauss’s Law”.
• OpenStax, University Physics Volume 2, Section 6.3, “Applying Gauss’s Law”.
• Benjamin Crowell, Light and Matter, Chapter 22, “Fields”.
• Physics LibreTexts, “Electricity and Magnetism”.