Equipotential surfaces

An equipotential surface is the set of points having one chosen electric potential. In a two-dimensional slice, it appears as a contour line. It is not a material wall, a charged particle path, or an electric field line. It is a geometric level set of a scalar field.

Equipotentials reveal field direction and relative strength because the electrostatic field is the negative gradient of potential. Their interpretation depends on contour levels, spatial coordinates, source singularities, and numerical resolution.

Equipotentials are level sets

For a potential function \(V(\mathbf r)\), the equipotential at value \(V_0\) is defined by

\[V(\mathbf r)=V_0\]

In three dimensions this equation usually defines a surface. In a two-dimensional plane it usually defines a curve. In one dimension it gives isolated points.

For a positive point charge,

\[V(r)=\frac{1}{4\pi\epsilon_0}\frac{q}{r}\]

constant potential means constant distance from the charge. The three-dimensional equipotentials are concentric spheres; a plane through the charge shows concentric circles. The circles are cross-sections of spheres, not evidence that the field itself is two-dimensional.

The electric field is normal to the surface

Along a small displacement tangent to an equipotential, \(dV=0\). The differential of potential is

\[dV=\boldsymbol{\nabla}V \cdot d\boldsymbol{\ell}\]

Because

\[\mathbf E=-\boldsymbol{\nabla}V\]

the electric field has no tangent component:

\[\mathbf E\cdot d\boldsymbol{\ell}=0\]

At a regular point, the field is perpendicular to the equipotential and points toward lower potential. The statement becomes ill-defined at a critical point where \(\boldsymbol{\nabla}V=\mathbf0\), because a zero vector has no direction.

Motion along an equipotential

The electrostatic work on a charge moving between two points on the same equipotential is

\[W_{\mathrm{field}} =-q\Delta V=0\]

This says only that the electrostatic force performs no net work. An external agent may still do work against gravity, friction, or a constraint. A charged particle also does not automatically follow an equipotential; the electric force points normally away from it for positive charge and oppositely for negative charge.

Common geometries

A uniform electric field in the positive horizontal direction can be represented by

\[V(x)=V_0-E_xx\]

Its equipotentials are parallel planes of constant \(x\), perpendicular to the field. Equal voltage intervals give equally spaced planes because the field magnitude is constant.

For equal positive and negative point charges, the perpendicular bisector between them has zero potential because the scalar contributions cancel. The electric field on that plane is generally nonzero and points from the positive charge toward the negative charge. This is a useful reminder that zero potential is a reference-dependent scalar value, not a zero-field condition.

For equal like charges, the midpoint can have zero field while its potential is nonzero. Near that critical point, contour topology can form a saddle-like pattern, and a field-angle test becomes unstable because the reference vector is nearly zero.

Contour spacing indicates field strength

The field magnitude is the rate at which potential changes with distance in the steepest direction. Between nearby equipotentials separated normally by \(\Delta n\) and differing by \(\Delta V\),

\[E_n\approx-\frac{\Delta V}{\Delta n}\]

If every contour uses the same voltage interval, closer contours suggest a larger field magnitude. If the contour levels are uneven, spacing alone cannot be compared. A plot must show numerical levels or a legend.

This relation is local. Curved contours and rapidly varying fields require sufficiently small separations for the finite ratio to approximate the derivative.

Conductors in electrostatic equilibrium

Inside a conductor in electrostatic equilibrium, the electric field is zero. Otherwise mobile charges would continue to move. Therefore the potential is constant throughout the conductor and over its surface:

\[\mathbf E_{\mathrm{inside}}=\mathbf0, \qquad V_{\mathrm{conductor}}=\mathrm{constant}\]

Just outside the surface, the electric field has no tangent component; a tangent field would drive surface charges. In vacuum, its normal component satisfies

\[\mathbf E_{\mathrm{outside}}\cdot\hat{\mathbf n} =\frac{\sigma}{\epsilon_0}\]

Here \(\hat{\mathbf n}\) is the outward unit normal and \(\sigma\) is the local surface charge density. The expression assumes the field inside is zero and the adjacent region is vacuum.

A conductor surface is a physical boundary and an equipotential, but most equipotential surfaces are not conductors.

Potential reference shifts relabel levels

Adding a constant \(C\) gives

\[V'(\mathbf r)=V(\mathbf r)+C\]

The geometric surface formerly labeled \(V_0\) is now labeled \(V_0+C\). Its shape and the electric field are unchanged. A correct contour pipeline shifts labels without moving vertices when every sampled potential receives the same offset.

Extracting contours from a grid

A numerical contour algorithm examines each grid cell and determines where the chosen level crosses its edges. Linear interpolation along an edge with endpoint values \(V_1\) and \(V_2\) uses the fraction

\[f=\frac{V_0-V_1}{V_2-V_1}\]

The crossing point lies fraction \(f\) of the way from the first endpoint to the second. This is valid only when the level is bracketed and the denominator is nonzero.

Cells with a vertex exactly on the level, an entire edge on the level, or alternating high and low corners require deterministic rules. Otherwise tiny rounding changes can alter connectivity. A marching-squares ambiguity can be resolved using the cell-center potential or another documented rule.

A narrow contour procedure

The extractor should retain the requested voltage with every segment.

import numpy as np
import matplotlib.pyplot as plt

coordinates = np.linspace(-2, 2, 201)
x, y = np.meshgrid(coordinates, coordinates)
radius = np.hypot(x, y)
voltage = np.ma.array(8.9875517923 / np.maximum(radius, 0.1), mask=radius < 0.1)
fig, ax = plt.subplots()
contours = ax.contour(x, y, voltage, levels=[5, 10, 15, 20])
ax.clabel(contours, fmt="%g V")
ax.set(xlabel="x (m)", ylabel="y (m)")
ax.set_aspect("equal")
plt.show()

Joining needs a spatial tolerance linked to grid scale. A tolerance that is too small leaves gaps; one that is too large connects distinct contours.

Orthogonality as a numerical check

At a contour vertex, estimate a tangent \(\hat{\mathbf t}\) from neighboring vertices and evaluate an independent electric field. The normalized residual

\[r_\perp =\frac{|\mathbf E\cdot\hat{\mathbf t}|} {\lVert\mathbf E\rVert}\]

should be small where the field is nonzero and the contour is resolved. Near a null, division by field magnitude is unstable, so report an absolute dot product or mark the angle undetermined.

Also evaluate the original potential function at each interpolated vertex and compare it with the requested level. Refine the grid and check whether both level error and geometry change decrease.

Singularities and boundaries

Point-charge potential diverges at the source. Contours cannot be reliably extracted from a cell containing that singularity using ordinary linear interpolation. Mask the cell, use an analytic local treatment, or replace the point with a finite source model.

Contours can meet the domain boundary. They should be reported as open segments rather than artificially closed. The boundary values and domain extent affect which pieces are visible but do not create new physical equipotentials.

Color-filled contours may hide narrow regions or make interpolation look smoother than the underlying grid. Preserve numerical levels, grid resolution, and exact probe values as text.

Verification and interpretation

Start with analytic fixtures. A point charge should produce circular contours in a planar slice. A uniform field should produce straight, equally spaced contours for equal voltage steps. Equal opposite charges should produce a zero-potential bisector with a nonzero normal field.

Add a uniform potential constant and confirm that the same geometric contours receive shifted labels. Halve the grid spacing while keeping the physical domain and levels fixed, then compare vertex-level error, curve position, connectivity, and field-orthogonality residuals.

An equipotential drawing is trustworthy only when it names voltage levels, coordinate plane, source configuration, reference convention, interpolation, resolution, and singularity treatment. Those details make it a quantitative view of a scalar field rather than decorative lines.

A worked Python notebook

Solve for the radius of an equipotential surface around a positive one nanocoulomb charge. The question for this calculation is precise: how does equipotential sphere radius change when we change chosen voltage level, 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.

Spherical equipotentials are specific to a single isolated point charge. 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 3 to 30 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 chosen voltage level, expressed in V. The output is equipotential sphere radius, expressed in m. 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 10 V, the expression gives 0.898755 m. 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 = 10.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "V")
print("Predicted equipotential sphere radius:", reference_output, "m")

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 2.99585 m. The midpoint, at 16.5 V, gives 0.5447 m. The upper endpoint gives 0.299585 m. Read these together with the assumptions: they describe different choices of chosen voltage level 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([3.0, 16.5, 30.0])
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} m")
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 chosen voltage level; the vertical axis shows equipotential sphere radius. 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(3.0, 30.0, 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("Chosen voltage level (V)")
ax.set_ylabel("Equipotential sphere radius (m)")
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 equipotential sphere radius change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 10 V is -0.0898755 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 = 10.135
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, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 10.135 V. The tangent estimate is 0.886622 m, while the full expression gives 0.886784 m. 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 = 27.0
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 chosen voltage level inside the stated interval, predict whether equipotential sphere radius 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: Spherical equipotentials are specific to a single isolated point charge. 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 m: 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(30.0) - evaluate(3.0))
for points in [101, 201, 401]:
    grid = np.linspace(3.0, 30.0, 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 7.4, “Determining Field from Potential”.
• OpenStax, University Physics Volume 2, Section 7.5, “Equipotential Surfaces and Conductors”.
• OpenStax, University Physics Volume 2, Section 6.4, “Conductors in Electrostatic Equilibrium”.
• Benjamin Crowell, Light and Matter, Chapter 22, “Fields”.
• Physics LibreTexts, “Electricity and Magnetism”.