Scalar potentials

Electric potential assigns one scalar value to each point in an electrostatic region. Potential difference tells how much electric potential energy changes per unit charge. Because a scalar is often simpler to add and differentiate than a vector field, potential provides a powerful route from sources to forces and energy.

Only potential differences are physically determined. Adding the same constant everywhere changes the chosen zero but does not change the electric field, force, or energy difference.

Potential difference from work

When the electrostatic field moves a test charge \(q\) from point \(A\) to point \(B\), the field's work is

\[W_{A\to B}^{\mathrm{field}} =q\int_A^B\mathbf E\cdot d\boldsymbol{\ell}\]

The change in electric potential energy is the negative of that work:

\[\Delta U =U_B-U_A =-W_{A\to B}^{\mathrm{field}}\]

Electric potential difference is potential-energy change per unit charge:

\[\Delta V =V_B-V_A =\frac{\Delta U}{q} =-\int_A^B\mathbf E\cdot d\boldsymbol{\ell}\]

The signs tell a physical story. A positive charge naturally moving in the field direction loses potential energy and moves toward lower potential. A negative charge naturally accelerating opposite the field direction can lose potential energy while moving toward higher potential.

Potential has SI units of joules per coulomb, called volts:

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

Electrostatic path independence

For an electrostatic field, the line integral between two points is independent of the path. Equivalently,

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

This is why one scalar potential can describe the field. If two numerical paths between the same endpoints give materially different integrals in a supposedly electrostatic calculation, the field, integration, or resolution is inconsistent.

Time-varying magnetic flux creates a nonconservative electric field with nonzero circulation. In that situation a single electrostatic scalar potential is not sufficient to describe the complete electric field. The present lesson assumes electrostatics.

Potential of a point charge

For a point charge \(q\) and zero potential chosen at infinite distance, the potential at distance \(r\) is

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

Potential retains the source charge sign. A positive point source has positive potential relative to infinity; a negative source has negative potential. Unlike the vector field, the scalar formula contains no direction.

Doubling distance halves the point-source potential magnitude. The corresponding field magnitude falls by a factor of four because differentiating the inverse-distance potential produces inverse-square behavior.

At \(r=0\), the ideal potential diverges. A strict evaluator should reject the point. A finite source distribution needs its own potential rather than an unexplained numerical softening.

Superposition is scalar addition

For point charges \(q_i\) at positions \(\mathbf r_i\), the total potential is

\[V(\mathbf r) =\frac{1}{4\pi\epsilon_0} \sum_i\frac{q_i} {\lVert\mathbf r-\mathbf r_i\rVert}+C\]

The constant \(C\) chooses the reference. Contributions add as signed scalars; they do not have components.

At the midpoint of equal opposite charges, the potentials cancel to zero. Their electric field vectors point in the same direction there and add. Therefore zero potential does not imply zero field.

For equal like charges, the field can cancel at the midpoint while the potentials add. Therefore zero field does not imply zero potential. Using both symmetry cases prevents field and potential rules from being confused.

Continuous charge distributions

For volume charge density \(\rho(\mathbf r')\), the potential is

\[V(\mathbf r) =\frac{1}{4\pi\epsilon_0} \int_V \frac{\rho(\mathbf r')} {\lVert\mathbf r-\mathbf r'\rVert} \,dV'+C\]

The primed coordinate labels the source element; the unprimed coordinate labels the observation point. Line and surface distributions replace volume density and volume measure with the corresponding line or surface quantities.

Potential integration is often easier than direct field integration because the kernel is scalar. Once a reliable potential is known, differentiation gives the field. Near sharp sources or boundaries, however, numerical differentiation can amplify discretization noise.

The additive reference is a gauge choice

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

then

\[V'_B-V'_A=V_B-V_A, \qquad -\boldsymbol{\nabla}V' =-\boldsymbol{\nabla}V\]

The reference changes absolute labels but no electrostatic prediction. Potential energy of a charge shifts by a constant proportional to that charge, while energy differences remain unchanged.

Choosing zero at infinity is convenient for bounded charge distributions whose potential approaches a finite limit there. It fails for ideal infinite sources such as an infinite line, whose potential difference grows logarithmically with distance. For such systems, choose a finite reference point and report only differences.

A finite reference example

For an infinite line with uniform line charge density \(\lambda\), the radial field has magnitude \(E=\lambda/(2\pi\epsilon_0r)\) for positive density. Between radii \(r_A\) and \(r_B\),

\[V_B-V_A =-\frac{\lambda}{2\pi\epsilon_0} \ln\left(\frac{r_B}{r_A}\right)\]

No potential at infinity was required. If \(r_B>r_A\) for a positive line, the logarithm is positive and the potential decreases outward, consistent with an outward electric field.

Potential energy of a test charge

Once the external potential is known, a test charge has potential energy

\[U=q_{\mathrm{test}}V\]

Between points,

\[\Delta U =q_{\mathrm{test}}(V_B-V_A)\]

The external potential excludes the test charge's own singular potential. If the test charge is large enough to move or polarize the sources, the fixed-source approximation fails and the full configuration energy must be recalculated.

Electron-volts are energy units, not potential units. One electron-volt is the magnitude of energy change for one elementary charge crossing one volt:

\[1\ \mathrm{eV} =1.602\,176\,634\times10^{-19}\ \mathrm J\]

A scalar-potential procedure

The calculation keeps reference choice separate from source summation.

import numpy as np

probe = np.array([0.0, 1.0])
sources = [(np.array([-1.0, 0.0]), 1e-9), (np.array([1.0, 0.0]), -1e-9)]
potential = 0.0  # Reference: zero at infinity.
for source, charge in sources:
    radius = np.linalg.norm(probe - source)
    if radius == 0:
        raise ValueError("Potential is singular at a point source.")
    potential += 8.9875517923e9 * charge / radius
print(f"Potential at the dipole midpoint plane: {potential:.6f} V")

To calculate a difference, evaluate both points with the same sources and the same reference, then subtract final minus initial. Never round the individual potentials before subtraction when their difference is small.

Verification and limitations

Check one positive and one negative point source at two distances. Verify inverse-distance scaling and sign. Test equal-like and equal-opposite source pairs at their midpoint, checking both field and potential.

Add a uniform reference constant and confirm that every potential shifts by that constant while differences and field estimates remain unchanged. Compare the negative finite difference of potential with an independently calculated field at points away from sources, and reduce the difference spacing to test convergence.

For a closed electrostatic loop, numerically integrate the field and verify approximately zero circulation. Report discretization error and avoid paths through singularities.

The scalar-potential model assumes electrostatics. Moving sources, induction, radiation, material polarization, conductor rearrangement, and quantum structure require additional state or different equations. Potential is powerful precisely because its reference and domain are explicit, not because it is an absolute height stored in space.

A worked Python notebook

Follow the potential of a positive one nanocoulomb charge as a function of distance. The question for this calculation is precise: how does potential relative to infinity change when we change radial 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.

Adding a constant reference offset changes potential values but leaves the electric field unchanged. 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 4 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 radial distance, expressed in m. The output is potential relative to infinity, 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 2 m, the expression gives 4.49378 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 = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted potential relative to infinity:", 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 2.25 m, gives 3.99447 V. The upper endpoint gives 2.24689 V. Read these together with the assumptions: they describe different choices of radial 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, 2.25, 4.0])
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 radial distance; the vertical axis shows potential relative to infinity. 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, 4.0, 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("Radial distance (m)")
ax.set_ylabel("Potential relative to infinity (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 potential relative to infinity change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 m is -2.24689 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 = 2.0175
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 2.0175 m. The tangent estimate is 4.45446 V, while the full expression gives 4.4548 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 = 3.5
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 radial distance inside the stated interval, predict whether potential relative to infinity 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: Adding a constant reference offset changes potential values but leaves the electric field unchanged. 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(4.0) - evaluate(0.5))
for points in [101, 201, 401]:
    grid = np.linspace(0.5, 4.0, 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

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