Continuous Charge Potentials

Coulomb kernel, source coordinates, and observation coordinates

A finite charge distribution in vacuum can use the potential reference $V (\infty) = 0$ . At an observation point $r$ , its potential is

V (r) = \frac{1}{4 π ε _{0}} D \int \frac{d q ( r ^{'} )}{∣ r - r ^{'} ∣} .

The primed coordinate $r^{'}$ belongs to the source element; the unprimed coordinate belongs to the point where the result is evaluated. Define

R = r - r^{'}, R = R .

The integration is over the source domain $D$ , not over space around the observation point. Each element contributes a signed scalar $d q / R$ . The source charge sign stays inside $d q$ ; a negative density produces a negative potential contribution without a separate direction rule.

Source and observation coordinates for the continuous-charge integral. The primed point labels a source element on the charged body; the unprimed point is where the potential is evaluated, and R is the length of the vector between them.

The integral has a simple dimensional check. The factor $(4 π ε_{0})^{- 1}$ has units $V m C^{- 1}$ ; multiplying $d q / R$ by it gives volts. A calculation that leaves $C m^{- 1}$ , $C m^{- 2}$ , or $C m^{- 3}$ in the final result has integrated a density without its appropriate geometric element.

The scalar kernel eliminates vector addition. It does not remove geometry. $R$ usually varies over the source, and the choice of a source parameter must make that variation explicit. A symmetric source may make every element have the same $R$ ; otherwise the integral retains the full dependence.

Outside a compact source, the leading far-field result is $V ≃ k Q / r$ , where $k = (4 π ε_{0})^{- 1}$ , $Q$ is the total charge, and $r$ is large compared with the source size. This is a limit check, not a replacement for the integral at ordinary distances. The first omitted terms retain information about the distribution's size, offset, and sign pattern.

Charge density and the geometric measure.

The differential charge must match the dimension of the source.

d q d q d q = λ (s) d s = σ (u, v) d A = ρ (u, v, w) d τ line source, surface source, volume source .

Here $λ$ , $σ$ , and $ρ$ have SI units $C m^{- 1}$ , $C m^{- 2}$ , and $C m^{- 3}$ . The symbols are arbitrary; the chosen parameterization determines the measure. A line can be parameterized by Cartesian coordinate, arc length, polar angle, or another monotonic coordinate. A circular arc of radius $a$ uses $d s = a d ϕ$ , while a cylindrical volume element uses $d τ = s d s d ϕ d z^{'}$ .

A parameterized surface $r^{'} (u, v)$ uses

d A = \frac{\partial r ^{'}}{\partial u} \times \frac{\partial r ^{'}}{\partial v} d u d v .

The cross-product magnitude is the local area scale. It is indispensable on a curved surface. Replacing $d A$ by $d u d v$ without checking the parameterization treats coordinate area as physical area and can change a result by a position-dependent factor.

A volume source retains its Jacobian with the density. A uniformly charged sphere can be assembled from shells with $d τ = 4 π a^{2} d a$ ; a finite cylinder can be assembled from rings with $d τ = s d s d ϕ d z^{'}$ . The factor $s$ in cylindrical coordinates is not optional. Annuli at larger radius contain more physical volume than annuli of the same radial width near the axis.

Reference choices and the range of the Coulomb integral.

The integral with $V (\infty) = 0$ requires a finite total charge concentrated in a bounded region, or a distribution whose potential contribution falls sufficiently fast at large distance. A finite line, ring, disk, shell, or charged volume satisfies that condition. The reference is then convenient because every source element has a finite distance from a sufficiently remote point.

An ideal infinite line or infinite plane does not admit a finite potential difference from a finite point to infinity. The field can remain finite, while the absolute reference $V (\infty) = 0$ is unavailable. Use a finite reference point $r_{0}$ instead, with

V (r) - V (r_{0}) = - r_{0} \int r E \cdot d ℓ .

An infinite plane of surface charge density $σ$ has field magnitude $∣ σ ∣/ (2 ε_{0})$ on either side. Choosing the plane as the reference gives

V (z) - V (0) = - \frac{σ}{2 ε _{0}} ∣ z ∣.

The result has no contradiction with the finite-disk expression. Letting a disk radius grow without changing $σ$ increases the total charge without bound. Potential relative to infinity then diverges, whereas a finite potential difference between two finite locations remains well defined.

Reference choice for an extended charge distribution. A bounded disk allows the zero reference at infinity, whereas an unbounded sheet needs a finite reference plane because the potential difference grows without limit with distance.

An additive constant changes every reported potential by the same amount. It leaves $E = - \nabla V$ , potential differences, and the force on a test charge unchanged. Record the reference in a laboratory table and in a numerical data file. A potential value without its reference cannot be compared directly with a separately referenced value.

Finite straight line: geometry before integration

Consider a uniformly charged segment on the $x^{'}$ axis from $x_{1}$ to $x_{2}$ . Let $P = (0, a, 0)$ , with $a > 0$ . The source coordinate is $r^{'} = (x^{'}, 0, 0)$ , so

R (x^{'}) = x^{'2} + a^{2}, d q = λ d x^{'} .

The line potential is

V (a) = k λ x_{1} \int x_{2} \frac{d x ^{'}}{x ^{'2} + a ^{2}} = k λ [ln (x^{'} + x^{'2} + a^{2})]_{x_{1}}^{x_{2}} .

The logarithm carries a dimensional argument if written in isolation. Its dimensionless ratio form is

V (a) = k λ ln [\frac{x _{2} + x _{2}^{2} + a ^{2}}{x _{1} + x _{1}^{2} + a ^{2}}] .

Finite-line geometry at an off-line point. Each source element lies at a different distance from P, so the potential integral must weight the whole spread of distances along the segment.

The coordinate origin can be shifted, but the physical endpoints and perpendicular distance cannot. A reliable setup writes $r$ , $r^{'}$ , and $R (parameter)$ before integrating. Starting with an antiderivative and then guessing its limits often hides a reversed interval, an omitted length factor, or a source coordinate used as an observation coordinate.

A centered segment of length $ℓ$ uses $x_{1} = - ℓ /2$ and $x_{2} = ℓ /2$ . The result becomes

V (a) = 2 k λ asinh (\frac{ℓ}{2 a}) .

At $a ≫ ℓ$ , $V ≃ k λ ℓ / a = k Q / a$ , where $Q = λ ℓ$ . At $a ≪ ℓ$ , the potential has logarithmic sensitivity to the perpendicular distance. The finite line remains a finite source, but its local behavior approaches that of a long line before the far-end geometry becomes important.

Recovering the line field from the potential.

On the perpendicular bisector, symmetry leaves only the component normal to the line. Differentiating the scalar result gives

E_{a} = - \frac{d V}{d a} = \frac{k λ ℓ}{a a ^{2} + ( ℓ /2 ) ^{2}} .

For positive $λ$ , $V$ decreases as $a$ increases, so the field points away from the line. A direct Coulomb-field calculation has the same sign and far-field limit,

E_{a} ≃ \frac{k Q}{a ^{2}} (a ≫ ℓ) .

An incorrect source distance can preserve units and a plausible sign while giving the wrong scaling. Differentiation exposes that error. The field must also respect source symmetry: no axial component appears on the perpendicular bisector of a centered uniform segment.

At an interior tabulated point, a centered finite difference is

E_{a} (a_{i}) ≃ - \frac{V ( a _{i} + h ) - V ( a _{i} - h )}{2 h} .

The spacing $h$ must be stated. A large $h$ smooths real curvature and creates truncation error. A very small $h$ subtracts nearly equal voltages, which amplifies voltage noise and coordinate uncertainty. Repeat the derivative at several spacings; the stable range, not the smallest available spacing, supports the reported field.

Ring charge: a symmetry reduction

Let a ring of radius $b$ lie in the $z = 0$ plane with total charge $Q$ . At an axis point $(0, 0, z)$ , every source element has the same distance $z^{2} + b^{2}$ . The potential is

V (z) = \frac{k}{z ^{2} + b ^{2}} \oint d q = \frac{k Q}{z ^{2} + b ^{2}} .

The ring is not equivalent to a point charge at ordinary axial distance. It has the same far-field leading term but a finite center potential $k Q / b$ . The potential is an even function of $z$ , and differentiation gives an odd axial field,

E_{z} (z) = - \frac{d V}{d z} = \frac{k Q z}{( z ^{2} + b ^{2} ) ^{3/2}} .

At the center, $E_{z} = 0$ while $V = k Q / b$ . Scalar potential cancellation and vector field cancellation are different statements; here every scalar element contribution has the same sign, while opposite vector components cancel at the center.

Charged-ring axis geometry. Every ring element sits at the same distance from a point on the axis, so the potential integral collapses to total charge over one common distance.

The axial potential and field provide two independent curves for a numerical model. The potential has zero slope at the center. The field magnitude rises from zero, reaches a maximum at $z = b / 2$ , then falls as $1/ z^{2}$ . A calculated axial field with a nonzero center value or an axial potential with unequal values at $+ z$ and $- z$ violates the ring geometry.

Ring-axis potential and axial field. The potential is even and peaks at the center, while the axial field is odd: it vanishes at the center, reaches extrema on either side, then decays in the far field.

Disk potential from concentric rings

A uniform disk of radius $R$ and surface density $σ$ can be assembled from rings. A ring of radius $s$ and radial thickness $d s$ has

d q = 2 π s σ d s, d V = \frac{2 π k σ s d s}{s ^{2} + z ^{2}} .

The disk-axis potential is

V (z) = 2 π k σ (z^{2} + R^{2} - ∣ z ∣) .

For $z > 0$ , differentiation gives

E_{z} (z) = 2 π k σ (1 - \frac{z}{z ^{2} + R ^{2}}) .

The sign reverses below the disk. The potential is continuous across the charged surface, whereas the normal field has the surface-charge jump. Direct differentiation away from $z = 0$ checks the smooth branch; the one-sided limits check the field discontinuity.

Disk built from concentric rings. The annular strip at radius s has area 2 pi s ds and contributes with the common-distance ring geometry before integration over radius.

The far-field check uses $Q = σ π R^{2}$ . Expanding the square root for $∣ z ∣ ≫ R$ gives $V ≃ k Q /∣ z ∣$ . Near the disk center on the positive side, $E_{z}$ approaches $σ / (2 ε_{0})$ when $R$ is much larger than the observation distance. That local limit connects the finite disk smoothly to the uniform infinite-plane field without using an invalid zero-at-infinity reference for the infinite plane.

Spherical shells and uniform spherical volumes

A thin spherical shell of radius $a$ and total charge $Q$ has potential

V (r) = {k Q / a, k Q / r, r \leq a, r \geq a .

The potential is continuous at $r = a$ . Differentiation gives zero field in the interior and $k Q / r^{2}$ outside. The derivative has different one-sided values at the surface because surface charge produces a normal-field jump. A continuous potential does not require a continuous field derivative.

Spherical-shell potential. Inside the shell the potential is constant; outside it falls as one over radius, and the curve stays continuous at the surface even though its slope changes there.

A uniform solid sphere with radius $a$ and total charge $Q$ has

V (r) = ⎩ ⎨ ⎧ \frac{k Q}{2 a} (3 - \frac{r ^{2}}{a ^{2}}), \frac{k Q}{r}, r \leq a, r \geq a .

The interior quadratic form follows from summing thin spherical shells: shells inside the observation radius contribute as enclosed charge divided by $r$ , while shells outside contribute their constant interior shell potential. Its radial derivative is $E_{r} = k Q r / a^{3}$ , which grows linearly from the center and joins the exterior field at the surface. The potential, field, and limiting point-charge form therefore test one another.

Uniform solid-sphere decomposition for a point at radius r. Shells inside r act through their enclosed charge; shells outside r each add their constant interior potential, and both parts are needed.

The shell and volume examples also separate charge support from field support. The potential can be evaluated inside a shell where no charge resides. A volume charge occupies the interior and yields a nonzero Poisson source there. The integral method describes both cases directly; the distinction appears through the source domain and density, not through an altered Coulomb kernel.

Arbitrary sheets, volumes, and coordinate Jacobians

Closed-form answers are unusual once the observation point leaves a symmetry axis. The source integral remains unchanged. A surface with parameterization $r^{'} (u, v)$ uses

V (r) = k u_{1} \int u_{2} v_{1} \int v_{2} \frac{σ ( u , v ) \frac{\partial r ^{'}}{\partial u} \times \frac{\partial r ^{'}}{\partial v}}{∣ r - r ^{'} ( u , v ) ∣} d v d u .

The source surface may be flat, curved, perforated, or composed of several patches. Each patch needs a nonoverlapping parameter range. A panel boundary is an accounting device; the physical surface has no extra charge there unless the density specifies one. A disk with a hole, for example, is represented by an annular range in radius or by an outer disk minus an inner disk. Using both descriptions in the same integral double-counts the omitted region.

A volume distribution requires its coordinate map and Jacobian before assigning integration limits. In spherical source coordinates,

r^{'} = s sin θ^{'} cos ϕ^{'} \overset{x}{^} + s sin θ^{'} sin ϕ^{'} \overset{y}{^} + s cos θ^{'} \overset{z}{^}, d τ = s^{2} sin θ^{'} d s d θ^{'} d ϕ^{'} .

The radial coordinate $s$ is deliberately different from the observation distance $r = ∣ r ∣$ . Reusing one symbol for both makes the kernel ambiguous. For an axisymmetric density $ρ (s, θ^{'})$ , the azimuthal integral may simplify, but the simplification must follow from the stated symmetry. Apply the azimuthal reduction only after establishing the symmetry of $ρ$ .

A finite cylinder with density $ρ (s, ϕ^{'}, z^{'})$ , radius $b$ , and length $L$ has the Coulomb integral

V (r) = k - L /2 \int L /2 0 \int 2 π 0 \int b \frac{ρ ( s , ϕ ^{'} , z ^{'} ) s}{∣ r - r ^{'} ( s , ϕ ^{'} , z ^{'} ) ∣} d s d ϕ^{'} d z^{'} .

The factor $s$ has the same status as the density and the kernel. Omitting it underweights outer annuli. Extending the limits through an empty axial bore overweights a hollow cylinder. Both errors can survive a total-charge check if the density is adjusted afterward, so the geometry should be checked independently.

Symmetry statements and their limits.

Symmetry simplifies a potential integral in three distinct ways.

Common distance. On the axis of a ring, every element has the same $R$ , so the kernel leaves the integral.
Paired source elements. Equal elements at mirrored locations can have equal potential contributions even when their field components cancel.
Rotational invariance. An axially symmetric source produces a potential that does not depend on the observation azimuth, though it can still depend on radius and axial coordinate.

Potential is scalar. Pairing source elements never creates opposite scalar signs unless the charge elements themselves have opposite signs. A neutral distribution can have a small or zero potential at a selected point because positive and negative source charges cancel algebraically. A uniformly positive symmetric distribution has positive finite potential at regular finite observation points away from ideal line- or point-charge support under the infinity reference.

State the exact symmetry used before discarding a coordinate. A disk potential is independent of observation azimuth on its axis; it is not independent of azimuth at an arbitrary off-axis point if the disk density is nonuniform. A spherical shell has radial potential only when its surface density is uniform. A numerical grid can test a claimed symmetry by sampling points related by the symmetry operation and comparing their potential residuals with the expected numerical uncertainty.

Analytical limits, continuity, and source singularities

Every analytic result benefits from limits that use no additional integration. A compact finite source with $Q \neq = 0$ approaches the point-charge potential $k Q / r$ at observation distances large compared with its largest dimension. A ring, disk, shell, and uniformly charged sphere all pass that check. When $Q = 0$ , the first nonzero multipole sets the leading far-field term. The next correction depends on source shape. A symmetric centered source often has no first-order displacement term; an off-center source generally does.

At a regular observation point away from idealized line or point charge support, the potential integral is finite and smooth enough for differentiation. At the support itself, the behavior depends on source dimension. An ideal line produces a logarithmic potential divergence as an observation point approaches it. A smooth charged surface can have finite potential on the surface because a small surface element contributes roughly $d A / R$ , whose local polar-area integral remains finite. Its normal electric field has different limiting values on the two sides. An ideal point charge has a $1/ R$ divergence.

Numerical methods must treat the source neighborhood consistently with the chosen physical model. A panel method that replaces a finite surface panel by a point charge at its center is accurate when the panel is small compared with its distance from the observation point. It is poor when the observation point lies close to that panel. A near panel can be integrated analytically, subdivided, or assigned a special quadrature rule. Dropping that panel changes the source charge and biases the result.

Potential continuity and field discontinuity provide a local check at a charged surface. Compute $V$ at points a decreasing distance above and below the surface. The two sequences should approach one common value for an ordinary finite surface density. With the normal directed from the $-$ side to the $+$ side, compute each normal derivative separately and check $E_{n, +} - E_{n, -} = σ / ε_{0}$ , equivalently $\partial_{n} V_{+} - \partial_{n} V_{-} = - σ / ε_{0}$ . A single centered difference that straddles the charge sheet averages the two limits and does not measure either physical one-sided field.

Discrete source panels and quadrature rules

Divide a finite line-charge source interval into panels $[s_{j}, s_{j + 1}]$ with width $Δ s_{j}$ . The midpoint approximation is

V_{N} (r) = k j = 1 \sum N \frac{λ ( s _{j + 1/2} ) Δ s _{j}}{r - r ^{'} ( s _{j + 1/2} )} .

Each term is a small panel charge divided by the distance from its panel midpoint. The approximation becomes exact only in the limit of refined panels for a smooth integrand. A uniform panel width is appropriate for a smooth source seen from far away. It is inefficient near a close observation point, density transition, sharp edge, or small geometric feature.

The composite trapezoid rule is

V_{trap} = k j = 1 \sum N \frac{Δ s _{j}}{2} [\frac{λ ( s _{j} )}{R ( s _{j} )} + \frac{λ ( s _{j + 1} )}{R ( s _{j + 1} )}] .

The trapezoid rule can improve a smooth line integral at fixed panel count, but it evaluates shared endpoints twice with half weights. The implementation must preserve that weighting. A naive endpoint sum with full weight at every panel boundary overcounts interior source positions. For a surface or volume, tensor-product midpoint and trapezoid rules follow the same principle but need every coordinate weight and Jacobian.

A ring or disk requires preservation of physical annular area. A radial midpoint rule for a uniform disk takes

Δ q_{j} = 2 π σ s_{j + 1/2} Δ s_{j}, V_{N} (z) = k j \sum \frac{Δ q _{j}}{z ^{2} + s _{j + 1/2}^{2}} .

Replacing $2 π s_{j + 1/2} Δ s_{j}$ with one constant area for every radial panel turns equal radial widths into equal charges, which is false. Outer rings have larger area. Check that $\sum_{j} Δ q_{j}$ equals $σ π R^{2}$ before comparing potential values.

Adaptive refinement and local error control.

Choose panel spacing from integrand variation and source geometry. The line-source kernel becomes sharply peaked when the observation point approaches the line. A practical refinement criterion compares one parent-panel estimate with the sum from two child panels. For a panel $j$ , let $V_{j}^{(1)}$ use one midpoint and $V_{j}^{(2)}$ use two equal subpanels. Refine when

V_{j}^{(2)} - V_{j}^{(1)} > τ_{j},

where $τ_{j}$ is an allocated local tolerance. Sum the local estimated errors conservatively when choosing a global tolerance. The criterion is not a proof of the true error, but it places panels where the numerical representation changes most.

Refinement by geometry alone can miss a density feature. A line with $λ (s)$ changing sharply near one endpoint needs panels there even if the observation point is remote. A piecewise density should split the integration domain at every discontinuity. On each smooth subinterval, a standard quadrature rule can converge normally. Integrating across a density jump with one high-order polynomial rule can create oscillatory or biased estimates.

A surface mesh requires both panel size and panel-aspect-ratio measurements. A long thin triangle close to a field point may have a small area but a poorly represented distance variation. Curved surfaces need enough panels to resolve curvature and density. A global count alone is not a convergence metric: two meshes with the same number of panels can have very different near-field resolution.

Convergence, extrapolation, and independent field checks

Uniformly refined meshes with characteristic spacing $h$ give a smooth quadrature result of the form

V_{h} = V_{⋆} + C h^{p} + O (h^{p + 1}),

where $V_{⋆}$ is the continuum result and $p$ is the observed convergence order. Three mesh levels give

p ≃ \frac{ln \frac{V _{h} - V _{h /2}}{V _{h /2} - V _{h /4}}}{ln 2} .

The estimate is meaningful only after the meshes represent the same physical source, observation point, reference, and density. Changing the source boundary while refining the mesh mixes geometry error with quadrature error. A result that changes irregularly may be limited by a near singularity, a density jump, roundoff, or an inconsistent panel construction rather than by the nominal quadrature order.

For stable $p$ , the Richardson-extrapolated estimate is

V_{⋆} ≃ V_{h /2} + \frac{V _{h /2} - V _{h}}{2 ^{p} - 1} .

Report both the raw finest-grid value and the extrapolated value. Extrapolation does not correct an incorrect reference, missing source region, or a panel model invalid near the observation point. It only estimates the leading smooth-discretization error under the observed refinement pattern.

Potential and field provide an independent validation pair. Compute the potential by source integration, then recover $E$ through numerical differentiation. Separately compute the electric field from the vector Coulomb integral on a small set of validation points. Potential sums are usually smoother, while field kernels and differentiation are more sensitive to close source elements. Agreement within propagated uncertainty supports both the source geometry and the sign convention.

E (r) = k D \int \frac{r - r ^{'}}{∣ r - r ^{'} ∣ ^{3}} d q (r^{'}) .

Do not differentiate a one-dimensional axis potential to obtain off-axis field components. An axis expression carries information only along that axis. The ring and disk examples permit an axial derivative because symmetry removes transverse components there. General field reconstruction needs a potential function in all relevant spatial directions or a direct vector integral.

Potential measurements and source-model calibration.

An instrument measures a potential difference between two connected terminals. A voltmeter probe near a charge distribution is part of the apparatus: its conductor, input resistance, cable capacitance, and reference lead can perturb the intended electrostatic configuration. A high-input-resistance electrometer reduces charge leakage, but it does not erase probe geometry or a nearby grounded enclosure.

Choose one reference conductor or reference electrode. Measure $Δ V = V (P) - V (P_{ref})$ at surveyed positions. Use the same reference in the model,

Δ V_{model} = k D \int (\frac{1}{∣ r _{P} - r ^{'} ∣} - \frac{1}{∣ r _{ref} - r ^{'} ∣}) d q .

The panel-wise difference form handles a finite experimental reference and background potential offsets. Evaluate the difference panel by panel when two nearby observation positions share a distant source contribution; that avoids subtracting two independently accumulated sums. The difference can still be much smaller than either absolute potential, so stable evaluation or sufficient precision remains necessary.

Separate source and position uncertainties. A line-density uncertainty scales every potential contribution from that line. An observation-position uncertainty is most important where the potential gradient is large. A reference-electrode displacement affects every difference reading coherently. Repeating the measurement at nearby positions estimates local slope only after the reference and source state remain stable between readings.

An imposed charge distribution requires total-charge verification by an independent charge measurement when possible. For example, integrate a measured line density, $Q_{line} = \int λ (s) d s$ , then compare with a charge sensor or known charging circuit. A potential fit can otherwise exchange a density-scale error for a distance-scale error, especially when all observation points lie on one narrow line.

In a numerical calculation, split the radius into $N$ annuli, form each $Δ q_{j} = 2 π σ s_{j + 1/2} Δ s$ , and sum the ring potentials. Repeat for $N = 8, 16, 32,$ and $64$ . The potential error should decrease smoothly until roundoff or an implementation detail dominates. Verify total panel charge at every $N$ , then compare the finest potential and derivative with the analytic values.

Signed potential residuals are

r_{i} = V_{meas, i} - V_{model, i} .

A constant residual across positions often indicates a reference offset. A residual that grows near the disk edge can indicate radius error, nonuniform surface density, or an off-axis probe coordinate. A residual that changes sign under source-charge reversal can be part of the source response; one that does not reverse is more likely an additive instrument or background term. The pattern identifies the next check more effectively than one percentage averaged over all positions.

Reduction record and calculation audit.

A potential integral is reproducible only when its geometry and numerical choices are recorded. The following record separates source description from reduction method.

Source model: density units, density function, physical boundaries, total charge check, and every excluded hole or gap.
Observation geometry: coordinates of every evaluation point, reference point, coordinate origin, and distance units.
Integral representation: source parameterization, Jacobian, integration bounds, symmetry reductions, and the selected potential reference.
Numerical method: panel construction, quadrature rule, near-panel treatment, refinement criterion, mesh sequence, and extrapolation rule if used.
Validation: far-field limit, total-charge check, symmetry samples, direct field comparison, and measured residual pattern.

The potential result should carry an uncertainty statement. Density calibration, source dimensions, observation position, reference electrode location, instrument offset, and discretization contribute differently. A common density-scale error correlates every calculated point. Random voltage noise can decrease with repeated readings. Panel refinement addresses numerical approximation only; it does not repair an incorrect density profile or an omitted conducting boundary.

A reported potential difference must state the two physical locations and the reference convention in the same sentence as the numerical value. For a reported field obtained by differentiation, state the difference stencil, sample spacing, and whether the derivative approached a charged surface from one side. These details preserve the distinction between an analytic Coulomb integral, a discretized model, and an instrument reading.

The same source measure and geometry must survive every analytical reduction and numerical approximation that follows.

Annuli, finite cylinders, and nonuniform radial densities.

An annulus tests disk algebra by subtraction. Let the uniform annulus have inner radius $R_{in}$ , outer radius $R_{out}$ , and surface density $σ$ . Subtract the potential of the missing inner disk from that of the outer disk. On the axis,

V (z) = 2 π k σ (R_{out}^{2} + z^{2} - R_{in}^{2} + z^{2}) .

The two $∣ z ∣$ terms cancel. The annulus-center potential is

V (0) = 2 π k σ (R_{out} - R_{in}) .

The center has no charge directly beneath it, yet charge on every annular ring contributes positive scalar potential when $σ > 0$ . The axial field is zero at the center by reflection symmetry. A calculation that produces a zero center potential has confused a hole in the source with cancellation between positive and negative charge.

The far-field charge check uses $Q = σ π (R_{out}^{2} - R_{in}^{2})$ . At $∣ z ∣$ much larger than $R_{out}$ , the annulus potential approaches $k Q /∣ z ∣$ . The hole affects the next shape-dependent correction, not the leading total-charge term. This check is essential for numerical polar grids because the sum of panel charges must equal the area difference, not the outer-disk area.

A finite uniform cylinder gives a genuine three-dimensional source example. Let its radius be $b$ , its length $L$ , its center be the origin, and its constant volume density be $ρ$ . At an axial observation coordinate $z$ , cylindrical source coordinates give

V (z) = 2 π k ρ - L /2 \int L /2 0 \int b \frac{s d s d z ^{'}}{s ^{2} + ( z - z ^{'} ) ^{2}} .

After radial integration,

V (z) = 2 π k ρ - L /2 \int L /2 [b^{2} + (z - z^{'})^{2} - ∣ z - z^{'} ∣] d z^{'} .

The remaining one-dimensional integral is convenient for numerical quadrature even when its closed antiderivative is not used. The factor $2 π k ρ [b^{2} + (z - z^{'})^{2} - ∣ z - z^{'} ∣] d z^{'}$ is the potential contribution from one thin disk slice. That slice contribution stays finite when the observation point lies inside the cylinder because the finite-radius disk slice resolves the local volume rather than replacing it by a point charge.

At distances large compared with both $b$ and $L$ , the finite cylinder must approach $k Q /∣ z ∣$ , with $Q = ρ π b^{2} L$ . At the symmetry center, the axial field is zero because source slices at $+ z^{'}$ and $- z^{'}$ create opposite axial field components. The potential is not zero there for $ρ > 0$ . The pair of checks resembles the ring and shell cases but now tests a volume Jacobian and two numerical integration variables.

A spherically symmetric nonuniform density has the general interior form below. If $ρ (s)$ occupies $0 \leq s \leq a$ , then

V (r) = 4 π k \frac{1}{r} 0 \int r ρ (s) s^{2} d s + r \int a ρ (s) s d s, 0 < r < a .

The first term comes from charge on shells inside the observation radius. The second comes from shells outside it, each giving its constant interior shell potential. Outside the support, the potential is $k Q / r$ , where

Q = 4 π 0 \int a ρ (s) s^{2} d s .

The decomposition gives a direct numerical method for radial density data. Tabulate $ρ (s)$ , integrate the two terms separately at each desired $r$ , and compare the field from $- d V / d r$ with $k Q_{enc} (r) / r^{2}$ . The field check uses

Q_{enc} (r) = 4 π 0 \int r ρ (s) s^{2} d s .

The field comparison distinguishes a radial-density integration error from an error in the exterior total-charge normalization.

Neutral compact sources and far-field interpretation.

The point-charge far-field check applies only when total charge $Q$ is nonzero. When a compact source has $Q = 0$ , the leading potential term can be dipolar,

V (r) ≃ k \frac{p \cdot r ^}{r ^{2}}, r ≫ source size .

$p = \int r^{'} d q$ is the electric dipole moment for the chosen origin. The $1/ r^{2}$ potential decay is faster than the monopole $1/ r$ decay. Changing the origin does not alter $p$ when $Q = 0$ , but it does alter the dipole moment of a charged distribution. A far-field comparison therefore needs the total-charge state and the stated origin.

Far-field hierarchy for compact sources. A source with nonzero total charge has a leading one-over-distance potential; an equal positive-negative pair is neutral, and its potential decays faster, as one over distance squared.

The far-field approximation should be tested with a dimensionless ratio such as $d / r$ , where $d$ is the largest source dimension. Plot the fractional residual

η (r) = \frac{∣ V _{exact} ( r ) - V _{approx} ( r ) ∣}{max ( ∣ V _{exact} ( r ) ∣ , V _{scale} )} .

The scale $V_{scale}$ prevents an artificial blowup near a potential zero in a neutral source. Report the range of $d / r$ for which the approximation meets a chosen tolerance. Calling a point “far away” without a ratio does not specify an approximation error.

Uncertainty propagation for an integrated potential.

Potential uncertainty has geometric, source, reference, and numerical components. A line model $V (a, λ, ℓ)$ has the first-order independent estimate

u_{V}^{2} ≃ (\frac{\partial V}{\partial λ} u_{λ})^{2} + (\frac{\partial V}{\partial a} u_{a})^{2} + (\frac{\partial V}{\partial ℓ} u_{ℓ})^{2} + u_{ref}^{2} + u_{num}^{2} .

The perpendicular-position derivative is

\frac{\partial V}{\partial a} = - E_{a} .

Position uncertainty contributes approximately $E_{a} u_{a}$ to potential uncertainty. Near a charged line or near a disk edge, that term can dominate. Treating every uncertainty as one generic percentage hides the parameter that controls the measurement design.

Density-scale uncertainty is often correlated across all observation points. If $λ$ is estimated from one calibration, increasing it moves every model potential together. Probe voltage noise may be nearly independent between samples. Reference-electrode drift can be common to an entire scan. A least-squares fit should retain those correlations or test sensitivity by repeating the complete calculation with shared parameters shifted together.

Numerical uncertainty requires a separate statement. A difference between two panel counts estimates discretization sensitivity only after total charge, reference, source boundary, and near-panel treatment are fixed. The uncertainty of a finite difference field contains the uncertainties of both voltage samples and of the spacing. Propagating only potential uncertainty while treating $h$ as exact understates a field result from a physical probe survey.

A source-reversal measurement uses

V_{odd} = \frac{V ( + Q ) - V ( - Q )}{2}, V_{even} = \frac{V ( + Q ) + V ( - Q )}{2} .

The odd part isolates a response proportional to the controlled source charge. The even part estimates static offset and background contributions that did not reverse. For independent readings with equal standard uncertainty $u$ , $u (V_{odd}) = u / 2$ . Include covariance and reversal-dependent systematic effects whenever the two readings share calibration, reference, or timing errors.

A reproducible numerical reduction.

The following sequence is appropriate for a nonuniform finite source whose analytic integral is unavailable.

Survey the source boundary and record a density model in SI units.
Choose an observation coordinate system and a finite reference electrode.
Write $r^{'} (u, v, w)$ , the physical measure, and the source domain.
Verify total numerical charge from the discrete panel weights.
Evaluate potential differences on a coarse mesh and map regions of rapid kernel variation.
Refine the source panels near close observations, density changes, curved edges, and every source discontinuity.
Repeat the computation on at least three refinement levels and estimate the stable discretization range.
Differentiate the converged potential only on a grid that resolves the required field component; use one-sided values at charge sheets.
Compare selected fields with direct vector Coulomb sums and compare far points with the appropriate monopole or dipole limit.
Retain panel coordinates, panel charges, reference convention, mesh sequence, raw voltage data, and the code or calculation record.

The sequence separates model error from quadrature error. A refined mesh cannot correct a source radius copied incorrectly from a drawing. An accurate density map cannot correct a probe coordinate measured from the wrong origin. A voltage fit with small residuals over one narrow scan can still fail at a second orientation or a different reference point. The validation points should therefore sample a near region, a symmetry location, an edge-sensitive region, and a far-field region.

A finite-line measurement can use the perpendicular bisector, one endpoint-near point, one farther point on the same normal, and one mirror-related point. For a disk, use the axis, a near-edge point, a point above the plane, and a point below the plane. Each position tests a different part of the source model. Report residuals in volts and in units of their combined uncertainty rather than declaring agreement from a visual overlay.

The source-integral approach applies to static charge distributions. A time-varying magnetic field can give an electric field with nonzero circulation, for which one single-valued electrostatic potential cannot describe the whole region. Static source conditions, fixed geometry, and a stated reference are therefore part of the model before any continuous-charge potential is evaluated.

Model discrimination from potential data.

Potential data can distinguish competing source models when the measurement positions are selected to expose their different spatial dependence. A point-charge approximation and a finite-line model can agree at a distant point because both have the same leading total-charge term. They separate at distances comparable with the line length. A finite-disk field can approach the infinite-sheet field near the disk center; potential comparisons with an infinite sheet require finite-reference differences. A spherical shell and a solid sphere have identical exterior potential for equal total charge but sharply different interior profiles.

For independent measurement and model uncertainties, normalized residuals are

z_{i} = \frac{V _{meas, i} - V _{model, i}}{u _{meas, i}^{2} + u _{model, i}^{2}} .

The denominator must include the uncertainty in the source dimensions and density parameters used by the model together with voltmeter repeatability. Add covariance terms when reference data, calibration, or fitted parameters are shared. A sequence of same-sign residuals whose magnitude grows toward a line endpoint indicates that a point-charge model is missing source extent. A shell model that fits exterior points and fails at interior points has the wrong charge support; changing total charge alone leaves that mismatch.

Do not tune every parameter against one short potential scan. Fix independently measured geometry first, then fit only parameters the experiment genuinely leaves unknown. For a nominally uniform disk, radius and probe-axis offset can trade against surface density over a narrow axial range. Add an off-axis measurement or a second axial distance range before treating all three as free fit parameters. The added geometry breaks the parameter correlation.

A fit also needs a stated physical boundary. A charged object placed near a grounded table, shielding plate, or metal support has a different potential from the same isolated source in free space. The continuous-charge integral describes the prescribed source distribution in the stated environment. If nearby conductors reorganize charge substantially, their induced response belongs in a boundary-value model rather than in a density fitted only on the original object.

Keep calculated potential, observed potential difference, and inferred field distinct in a final data table. $V$ and $Δ V$ both use volts but have different reference conventions; $E$ uses $V m^{- 1}$ . A calculation can reproduce potential differences while retaining a common additive offset in absolute potential. A differentiated field can disagree because small potential noise becomes large after division by probe spacing.

Report the source geometry, density model, reference, observation positions, quadrature resolution, uncertainty convention, and validation tests. Limit the conclusion to the measured geometry and the stated source, reference conductor, and surrounding boundaries.