Electric Field Maps

Tangent curves of a vector field

An electric field line is a curve whose tangent has the local direction of $E$ . If the curve is written as $r (s)$ , with increasing parameter $s$ chosen along the positive field direction, then

\frac{d r}{d s} = λ (s) E (r (s)), λ (s) > 0.

The positive function $λ$ changes only the rate at which the curve is traced. It does not change its shape. A field line therefore represents direction at a continuous set of points, whereas a vector plot represents direction and magnitude at selected sample points. Both drawings are derived from the same vector field; neither adds a separate physical law.

Representation	Plotted quantity	Directly supported inference	Unsupported inference
Vector map	$E$ at sampled positions	local direction; magnitude when an arrow scale is stated	behavior between unresolved samples
Field line	curve tangent to $E$	direction and source-to-sink connectivity	an SI field value from line count alone
Equipotential contour	positions with one value of $V$	local normal direction of $E$	a uniform magnitude from contour shape alone

The three displays use the same underlying data but retain different information. Vector arrows preserve a sampled magnitude scale; streamlines preserve tangent direction; equipotential contours support a gradient estimate. A map should identify which display is being used before a reader infers a numerical quantity from it.

Vector samples, streamlines, and equipotential contours of one two-dimensional electric field. Vector samples retain local direction and a stated arrow scale; a streamline follows the tangent directions; equal-potential contours are normal to the field recovered from a measured potential surface.

At any point where $E \neq = 0$ , one field direction is defined. Two field lines cannot cross at such a point because crossing curves would require two different tangent directions at one position. A zero-field point is an exception to the tangent construction: no direction is available there. Lines may approach, leave, or terminate at a point where the field vanishes in a drawn map, but the picture must not imply a preferred direction at the point itself.

Tangent construction for an electric field line. Each blue arrow gives the local direction of the prescribed electric field, and the gray curve has a tangent parallel to those arrows at its sampled points; a crossing would require two directions at one location.

The direction convention is fixed by a positive test charge. Lines leave a positive point source and enter a negative point source. An electron placed at a point on the same drawing experiences force opposite to the arrow, because $F = q E$ and its charge is negative. The line direction remains unchanged when the probe sign changes.

A single point charge $q$ has radial symmetry and field

E (r) = \frac{1}{4 π ε _{0}} \frac{q}{r ^{2}} \overset{r}{^} .

Every radial ray is a field line. The magnitude changes along each ray as $1/ r^{2}$ , so arrows placed at equal radial intervals should shorten with distance on a quantitative vector map. A field-line drawing often uses the same number of rays at all radii; this convention retains radial direction but does not encode the inverse-square magnitude by line count along a single ray.

Radial electric-field map of a positive point source. Arrows point outward at every sample position, while the shorter outer arrows decrease with inverse distance squared; the source marker itself is excluded from the point-charge approximation.

Line density and the limits of a drawing convention.

Line density becomes quantitative only after a drawing convention has been declared. For equal area elements normal to the local direction, a calibrated diagram may use

\frac{d N}{d A} = C ∣ E ∣,

where $d N$ is the number of drawn lines, $d A$ is the area element, and $C$ is the stated drawing scale. Without $C$ , line density gives only a local comparison within one figure. A count of twelve rays for $q$ and twenty-four for $2 q$ is a valid proportional convention; it is not a measurement of charge or flux.

Map feature	Valid reading	Required condition
Increasing separation of radial lines	decreasing magnitude away from a point source	same solid-angle sectors and projection
More lines through equal local areas	larger relative magnitude	one unchanged line-count scale
Longer vector arrows	larger magnitude	labelled arrow-length scale

A point source has $d A = r^{2} d Ω$ . The same rays cross a larger area at larger $r$ , matching the inverse-square dependence. Electric flux quantifies the field; a field-line sketch records its geometry.

Line density must be compared only at neighboring regions of the same drawing, using the same projection and scale. A three-dimensional source projected onto a page can make equally separated spatial lines appear crowded. A plot that changes its line count abruptly for visual clarity has no magnitude information across that change. These limitations matter when using textbook sketches to reason about relative strength near conductors or between plates.

Field lines are not individual physical objects. A line does not carry a fixed portion of charge or energy, and a sparse sketch is not evidence that the field is small. Quantitative work begins with $E$ , a charge distribution, or a measured potential difference. The drawing checks direction, symmetry, and the plausibility of a calculated result.

Source, sink, and zero-field geometry.

Electric field lines originate on positive charge and terminate on negative charge, or extend to or arrive from infinity when the charge distribution has nonzero net charge. Electrostatic field lines do not form closed loops. The electrostatic field has zero circulation around a closed path,

\oint E \cdot d ℓ = 0,

so a line that returned to its starting point with a fixed positive tangent would imply a nonzero component of $E$ along the entire closed route. Time-varying magnetic fields change this result; the present statement is restricted to electrostatics.

Equal positive charges provide a simple zero-field geometry. At the midpoint, the two fields have equal magnitude and opposite directions, so their vector sum vanishes:

E_{left} + E_{right} = 0 .

The electric potential at the midpoint is generally positive rather than zero, because potential contributions add as scalars. A field map should therefore label a zero-field site only after vector components have been added. A visual gap between lines is suggestive but not a proof of cancellation.

Superposed field near two equal positive charges. The midpoint has zero electric field because equal horizontal components oppose one another, whereas points on the vertical symmetry line retain upward or downward components from both sources.

The same drawing has a directional distinction near the midpoint. A small horizontal displacement produces a field toward the nearer positive charge, while a small vertical displacement produces a field away from the two-charge axis. The field directions form a saddle-like pattern. A test charge can have zero force at the central point without being stably confined in every direction. Field-line sketches are valuable for recognizing such component changes before a full stability calculation.

Superposition before drawing curves

Every line in a multi-source diagram belongs to the total field,

E_{total} (r) = i \sum E_{i} (r) .

Field lines must never be copied from each source separately and then overlaid as though all curves described one field. At a selected grid point, calculate or infer the vector sum first, then draw a tangent arrow. Repeating that construction through the region yields a vector map. Smooth field lines can be traced through the arrows afterward.

For equal and opposite point charges, the field between the charges is directed from positive to negative. Outside the pair, components curve away from the intercharge axis. The resulting line pattern represents a dipolar source, although a later lesson develops the electric dipole's moment, torque, energy, and far-field approximation separately. Here the important point is geometric: each tangent reflects the sum of two source fields at its own location.

Superposition map for an equal positive-negative source pair. Vectors in the region between sources point from plus to minus, while off-axis vectors tilt because both source contributions have horizontal and vertical components.

At an off-axis sample point, add components rather than arrow lengths:

E_{net} = i \sum E_{i}, E_{x} = i \sum E_{i, x}, E_{y} = i \sum E_{i, y} .

Cancellation: equal magnitudes cancel only when their directions oppose.
Mirror pair: components normal to a symmetry axis can cancel while parallel components add.
Off-axis point: the net direction usually differs from the radial direction of either individual source.

The component sum determines the arrow; a streamline is drawn only after that local vector has been determined.

Uniform regions and edge fringing.

Two broad, oppositely charged conducting plates form an approximately uniform field in their central region. There, field lines are nearly straight, parallel, and equally spaced. A uniform field has the form

E = E_{0} \overset{n}{^},

where both $E_{0}$ and the unit direction $\overset{n}{^}$ are independent of position within the stated region. The approximation requires plate dimensions much larger than their separation and a sample location sufficiently far from plate edges.

At an edge, charge distribution and field direction change. Lines bend outward through the surrounding space, and the field develops components parallel to the plates. A calculation based on one constant acceleration or one fixed field direction cannot be extended through that fringe region without additional modelling. The central field is a local approximation with a geometric domain, not a property of every point between finite plates.

Parallel-plate field map. Central arrows are parallel and evenly separated, marking the uniform-region approximation; edge arrows bend outward through the fringing region, where field direction and magnitude vary near the finite plate ends.

The surface of a conductor in electrostatic equilibrium imposes a boundary condition. The tangential component of the exterior field at the surface vanishes; otherwise mobile charge would move along the surface. Field lines therefore meet the surface at right angles. Their spacing can vary from one location to another because surface charge density is nonuniform, especially near sharp curvature or nearby conductors.

Field directions beside a conducting surface in electrostatic equilibrium. Arrows meet the surface normally because a tangential electric-field component would drive mobile surface charge; closer normal arrows indicate a larger local field magnitude in the mapped exterior region.

Constructing and testing a vector map

A measured or calculated map begins with a coordinate grid. At each grid point, record $E_{x}$ , $E_{y}$ , and, when needed, $E_{z}$ . Plot an arrow with the measured or calculated direction and a length proportional to a stated scale. The arrow scale must appear in the figure or caption; without it, longer arrows are merely decorative.

An instrument that measures one component requires repeated measurements with the probe axis aligned to each coordinate direction or rotate the apparatus through calibrated orientations. A zero reading can mean a zero vector component, a sensor axis perpendicular to the field, a cancelled source contribution, or an instrument below its resolution. State which interpretation is supported by the measurement procedure.

Use superposition and symmetry as independent checks. A map for equal source charges should mirror across the appropriate symmetry plane. Reversing all source charges reverses every vector. Doubling one source charge doubles only that source's contribution, so its effect on the total map is found by vector addition rather than by doubling all arrows. Measurements at paired locations provide a practical test of these relations.

Field lines are then traced so that each curve remains tangent to the local vector arrows. A line should be terminated at a source, a conductor surface, the boundary of the mapped region, or a zero-field site where its direction becomes undefined. Drawing a smooth curve through gaps without nearby vector samples can conceal a rapidly varying field. Increase grid resolution near charges, sharp conductor features, and regions where neighboring arrows change direction rapidly.

Geometric checks for field-line sketches.

An electrostatic field-line diagram should pass the following checks before it is used to support a calculation.

Direction: arrows leave positive free charge and enter negative free charge; at a conductor surface they are normal to the surface.
Uniqueness: curves do not cross except at a labelled zero-field site, where tangent direction is undefined.
Superposition: multi-source curves follow the vector sum rather than the individual source patterns overlaid on one page.
Scale: line density is described as a relative visual convention unless a calibration connects it to a numerical field or flux.
Domain: uniform arrows are restricted to a region where the source geometry supports the approximation; edge and boundary effects are shown separately.

These checks distinguish a field map from a charge sketch. The resulting direction, symmetry, and boundary information precedes any quantitative calculation of force, potential, flux, or charged-particle motion.

Topology near isolated positive and negative sources.

An isolated positive source is a source of electric field lines: arrows leave the charge and continue until they end on negative charge or at the boundary of the modelled region. An isolated negative source is a sink: arrows enter it from negative charge elsewhere or from infinity. This source-sink language describes the direction of the total electrostatic field. It does not mean that material flows along a line.

A separated positive and negative pair has field lines that, in the finite two-charge model, begin at the positive charge and end at the negative charge. Lines near the intercharge axis are nearly straight, while outer lines curve through a wider region. Their curvature follows the vector sum of the two source fields at each point. A line drawn radially outward from the positive source can be correct very near that source and incorrect farther away after the negative source contribution becomes comparable.

Topology can be checked without setting an arbitrary line count. Lines do not begin or end in empty space inside a charge-free map unless the drawing boundary has been reached. They do not cross at a point where the field is nonzero. A large empty region can contain a weak field, while a crowded region can result from a drafting convention. The reliable geometric information is tangent direction, connectivity from sources to sinks, and the way the pattern changes under superposition.

At a zero-field site, no tangent direction is defined. A diagram may show curves approaching such a site from several directions, but it should mark the cancellation rather than treating the site as an ordinary crossing. Equal positive charges and equal negative charges provide common examples of this geometry. The sign of the electric potential at such a site is a separate scalar question and cannot be read from the line topology alone.

Conductors, cavities, and induced surface charge

In electrostatic equilibrium, the electric field inside the conducting material is zero. A field-line map must therefore leave the metal interior free of lines. At an outer conductor surface, exterior lines are normal to the surface. Their direction depends on the local induced surface charge: arrows leave positive surface regions and enter negative surface regions. The density of drawn arrows can suggest stronger or weaker exterior field, but only after one drawing convention has been fixed.

A cavity changes the boundary geometry. Consider an initially neutral isolated conductor with a positive charge placed inside an empty cavity. The cavity field lines leave the internal charge and end on negative induced charge on the cavity wall. The conductor then carries positive induced charge on its outer surface so that its total charge remains zero. Exterior lines leave that outer surface and extend to the boundary of the model or to negative charge elsewhere. No line passes through the metal from the cavity to the outside.

Qualitative induced-charge map for a positive charge inside a cavity of an initially neutral conductor. Cavity lines end on negative induced inner surface charge; positive induced outer-surface charge produces exterior field lines, while the metal itself contains no electrostatic field lines.

The diagram establishes topology only. The surface-charge pattern depends on cavity shape, charge position, and the conductor's net charge. A charge off the cavity centre produces a nonuniform inner pattern, with stronger field and denser induced charge on the nearer wall. The total induced inner charge is opposite to the enclosed free charge, while its density remains nonuniform in general.

A grounded conductor follows a different outer-charge condition because charge may flow between the conductor and Earth. The cavity boundary remains a conductor boundary, but the outer surface charge is determined by the grounding connection and the surrounding charge distribution. A field-line sketch should name the conductor condition—isolated neutral, specified net charge, or grounded—before its outer lines are interpreted quantitatively.

Symmetry and field-map audit.

Symmetry applies only when the complete source and boundary configuration has the symmetry. Equal source charges in otherwise empty space give a mirror relation across their perpendicular bisector. The addition of a nearby conducting wall, a grounded support, or an unequal charge breaks that relation. A map should not be mirrored because two visible charges look similar while hidden boundary conditions differ.

Audit the source data before testing local vectors. State the coordinate system and all conductors or prescribed charges included in the model. Identify the mirror planes, rotation axes, or translational symmetries that remain after the boundaries are included, then compare the required components at paired points. At conductor boundaries, exterior vectors are normal to the surface; inside ideal electrostatic conductor material, they vanish.

Symmetry identifies zero components and relates values at paired locations; it usually does not determine field magnitude at an arbitrary point. A vector map still requires source strengths, distances, measurements, or a solved boundary-value model. When a drawing fails a symmetry check, inspect the source list and boundary assumptions before altering individual arrows by eye.

Boundary audit for conductor maps.

A conductor map is checked region by region. Mark each connected metal region, each cavity, every free charge, and every electrical connection before tracing lines. In the metal, the electrostatic field is zero. In an empty cavity containing no charge, the field is also zero when the enclosing conductor is in electrostatic equilibrium; there is then no field-line pattern to draw inside that cavity. A free charge in the cavity changes this conclusion by requiring induced charge on the inner wall.

At every conductor interface, inspect direction before density. Exterior arrows must meet an ideal conductor normally. A drawn arrow that grazes the surface represents a tangential component and contradicts electrostatic equilibrium. Next inspect connectivity: a cavity arrow may end on its inner wall, while an exterior arrow begins or ends on the outer surface according to the induced charge there. No arrow should cross from cavity to exterior through the metal.

The conductor condition completes the boundary data. An isolated neutral conductor, an isolated conductor with specified net charge, and a grounded conductor can have different outer maps even when their cavities contain the same free charge. The boundary geometry alone does not determine the outer surface charge. This distinction prevents a qualitative sketch from silently assuming a charge reservoir that the physical system does not contain.

Cavities, equipotential geometry, and calibrated field maps.

A conductor cavity separates interior and exterior field maps. In electrostatic equilibrium, the electric field in the conducting material is zero. Field lines drawn inside a cavity can begin or end on charges placed in that cavity and on induced charge on the inner wall, but they cannot continue through the metal to the exterior. Exterior field lines are determined by charge on the outer surface together with external charges. A drawing that joins an inner-wall line to an outer-surface line through the metal contradicts the zero-field condition in the conductor.

Grounding changes the charge constraint. A grounded conductor is held at a specified potential through a connection to a large charge reservoir. Its total conductor charge can change as charge passes through that connection. An isolated conductor cannot exchange charge with the environment; its induced inner and outer surface charges must respect its specified total charge. The cavity field may have the same local geometry in two cases while the outer field maps differ because the outer charge constraint is different.

A Gaussian surface drawn within the conducting material has $E = 0$ , so its enclosed charge must vanish:

\oint_{S} E \cdot d A = 0 ⟹ Q_{inner} = - Q_{cavity} .

This condition fixes the total induced charge on the cavity wall. It does not fix its spatial distribution, which depends on the cavity shape and source position.

Conductor condition	Quantity fixed by the model	Consequence for the outer map
isolated, neutral	total conductor charge $= 0$	outer induced charge balances the inner-wall charge
isolated, specified charge	total conductor charge is prescribed	outer charge follows from the prescribed total and $Q_{inner}$
grounded	potential is prescribed	charge may enter or leave through the grounding connection

Equipotential curves prepare the geometric interpretation of a field map. The electric field points in the direction of greatest decrease of potential, so it is perpendicular to an equipotential curve at each point where the potential is smooth. Field lines therefore cross equipotential curves at right angles. This local geometric rule supplements the quantitative potential calculation. The spacing of adjacent equipotential curves also matters: closely spaced equal-potential increments indicate a larger field magnitude than widely spaced increments.

The relation can be written

E = - \nabla V .

For field-line sketches, use it as a consistency check. A field line tangent to an equipotential curve is incorrect unless the field is zero at that point. A conductor in electrostatic equilibrium is itself an equipotential body; exterior field lines meet its surface normally. The statement applies to the surface geometry and does not imply that the exterior field magnitude is uniform around a curved conductor.

Field lines and equipotential curves. Electric-field directions cross equal-potential curves normally, while closer equipotential spacing corresponds to a larger local field magnitude for the same potential interval.

A quantitative field map requires a measured potential grid and a calibrated spatial coordinate system. In a conducting-sheet analogue, set electrode potentials with a stable supply, measure voltage at a grid of positions with a high-input-resistance probe, and record both the probe coordinates and voltage uncertainty. The electric field is estimated from potential differences over known separations. The grid spacing must be small compared with the scale over which the potential changes; a coarse grid can miss a narrow high-field region near an electrode edge.

Calibrate the position scale before collecting voltage data. A ruler mark, camera calibration target, or motor-stage encoder gives physical coordinates, but its origin must be related to the electrode geometry. Check the voltmeter zero, supply stability, and probe loading with reference points. A probe that draws appreciable current changes the sheet potential it is meant to measure, producing a systematic map error rather than random noise. Record the probe input resistance and the measurement settling time with the map.

At an interior grid point, centred differences give the local components:

E_{x} (x_{i}, y_{j}) \approx - \frac{V _{i + 1, j} - V _{i - 1, j}}{2Δ x}, E_{y} (x_{i}, y_{j}) \approx - \frac{V _{i, j + 1} - V _{i, j - 1}}{2Δ y} .

For independent equal voltage uncertainties $σ_{V}$ , the first estimate has the approximate random uncertainty

σ_{E_{x}} \approx \frac{σ _{V}}{2 Δ x} .

Halving $Δ x$ improves spatial resolution but increases the contribution of meter noise to the derivative. At a boundary, use a labelled one-sided difference and report its larger truncation error. Interpolation belongs after the derivative rule has been chosen; a smooth contour does not improve a sparse measurement.

Record	Quantitative role	Diagnostic check
electrode geometry and coordinates	sets $Δ x, Δ y$ and the model domain	compare a reference separation with the survey scale
repeated voltages	estimates $σ_{V}$ and drift	reverse scan order and compare residuals
probe input resistance and settling time	bounds loading and time-dependent bias	repeat a reference point before and after a scan
derivative and interpolation rule	determines plotted $E$	retain raw grid values with the map

The charge placed in a cavity also imposes a charge condition on the inner wall. A Gaussian surface lying inside the conducting material has zero electric flux because the field there is zero. Its enclosed charge must therefore vanish. The inner-wall induced charge is the negative of the total free charge in the cavity. This statement sets the total induced inner charge; it does not make the inner-wall density uniform. The density depends on the cavity geometry and the position of the cavity charge.

An isolated conductor's outer surface carries whatever additional charge is required by the conductor's specified total charge after the inner-wall charge is accounted for. For a grounded conductor, charge can enter or leave until the specified potential condition is satisfied. In both cases, the local cavity map must be checked against the inner-wall charge condition and the exterior map against the stated outer charge constraint. Cavity, metal, and exterior must therefore be analysed as distinct field-line regions.

Equipotential measurements provide a second check on a drawn map. Mark equal measured voltage intervals, then compare the measured field-arrow direction with the local normal to those intervals. A large tangent component indicates a coordinate error, an interpolation artifact, or a map outside electrostatic conditions. The potential grid also sets the arrow-length scale: use the same voltage interval and same spatial scale across the map before comparing field magnitude in different regions.

Uncertainty in a derivative grows when the grid spacing is made very small without improving voltage precision. A potential difference divided by a short distance can amplify meter noise; a very large spacing smooths away real field variation. Select a spacing that resolves the electrode geometry while leaving voltage differences above the repeatability floor. A coarse pilot map can identify high-gradient regions, after which a denser local grid can be measured with the same calibration protocol.

A final calibration check uses a region with an independently predictable geometry, such as the central portion between broad, nearly parallel electrodes. Compare the measured potential slope there with the applied voltage divided by electrode spacing. Agreement validates the coordinate and voltage scales within uncertainty; disagreement requires checking supply leads, probe loading, electrode spacing, and the assumed uniform region. Use this reference comparison to anchor the field scale. Less symmetric regions still require their own mapped measurements.

Numerical streamlines and adaptive superposition maps

A numerical field-line map begins with vector values on a grid, not with hand-drawn curves. At a seed point $r_{0}$ , the streamline tangent is the local field direction. A simple forward step is

r_{j + 1} = r_{j} + h \frac{E ( r _{j} )}{∣ E ( r _{j} ) ∣},

where $h$ is a chosen spatial step. The normalization advances the curve by a fixed distance rather than by an amount proportional to field magnitude. Magnitude is shown separately through arrow density, colour scale, or tabulated values. Without this normalization, a large field creates long plotting steps and can skip curvature or pass through a nearby charge location.

The vector at an off-grid streamline point must be interpolated from neighbouring grid values. Bilinear interpolation is adequate for a smooth rectangular grid when the spacing resolves the field variation. Near a point charge or sharp electrode edge, collect data at a spacing that resolves substantial within-cell variation. Record the interpolation rule, grid spacing, and stopping conditions. A streamline should stop at a charged source, a conducting surface, or the outer edge of the mapped region; extending it through an excluded region creates a numerical curve with no physical interpretation.

Step size controls geometric error. A large step follows the initial tangent too far and cuts across a curved line; a very small step increases cost without improving a field that is already uncertain at the grid scale. Compare maps made with $h$ and $h /2$ . If the curves or their intersections with a surface move appreciably, the integration has not converged. A higher-order stepping method can reduce integration error, but it cannot repair an under-resolved input grid or a poorly calibrated field measurement.

Adaptive sampling directs measurements where they affect the map most. Estimate field change across each cell from neighbouring vector differences or potential differences. Subdivide cells where direction rotates rapidly, magnitude changes strongly, or a field-line seed approaches a conductor edge. Retain a coarse grid where the field is nearly uniform. The refinement criterion must be numerical and stated in the report; otherwise dense sampling can follow an observer's expectation rather than the measured gradient.

Use a component residual for each measured or computed vector sample:

r_{i} = E_{map, i} - E_{ref, i}, z_{i} = \frac{∣ r _{i} ∣}{σ _{E, i}},

where $E_{ref, i}$ is a direct Coulomb or potential-gradient estimate and $σ_{E, i}$ includes the stated source, coordinate, and voltage uncertainty. The residual direction identifies a sign or coordinate error; its normalized magnitude tests agreement against the measurement model.

Check	Quantity compared	Acceptance condition
local vector	$E_{map, i}$ and $E_{ref, i}$	components agree within the stated uncertainty
topology	line direction and endpoints	lines leave positive charge and terminate on negative charge, conductor charge, or the mapped boundary
symmetry	paired vector components	required even and odd component relations hold for the complete source-and-boundary model
scale	one reference arrow	potential-gradient or Coulomb estimate agrees before the full map is interpreted
convergence	selected streamline positions	grid and step refinement shift features by less than the reported coordinate bound

For the two-charge map, the axial vectors between the sources point from positive to negative and reinforce. Above and below the axis, the vertical component changes sign by reflection symmetry. Reversing one source charge reverses that source contribution at every grid point. These checks are performed on components, not on the visual line shape.

Seed placement controls visual coverage rather than field magnitude. A symmetric map can assign a fixed enclosed-charge increment to each seed line; a grid map can place seeds at regular points on a contour around a source. State the seed rule whenever line density changes across a figure. Otherwise a dense bundle near an electrode can be mistaken for a calibrated magnitude scale.

Run convergence separately from the drawing. Recompute selected streamlines with smaller integration steps and refined local cells. Compare surface intersections, turning positions, and distance from excluded charge cells. A stable result remains within the coordinate uncertainty after both refinements. When the shift exceeds that bound, retain the coarse record, mark the region unresolved, and do not replace the uncertainty with a smooth curve.

Record the coordinate origin, grid locations, source values, interpolation rule, seed rule, refinement threshold, stopping conditions, and at least one component calculation. Repeated voltage readings or source perturbations supply the uncertainty used in the residual. These data distinguish physical map features from display choices and permit an independent reconstruction of the streamlines.

Critical sites, resolution bounds, and analytic-map checks.

A zero-field critical site requires special treatment in a streamline map. The unit direction $E /∣ E ∣$ is undefined where the field magnitude is zero, so a tracing routine cannot step through that point using the ordinary direction rule. Near a critical site, small measurement or interpolation errors can rotate the estimated direction substantially because the denominator is small. Mark the region as a critical neighbourhood, stop streamlines before entering its uncertainty radius, and use a separate component analysis to classify the local vector pattern.

For two like charges of unequal magnitude, a zero-field point can occur on the line between them where the oppositely directed field magnitudes balance. Its position is found from the component equation, not from a visual crossing of field lines.

Map resolution should be reported as an error bound rather than as a visual density choice. A grid with spacing $Δ x$ cannot localize a feature more accurately than the coordinate calibration and the cell scale without additional interpolation assumptions. If a streamline crosses a conductor surface between grid nodes, bracket the crossing with neighbouring cells and report the interval. Refinement should reduce that interval; if it does not, voltage noise, source-position uncertainty, or a non-smooth geometry dominates the error.

Use two independent bounds. The numerical bound compares results after halving grid spacing and streamline step size. The measurement bound perturbs voltage readings, charge values, and sensor coordinates within their calibration uncertainty, then recomputes the map. A feature is well resolved only when both bounds are smaller than the physical scale assigned to the feature. A dense graphic with no uncertainty study is not a high-resolution measurement.

An analytic-versus-measured check separates physical superposition from plotting choices. Use a geometry with known source values, such as two point charges at surveyed coordinates. Calculate the net field at selected grid points by adding the two Coulomb vectors component by component. Measure the corresponding field from a calibrated potential grid or field probe. Compare both magnitude and direction; a map can have similar-looking lines while its component signs are wrong.

For example, choose three points: one on the line joining the charges, one above the midpoint, and one far from both sources. At each point, tabulate $E_{x}$ , $E_{y}$ , magnitude, and direction from the analytic calculation and from the measurement. The line point checks the sign of axial contributions, the off-axis point checks both components, and the distant point checks the overall source scale. Residuals should be normalized by the combined calculation and measurement uncertainty rather than by the field magnitude alone, which becomes unstable near a critical site.

The validation record should preserve the source coordinates, analytic component calculation, measurement calibration, grid and step refinements, critical-site stop rule, and the uncertainty used for each comparison. Report unresolved regions instead of forcing streamlines through them. With these bounds, the curves represent geometry while the validation table and error bounds state what the data support quantitatively.

Critical-site classification also needs a coordinate check. Recalculate the local components after rotating or translating the grid origin; the physical location and the balance of the two source contributions must be unchanged even though displayed coordinates differ. A result that depends strongly on the selected grid origin usually signals a component-indexing or interpolation error. Keep the local component table with the map so the zero-field claim can be reproduced without tracing a curve through the uncertain region.

Refinement should stop when its shift is smaller than the measurement bound or the required reporting precision. Continuing to subdivide beyond that point creates more computed points but does not create new information. Conversely, a refinement shift larger than the stated uncertainty means the original grid was not adequate for the claimed feature. Report both the coarse estimate and the refined estimate; their difference documents numerical convergence directly.

For the analytic-versus-measured comparison, use a blind subset of grid points when possible. Set calibration and interpolation settings from reference points, then test the remaining points without retuning the map. A directional residual can reveal a coordinate-axis reversal even when magnitude residuals are small. A magnitude residual with correct direction can indicate a voltage-scale or charge-scale error. Keeping these residual types separate retains diagnostic detail beyond a single pass-or-fail number.