--- title: Potential Gradients and Equipotentials module: Electric Potential moduleNumber: 3 lessonNumber: 3 order: 303 summary: | Given the potential everywhere, how do we recover the field? The field is the negative gradient, $\vec E=-\nabla V$: it points down the steepest local drop in potential, and its magnitude is set by how fast $V$ changes, not by the shape of a contour. We read off components with directional derivatives, reconstruct fields from measured potential grids using centered differences, and use closed-loop integrals and grid refinement to test whether a reconstructed field is physically consistent. topics: [Electric Potential] draft: false sources: - book: Tipler & Mosca ref: "Ch. 23 — Electric Potential; §§23-2–23-4" --- ## Potential as a spatial map An electrostatic potential is a scalar field $V(\vec r)$ over a region. The local change for a small displacement $\d \vec\ell$ is $$ \d V=\frac{\partial V}{\partial x}\d x+ \frac{\partial V}{\partial y}\d y+ \frac{\partial V}{\partial z}\d z =\nabla V\cdot \d \vec\ell. $$ Electrostatic work gives the same differential as $\d V=-\vec E\cdot\d\vec\ell$, hence $$ \vec E=-\nabla V. $$ The negative gradient points in the direction of greatest local decrease in potential. Potential differences over known displacements determine field components; one potential value does not. Two neighboring readings, their separation, and the direction of the separation are the minimum local information needed to estimate a field component. $$ % caption: Equipotential contours in a plane. At $P$ the field $\vec E=-\nabla V$ is normal to the contour and points toward lower potential; a displacement tangent to the contour gives no first-order change in $V$. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black] (0.35,0.45) .. controls (1.25,1.00) and (1.20,2.55) .. (0.45,3.15); \draw[black] (1.35,0.20) .. controls (2.30,0.90) and (2.25,2.72) .. (1.35,3.42); \draw[black] (2.55,0.12) .. controls (3.48,0.92) and (3.45,2.85) .. (2.48,3.55); \draw[black] (3.78,0.25) .. controls (4.58,1.05) and (4.52,2.72) .. (3.70,3.40); \node[below] at (0.55,0.60) {$12\ \mathrm V$}; \node[below] at (1.55,0.38) {$8\ \mathrm V$}; \node[below] at (2.75,0.30) {$4\ \mathrm V$}; \node[below] at (3.95,0.50) {$0\ \mathrm V$}; \draw[black] (2.02,1.42)--(2.98,2.22); \node[black,below right] at (2.02,1.42) {tangent}; \filldraw[draw=acc,fill=acc!10] (2.50,1.82) circle (2pt); \node[above left] at (2.52,1.88) {$P$}; \draw[->,acc,very thick] (2.50,1.82)--(3.45,1.72) node[right] {$E$}; \end{tikzpicture} $$ The Cartesian components follow directly: $$ E_x=-\frac{\partial V}{\partial x},\qquad E_y=-\frac{\partial V}{\partial y},\qquad E_z=-\frac{\partial V}{\partial z}. $$ For example, a local potential $V(x,y)=ax^2-by^2$ has $\vec E=-2ax\,\hat\imath+2by\,\hat\jmath$. The origin has zero field, while the potential has opposite curvature along the two coordinate axes. The zero derivative at one point does not establish stable three-dimensional electrostatic equilibrium; the full local geometry determines the response to a displacement. **Directional derivatives and measured components.** Let $\hat t$ be a unit vector along a selected measurement direction. The rate of potential change along that direction is $$ \frac{\d V}{\d s}=\nabla V\cdot\hat t =-\vec E\cdot\hat t. $$ A pair of probes separated by $\Delta s\,\hat t$ estimates the component $E_t\simeq-\Delta V/\Delta s$. The result is a projection, not necessarily the field magnitude. A probe line perpendicular to the field gives nearly zero voltage difference even in a region of strong field. A complete two-dimensional field map requires at least two independent directional differences or a fitted potential surface. Finite separation sets a resolution limit. If the potential varies substantially over the probe spacing, the difference quotient gives an average slope across the probe pair. Reducing separation improves spatial resolution but can make the measured voltage comparable with instrument noise. A reported gradient therefore needs both a spacing and an uncertainty estimate. Its coordinate location and the probe orientation also belong in the field report. The sign convention follows the order of the probe labels. If $V_B **Worked example.** Two probes $2.0\ \mathrm{cm}$ apart along the $x$-direction read > $V_A=6.0\ \mathrm V$ at $A$ and $V_B=2.0\ \mathrm V$ at $B$, with $B$ in the $+x$ > direction from $A$. A second pair, the same spacing along $y$, reads no difference. > Estimate the field. > > The measured component along the probe line is the negative slope: > > $$ > E_x\simeq-\frac{V_B-V_A}{\Delta s}=-\frac{2.0-6.0\ \mathrm V}{0.020\ \mathrm m} > =+2.0\times10^2\ \mathrm{V/m}. > $$ > > The transverse pair gives $E_y\simeq0$, so the local field is $200\ \mathrm{V/m}$ > in $+x$. This is a projection: had the field been strong but perpendicular to the > $x$ probe line, that pair alone would have reported nearly zero. Two independent > directions are the minimum for a plane field. ## Equipotential surfaces and conductors An equipotential surface has $V=\mathrm{constant}$. Every tangent displacement within it has $\d V=0$, so $\vec E$ is normal to the surface. Equipotential surfaces cannot cross: one spatial point cannot have two potential values in the same source configuration. Their spacing converts a map into a field estimate. For adjacent contours differing by $\Delta V$ with normal spacing $\Delta n$, $$ |\vec E|\simeq\frac{|\Delta V|}{\Delta n}. $$ This approximation becomes accurate when contours are close enough that curvature and field magnitude change little across their separation. Closely packed contours indicate a large field magnitude. Their labels alone do not fix the direction; the direction follows from decreasing potential and from the chosen normal. A conductor in electrostatic equilibrium has one potential throughout its connected material and across its surface. A tangential electric field would move mobile charges and change the distribution, so equilibrium requires zero tangential field. The normal exterior field can be nonzero and is set by surface charge and adjacent boundaries. Consequently, a conductor surface is an equipotential, while a generic equipotential surface need not be a conductor. Near a sharp conductor feature, exterior equipotential surfaces can crowd together. The normal field and surface charge density are then large locally. Local geometry and the conductor boundary condition establish that result. The complete boundary-value problem, including surrounding conductors and prescribed potentials, determines the field and charge distribution across the full object. ## Coordinates adapted to symmetry The gradient formula changes form with coordinates because each coordinate has a different physical scale factor. For cylindrical coordinates $(s,\phi,z)$, $$ \nabla V= \hat s\frac{\partial V}{\partial s} +\hat\phi\frac{1}{s}\frac{\partial V}{\partial\phi} +\hat z\frac{\partial V}{\partial z}. $$ A cylindrically symmetric potential has no angle or axial dependence, so $\vec E=-\d V/\d s\,\hat s$. A long uniformly charged line provides a standard example. Between two finite radii, its potential difference has the logarithmic form $$ V(s_b)-V(s_a)=-\frac{\lambda}{2\pi\varepsilon_0} \ln\frac{s_b}{s_a}, $$ and differentiation gives $E_s=\lambda/(2\pi\varepsilon_0s)$. Cylindrical area growth produces the logarithm while the definition of voltage remains unchanged. In spherical coordinates, a potential with only radial dependence has $\vec E=-\d V/\d r\,\hat r$. For $V=kQ/r$, differentiation gives the inverse-square field. The direction follows the sign of $Q$: positive $Q$ gives an outward field, while negative $Q$ reverses the radial vector. A radial graph of potential therefore carries both a slope and a coordinate-direction convention. Coordinate selection must follow the physical symmetry, not the shape of a drawing alone. A finite wire is not cylindrically symmetric at distances comparable with its length, and a nearby grounded plane breaks the radial symmetry of a charged sphere. In those cases, $V$ can still be differentiated to obtain field, but the potential must first include the actual boundary geometry. ## Path integrals and the closed-loop condition Potential difference is defined by a line integral: $$ V(B)-V(A)=-\int_A^B\vec E\cdot \d \vec\ell. $$ In an electrostatic field, the value is path independent. A route can therefore be broken into pieces chosen to simplify the integrand. In a uniform horizontal field, vertical pieces contribute zero because their displacements are perpendicular to $\vec E$. In a radial point-charge field, circular arcs contribute zero and a radial segment carries the full potential change. Path choice can shorten the calculation while the physical voltage difference remains fixed. The same condition appears on a closed path: $$ \oint\vec E\cdot \d \vec\ell=0. $$ Traversing a contour map around a small rectangle provides a local check. The voltage drop along one side is reversed by the corresponding return side when the field derives from a single electrostatic potential. Measured differences that fail to close within uncertainty can arise from probe offsets, changing source charge, or an induced nonconservative field. A nonzero loop residual requires checking those effects before averaging readings. $$ % caption: Two routes between $A$ and $B$ in a uniform field. Horizontal segments carry the potential change and vertical segments carry none; the closed rectangular loop has zero net line integral for a stationary source. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \foreach \y in {0.20,0.95,1.70,2.45} {\draw[->,black] (0.20,\y)--(5.30,\y);} \draw[acc,very thick,->] (1.05,0.65)--(4.35,0.65)--(4.35,2.05)--(1.05,2.05)--cycle; \filldraw[draw=black,fill=black!8] (1.05,0.65) circle (2pt); \node[below] at (1.05,0.58) {$A$}; \filldraw[draw=black,fill=black!8] (4.35,2.05) circle (2pt); \node[above] at (4.35,2.12) {$B$}; \node[black,right] at (4.95,2.75) {$E$}; \end{tikzpicture} $$ The closed-loop condition is restricted to electrostatics. A changing magnetic flux can give a nonzero loop integral even in empty space. The measured electromotive force then depends on the stated circuit path and time dependence. Potential maps remain valuable locally, but a single global scalar potential map cannot encode the complete induced electric field around the loop. ## Contour spacing, interpolation, and map resolution Contour labels provide a finite-difference estimate of field. If consecutive contours differ by $2\ \mathrm V$ and are separated normally by $0.50\ \mathrm{cm}$, the local field magnitude is approximately $400\ \mathrm{V/m}$. The estimate applies near the midpoint between contours. It should not be extended across a broad region where the contour spacing changes substantially. Interpolation requires a model of variation between sampled points. Linear interpolation across a small cell gives a constant estimated gradient in that cell. A smooth fitted surface can reduce random measurement noise but may hide a steep physical gradient near an electrode edge. The interpolation method, grid spacing, and reference potential determine what a published field map actually represents. Equipotential lines are frequently drawn with visually uniform spacing for clarity. Such drawings communicate direction and topology, not calibrated magnitude, unless their labels and geometry state equal potential increments. A reader should derive field magnitude from labeled differences and measured normal distance, not from the ink density of a schematic diagram. Conducting boundaries supply especially reliable contour information: each connected conductor has one known potential, while the external contours must meet its surface normally. A numerical potential solution can be checked by verifying that its contours neither cross nor end in the empty region and that its gradient has the specified conductor boundary behavior. These geometry checks complement numerical residuals from the governing field equation. ## Conducting boundaries and normal derivatives At an electrostatic conductor surface, the potential has one common value. Resolve the exterior field into a normal component $E_n$ and tangent components. The tangent components vanish at equilibrium, because a nonzero tangent field would drive mobile charge along the surface. The remaining normal component is related to the surface-charge density by $$ E_n=-\frac{\partial V}{\partial n} =\frac{\sigma}{\varepsilon_0} $$ for vacuum immediately outside the conductor, with the normal chosen outward. The relation connects a measurable potential gradient with charge density. It does not assert that charge density is uniform: surface curvature, neighboring conductors, and imposed voltages change the local normal derivative. An equipotential contour meets a conductor surface tangentially, while field arrows meet the surface normally. In a narrow gap between oppositely biased conductors, contours are closely spaced and nearly parallel. Near the outer edges they spread and curve. The central estimate $E\simeq\Delta V/d$ then applies only where the gap is small relative to electrode width and the field is nearly normal to the faces. $$ % caption: Geometry at a charged conducting surface held at $V_0$. Exterior equipotentials meet the metal tangentially and $\vec E$ is normal; tighter contour spacing near the curved tip marks a larger normal derivative and larger surface charge density. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \fill[black!8] (0.55,0.45) .. controls (2.45,0.55) and (2.50,2.72) .. (0.55,2.82) -- cycle; \draw[black,thick] (0.55,0.45) .. controls (2.45,0.55) and (2.50,2.72) .. (0.55,2.82); \draw[black,dashed] (2.75,0.38) .. controls (3.35,1.10) and (3.35,2.20) .. (2.75,2.90); \draw[black,dashed] (3.55,0.26) .. controls (4.30,1.08) and (4.30,2.22) .. (3.55,3.02); \draw[black,dashed] (4.52,0.18) .. controls (5.35,1.08) and (5.35,2.22) .. (4.52,3.10); \draw[->,acc,very thick] (2.37,1.65)--(3.18,1.65) node[right] {$E$}; \node[left] at (1.20,1.65) {$V_0$}; \end{tikzpicture} $$ A conductor connected to a reference source has a prescribed potential boundary condition. An isolated conductor instead has a prescribed total charge together with an unknown constant potential. Both cases require the exterior field solution to determine local $\sigma$. Different recorded potentials at two points on one connected equilibrium conductor indicate either a measurement error, a time-dependent current, or an inconsistent boundary model. **Finite differences on a potential grid.** Measured or simulated potential values are often available on a rectangular grid. At an interior grid point with spacings $h_x$ and $h_y$, central differences estimate the field components: $$ E_x\simeq-\frac{V(x+h_x,y)-V(x-h_x,y)}{2h_x},\qquad E_y\simeq-\frac{V(x,y+h_y)-V(x,y-h_y)}{2h_y}. $$ The estimate is attached to the grid point at the center of the two samples. A forward difference uses one neighboring value and has larger leading truncation error for a smooth potential. Near a conductor boundary, one-sided differences may be necessary because no sample exists inside the metal. The resulting normal derivative should be compared with known surface charge or imposed voltage data. Grid resolution and random voltage noise compete. Halving $h_x$ improves the ability to resolve a rapidly changing potential, but the voltage difference in the numerator becomes smaller and can be dominated by meter noise. Repeated readings, smooth physically justified fits, and independent field probes provide ways to separate a real gradient from measurement scatter. A plotted arrow field should record its finite-difference scale and identify it as a spatial average. $$ % caption: Central-difference reconstruction of the field at $P$. The four neighbor samples give the two coordinate differences; smaller grid spacing improves resolution but shrinks the measured voltage in each difference. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \foreach \x in {0.60,2.20,3.80} {\draw[black] (\x,0.42)--(\x,3.00);} \foreach \y in {0.42,1.71,3.00} {\draw[black] (0.60,\y)--(3.80,\y);} \filldraw[black] (0.60,1.71) circle (1.6pt); \filldraw[black] (3.80,1.71) circle (1.6pt); \filldraw[black] (2.20,3.00) circle (1.6pt); \filldraw[black] (2.20,0.42) circle (1.6pt); \filldraw[draw=acc,fill=acc!10] (2.20,1.71) circle (2.6pt); \node[above right,fill=white,inner sep=1pt] at (2.28,1.76) {$P$}; \node[left] at (0.52,1.71) {$V_W$}; \node[right] at (3.88,1.71) {$V_E$}; \node[above] at (2.20,3.08) {$V_N$}; \node[below] at (2.20,0.34) {$V_S$}; \draw[<->,acc,thick] (0.60,0.02)--(2.20,0.02) node[midway,below] {$h_x$}; \draw[<->,acc,thick] (4.20,0.42)--(4.20,1.71) node[midway,right] {$h_y$}; \end{tikzpicture} $$ An electrostatic map has an additional internal consistency condition. Field components reconstructed from $V$ must have zero circulation around every small grid loop to within sampling uncertainty. Equivalently, a finite-difference curl estimate should vanish in a stationary electrostatic region. A systematic nonzero loop value points to calibration drift, source variation, or an electromagnetic induction effect outside the static-potential model. **Constrained motion and local potential shape.** A charge restricted to a track parameterized by $s$ has potential energy $U(s)=qV(s)$ and the force along the track is $$ F_s=-\frac{\d U}{\d s}=-q\frac{\d V}{\d s}. $$ A stationary point has zero constrained force. For positive $q$, a local minimum of $V(s)$ produces restoring force along the track; a local maximum produces force away from the point. Negative $q$ reverses the potential-energy curvature. The track or mechanical support is part of the physical system: the one-dimensional classification says nothing about displacement directions that the constraint forbids. In empty three-dimensional electrostatic space, a potential satisfying the charge-free field equation cannot have a strict local maximum or minimum. A field null can instead be a saddle, with restoring behavior in one direction and anti-restoring behavior in another. Charged-particle traps use time dependence, magnetic fields, material boundaries, or mechanical constraints to supply the missing stabilization. The field map and the energy graph must use the same charge sign. A contour map shows $\vec E=-\nabla V$ independent of test charge. A force map for electrons reverses the arrows from a field map, and an electron's potential-energy contours reverse the ordering of $V$ contours. Keeping the source potential, test-charge force, and test-charge energy as separate quantities prevents sign changes from being applied twice. **Reference offsets and physically measured differences.** Potential has an arbitrary additive constant. Replacing a solution by $$ V'(\vec r)=V(\vec r)+C $$ leaves its gradient unchanged, so it leaves $\vec E$, force, and every potential difference unchanged. A reference conductor assigned $V=0$ fixes one convenient value of $C$. It does not declare every unconnected conductor uncharged, and it does not make the local field vanish at the reference conductor's surface. Potential maps from separate measurements can be compared only after their references have been aligned. A probe connected to a different reference lead introduces an offset in every reading. The offset has no effect on locally computed field if it is constant across the grid, but it changes an incorrectly reported absolute potential. A drifting reference produces position-independent shifts over one scan and time-dependent differences between scans; a field estimate from neighboring simultaneous readings can be more reliable than a comparison of two absolute maps acquired at different times. Voltage sources impose potential differences between terminals, not necessarily the potential of either terminal relative to distant space. Grounding one terminal usually establishes a shared reference and a charge-transfer path. The solution still depends on every conductor, dielectric, and free-charge boundary in the region. Recording the reference node and the time at which it was connected is part of a reproducible potential measurement. **Level surfaces and normal geometry in three dimensions.** An equipotential surface is a level surface of the function $V(x,y,z)$. At a point where $\nabla V\ne0$, the gradient is perpendicular to every tangent direction of that surface. A tangent displacement $\d \vec\ell_t$ obeys $$ \nabla V\cdot \d \vec\ell_t=0. $$ The unit normal can be written $\hat n=\nabla V/|\nabla V|$, and the electric field is anti-parallel to it for a positive outward gradient convention. At a regular point, $\vec E$ is normal to the equipotential and points toward lower potential. The magnitude requires the normal rate of change, not curvature alone. A point charge has spheres $r=\mathrm{constant}$ as level surfaces. Their curvature decreases with radius, but field magnitude also decreases because the potential slope $d(kQ/r)/\d r$ decreases. For a uniform field, the level surfaces are planes with zero curvature and constant spacing. Curvature and field strength are therefore distinct geometric properties. Contour crowding in the boundary-value solution, rather than curvature alone, determines the large field near a sharp conductor tip. The tangent-plane approximation is local. Over a large displacement on a curved equipotential, the tangent direction changes along the route. The total potential change remains zero for a path contained on the same equipotential surface, but a straight chord between two points can leave the surface and acquire a nonzero potential difference. ## Critical points, saddles, and contour topology A critical point has $\nabla V=\mathbf0$, hence zero electric field. The contour pattern near such a point identifies whether potential rises, falls, or changes oppositely along different directions. For $$ V(x,y)=a(x^2-y^2), $$ the origin is a saddle. Along the $x$ axis, potential increases away from the origin; along the $y$ axis, it decreases. The field components are $E_x=-2ax$ and $E_y=2ay$. A positive charge is pushed inward along one axis and outward along the other, so the zero-field point cannot confine it in the full plane. $$ % caption: Saddle equipotentials of $V=a(x^2-y^2)$. The field vanishes at the origin; contour branches curve oppositely along the two axes, so a positive charge is restored along one axis and pushed away along the other. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0,1.62)--(5.55,1.62) node[right] {$x$}; \draw[->,black] (2.78,0)--(2.78,3.24) node[above] {$y$}; \draw[black,thick] (0.45,0.52) .. controls (1.35,0.72) and (2.12,1.15) .. (2.68,1.53); \draw[black,thick] (2.88,1.71) .. controls (3.44,2.09) and (4.22,2.52) .. (5.12,2.72); \draw[black,thick] (0.45,2.72) .. controls (1.35,2.52) and (2.12,2.09) .. (2.68,1.71); \draw[black,thick] (2.88,1.53) .. controls (3.44,1.15) and (4.22,0.72) .. (5.12,0.52); \draw[->,acc,thick] (2.25,1.62)--(1.40,1.62); \draw[->,acc,thick] (3.31,1.62)--(4.16,1.62); \filldraw[draw=acc,fill=acc!10] (2.78,1.62) circle (2.2pt); \node[below right] at (2.90,1.50) {$E=0$}; \end{tikzpicture} $$ Saddle behavior is consistent with the charge-free potential equation and with the absence of stable free-space electrostatic traps. A zero contour can pass through a field null, but zero potential and zero field remain independent conditions. The contour map distinguishes them: a zero contour is one level set, while a field null requires all first derivatives to vanish simultaneously. **Common equipotential geometries.** Three source geometries provide direct comparisons. An isolated point charge has concentric spherical equipotentials and radial field. A long line has concentric cylindrical equipotentials and radial field in a transverse cross-section. A uniform field has parallel planar equipotentials. In each case, field lines cross the equipotential surfaces normally, but the field magnitude follows a different spacing rule because the potential has a different dependence on distance. A positive point charge has $$ V(r)=\frac{kQ}{r},\qquad |\vec E|=\frac{kQ}{r^2}. $$ Equal potential intervals occur at radii that grow farther apart. For a uniform field in the $x$ direction, $$ V(x)=V_0-Ex, $$ so equal voltage intervals have equal spatial separation. The diagrammatic rule that contour density represents field strength applies only when contour labels increase by equal voltage amounts. Unlabeled contour density may reflect an artist's choice of scale instead of a quantitative gradient. A dipole has more intricate topology. Its zero-potential surface passes through the midpoint perpendicular to the dipole axis, but the field on that surface is generally nonzero. Far from a neutral dipole, positive and negative potential regions approach one another while their magnitudes fall faster than the potential of a net charge. A contour map must therefore be read together with sign labels; the geometric location of a zero contour does not mark a force-free surface. $$ % caption: Dipole potential cross-section. The perpendicular midplane is a $V=0$ contour, yet field arrows cross it; positive and negative contour families meet at the zero level without the field being zero away from the center. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \filldraw[draw=black,fill=black!8] (1.50,1.55) circle (0.15); \node at (1.50,1.55) {$+$}; \filldraw[draw=black,fill=black!8] (4.55,1.55) circle (0.15); \draw[black!70,thick] (4.44,1.55)--(4.66,1.55); \draw[black,thick] (0.55,0.45) .. controls (1.18,0.82) and (2.18,1.03) .. (2.92,1.55); \draw[black,thick] (0.55,2.65) .. controls (1.18,2.28) and (2.18,2.07) .. (2.92,1.55); \draw[black,thick] (5.50,0.45) .. controls (4.87,0.82) and (3.87,1.03) .. (3.13,1.55); \draw[black,thick] (5.50,2.65) .. controls (4.87,2.28) and (3.87,2.07) .. (3.13,1.55); \draw[black,dashed] (3.02,0.24)--(3.02,2.86); \draw[->,acc,very thick] (2.20,1.55)--(3.80,1.55); \node[acc,above] at (3.90,1.62) {$E$}; \node[below] at (3.02,0.24) {$V=0$}; \end{tikzpicture} $$ ## Reconstructing potential from field data When field components are measured directly, potential differences follow by integrating the component parallel to a selected path. A rectangular grid admits a simple discrete construction. Start from a reference node with assigned potential, add $-E_x\Delta x$ along horizontal links and $-E_y\Delta y$ along vertical links, then compare estimates reaching the same node by different routes. Agreement within uncertainty tests the electrostatic assumption and the instrument calibration. Measured data usually contain noise, so path sums around distinct routes need not match exactly. A reconstruction reports the loop residuals and fits a scalar potential whose discrete gradient best matches the observed components. The fitted potential removes random inconsistencies only when the deviations are compatible with measurement noise. A consistent nonzero circulation signals time variation or a systematic error and should remain visible in the analysis. The field values require coordinate registration. A camera image of electrode shapes or a stage position readout provides the physical $x,y$ coordinates; pixel spacing alone is not a voltage-map scale. Probe orientation matters for vector field sensors, while potential probes record a scalar difference independent of their rotation only when their sensing electrode is small compared with the local contour curvature. Potential reconstruction cannot determine a unique global offset from field data. The reference node sets that offset. If a conducting boundary has known voltage, using it as the reference improves numerical conditioning because many paths begin from a physically fixed value. For a floating isolated conductor, the unknown constant potential becomes an additional boundary parameter tied to total charge. **Gradient magnitude, scale, and uncertainty.** The local field magnitude is the norm of the gradient, $$ |\vec E|=\sqrt{ \left(\frac{\partial V}{\partial x}\right)^2+ \left(\frac{\partial V}{\partial y}\right)^2+ \left(\frac{\partial V}{\partial z}\right)^2}. $$ An uncertainty in each derivative propagates nonlinearly into this magnitude. Near a field null, small noisy component estimates can dominate the reported direction, so a magnitude with an uncertainty interval is more defensible than a long arrow with arbitrary orientation. Away from a null, relative errors in the largest component often set the magnitude error, while smaller components can still matter for direction. Dimensional checks help distinguish a potential map from a field map. Potential differences divided by distance have units of volts per meter, equal to newtons per coulomb. A contour spacing in millimeters must be converted to meters before a field magnitude is reported. A factor-of-one-thousand error in map scale preserves the contour topology while destroying every quantitative field estimate. Spatial averaging also changes the reported magnitude. A probe with finite face area measures a weighted average of potential or field over that area. In a region where contours are nearly straight and widely spaced, the difference from the midpoint value can be negligible. Near a sharp tip or narrow gap, the same probe can average across a large gradient. The probe dimensions belong beside the contour spacing in any quantitative map. ## Worked analysis of a contour interval > **Worked example.** In the central region between two broad plates, the potential > map shows contours at $12$, $8$, $4$, and $0\ \mathrm V$, with adjacent contours > $1.00\ \mathrm{cm}$ apart along the normal running from the higher-potential plate > toward the lower. Estimate the field. > > A centered difference across one contour interval gives the normal component > directly: > > $$ > E_n\simeq-\frac{\Delta V}{\Delta n}=-\frac{0-4\ \mathrm V}{0.0100\ \mathrm m} > =+4.00\times10^2\ \mathrm{V/m}. > $$ > > The sign is measured against the chosen normal: a positive charge is pushed that > way, an electron the opposite way with the same magnitude $eE_n$. The map itself > carries no test-charge sign; force and potential energy appear only once a charge > is named. At the edge of the same plates, contour spacing and orientation change. Reusing the central $400\ \mathrm{V/m}$ estimate there ignores fringing. A local gradient may be estimated from two nearby contours if the normal distance is measured perpendicular to them, not along an arbitrary horizontal ruler. The estimate can require two components when the contours are visibly tilted relative to the measurement axes. $$ % caption: Quantitative contour interval between broad plates. Equal $4\ \mathrm V$ steps $1.00\ \mathrm{cm}$ apart give a central field of $400\ \mathrm{V/m}$; the curved outer contours mark where this one-dimensional estimate fails. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,very thick] (0.55,0.35)--(0.55,3.00); \draw[black,very thick] (4.95,0.35)--(4.95,3.00); \foreach \x/\v in {1.30/12,2.10/8,2.90/4,3.70/0} { \draw[black,dashed] (\x,0.55)--(\x,2.80); \node[above] at (\x,2.80) {$\v\ \mathrm V$}; } \draw[<->,black,thick] (2.10,0.18)--(2.90,0.18) node[midway,below] {$1$ cm}; \draw[->,acc,very thick] (2.30,1.55)--(3.20,1.55) node[right] {$E$}; \draw[black,dashed] (1.30,0.55) .. controls (0.98,0.85) and (0.98,2.50) .. (1.30,2.80); \draw[black,dashed] (3.70,0.55) .. controls (4.05,0.85) and (4.05,2.50) .. (3.70,2.80); \end{tikzpicture} $$ The uncertainty in this result includes voltage resolution and distance resolution. With contour increment uncertainty $\delta(\Delta V)$ and normal-distance uncertainty $\delta n$, the fractional field uncertainty has the approximate scale $$ \frac{\delta E}{|E|}\simeq \frac{\delta(\Delta V)}{|\Delta V|}+\frac{\delta n}{n} $$ when the errors are treated conservatively. Repeated scans can estimate random variation. Offset drift between potential readings affects differences only when it changes during the interval or differs between probes; a fixed common offset cancels from a local contour difference. **Boundary data and contour topology.** Electrostatic potential in a charge-free region is constrained by values on its boundaries. Conducting surfaces provide constant-potential boundaries. The contour geometry between two conductors must connect those values smoothly, meet each conductor tangentially, and avoid crossings. A contour that ends in empty space or changes label without crossing a charge layer signals a plotting or numerical error. The number of conductors matters. Two isolated conductors held at specified values give a potential difference, but the absolute offset is fixed only after a reference condition is stated. A floating conductor has a constant potential that is usually unknown until its total charge condition is applied. Treating a floating conductor as grounded changes both the contour map and the induced charge distribution. Contour topology also records source sign and symmetry. Around an isolated positive charge, potential levels nest with larger values inward. Around a neutral dipole, positive and negative level families join through a zero-level surface. A potential map with the wrong nesting order can indicate a reversed electrode polarity or a reference-sign error even before a numerical field derivative is computed. **Interfaces and derivative jumps.** Potential remains continuous across an ideal surface charge layer of finite charge density, while its normal derivative can jump. With a unit normal from region 1 to region 2 in vacuum, $$ \left.\frac{\partial V}{\partial n}\right|_1- \left.\frac{\partial V}{\partial n}\right|_2 =\frac{\sigma}{\varepsilon_0}. $$ The derivative difference represents the jump in normal electric field required by the surface charge. A contour map drawn at coarse scale may look smooth across the layer because $V$ itself is continuous. A derivative estimate taken from either side gives different normal slopes. This is why field arrows can change magnitude abruptly at a charged sheet while potential contours remain connected. At a dielectric interface, the field derivative also depends on permittivity and bound polarization charge. The normal displacement condition isolates free surface charge. A potential contour map alone gives gradients, but charge inference needs the material model on each side. A map interpreted with vacuum permittivity across a dielectric boundary gives the wrong free-charge density. $$ % caption: Normal potential slopes on the two sides of a charged sheet. Contours stay connected across it while unequal normal spacing shows the field jump from surface charge or a change of dielectric. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \fill[black] (2.72,0.25) rectangle (2.98,2.95); \draw[black,thick] (2.72,0.25)--(2.72,2.95) (2.98,0.25)--(2.98,2.95); \foreach \x in {0.70,1.40,2.10} {\draw[black,dashed] (\x,0.45)--(\x,2.75);} \foreach \x in {3.36,4.22,5.25} {\draw[black,dashed] (\x,0.45)--(\x,2.75);} \draw[->,acc,very thick] (1.95,1.90)--(2.58,1.90); \draw[->,acc,very thick] (3.12,1.90)--(4.00,1.90); \node[acc] at (2.41,2.35) {$E_1$}; \node[acc] at (3.79,2.35) {$E_2$}; \node[below] at (2.85,0.22) {sheet}; \end{tikzpicture} $$ Derivative jumps are local boundary statements. The potential elsewhere follows the entire source and conductor geometry. A measured abrupt contour-spacing change can also result from a poorly resolved thin electrode, a probe touching a conducting surface, or a discontinuity in calibration. Comparing observations from both sides and stating the material boundary distinguishes these possibilities. > **Worked example.** A charge-free laboratory region has potential (SI units) > > $$ > V(x,y)=120x-30x^2+40xy+10y^2. > $$ > > Find the field at $(x,y)=(1.0\ \mathrm m,\,0.50\ \mathrm m)$, and check it against a > short displacement. > > The field is minus the gradient: > > $$ > \begin{aligned} > E_x&=-\frac{\partial V}{\partial x}=-(120-60x+40y),\\ > E_y&=-\frac{\partial V}{\partial y}=-(40x+20y). > \end{aligned} > $$ > > At the stated point, > > $$ > E_x=-80\ \mathrm{V/m},\qquad E_y=-50\ \mathrm{V/m},\qquad > |\vec E|=\sqrt{80^2+50^2}=94\ \mathrm{V/m}, > $$ > > pointing toward decreasing $x$ and $y$. A positive charge feels a force along > $\vec E$, an electron opposite to it; the potential value at the point is not needed > once the derivatives are known. Now step $\d\vec\ell=(0.010\ \mathrm m)\hat\imath$ > from that point: > > $$ > \d V=\frac{\partial V}{\partial x}\,\d x=(80\ \mathrm{V/m})(0.010\ \mathrm m) > =0.80\ \mathrm V, > $$ > > so the field does $-0.80\ \mathrm J/C$ of work per unit positive charge over the > step, consistent with $E_x=-80\ \mathrm{V/m}$. A larger displacement would need the > potential at both endpoints, since the components vary with position here. The mixed term $40xy$ has a visible geometric effect. Changing $y$ changes $E_x$, and changing $x$ changes $E_y$, so the field direction rotates across the map. The contours are curved instead of parallel straight lines. A component calculation that differentiates only the square terms misses this coupling and gives an inconsistent tangent direction for the contour through the selected point. **A boundary-to-field workflow.** Electrostatic map construction benefits from a fixed sequence of statements and checks: 1. **State the region.** List source charges, conducting surfaces, dielectric boundaries, and the coordinate system. Mark which conductors are grounded, held at specified voltage, isolated, or floating. 2. **Set the reference.** Assign the zero of potential or record a specified potential difference. A reference offset changes displayed voltage values but leaves the gradient unchanged. 3. **Use symmetry only when justified.** Spherical, cylindrical, or planar expressions require the matching source and boundary symmetry. A finite edge, nearby conductor, or off-axis probe can invalidate a reduced one-coordinate formula. 4. **Determine potential or measured differences.** Sum source contributions, integrate a prescribed field, or solve from boundary data. Keep source charge signs in the potential terms and reserve test-charge sign for energy and force. 5. **Differentiate with stated coordinates.** Cartesian, cylindrical, and spherical gradients carry different scale factors. The derivative direction must be attached to the selected coordinate unit vectors. 6. **Check contours and boundaries.** Equipotentials meet conductors tangentially; field is normal to them. Check potential continuity and any known normal-field jump at charged interfaces. 7. **Check dimensions and limits.** Gradients must have volts per meter. Test a far-distance limit, a symmetry-axis limit, and a short-distance limit appropriate to the source model. 8. **Report resolution.** For measured maps, state grid spacing, probe size, reference lead, voltage accuracy, and whether field values are local derivatives or finite averages. Differentiate only a potential consistent with the sources, materials, and boundary conditions. Insert the test-charge sign after establishing the potential gradient. **Limits of a static potential map.** The relation $\vec E=-\nabla V$ with one globally defined scalar potential requires an electrostatic field in the region being analyzed. Source charges must be stationary on the measurement time scale, and changing magnetic flux must be negligible over the paths used for potential differences. A circuit carrying steady current can have approximately time-independent electric potential along resistive elements, while a rapidly changing current loop can generate an induced field whose closed-loop integral is nonzero. In a time-dependent case, the local electric field may be decomposed into a scalar- potential contribution and an induction contribution. An endpoint voltage reading then depends on lead routing and measurement arrangement as well as on the endpoint locations. Closing a measurement loop can enclose changing magnetic flux. The electrostatic contour interpretation must be restricted to the portion of the field that is conservative under the stated conditions. Material response also limits direct interpretation. In a linear isotropic dielectric, the potential gradient gives the electric field, while free surface charge is related to the normal displacement through the material constitutive relation. In anisotropic material, the displacement need not be parallel to the electric field. A contour map continues to show the scalar potential, while charge inference requires the material constitutive relation. At atomic distances, a continuum potential map averages microscopic charge and polarization. At very high fields, field emission, ionization, dielectric breakdown, or nonlinear material response can alter the source distribution. A contour map calibrated at low voltage can cease to represent the same boundary-value problem after breakdown begins. The map should therefore report voltage range, environment, and the evidence that the source configuration remained stationary during data collection. **Compact verification examples.** For $V(x)=V_0-Ex$, differentiating gives constant $E_x=E$. A graph with a changing slope cannot describe a uniform field, even if the plotted voltage points lie on a straight-looking line at low resolution. For $V(r)=kQ/r$, the gradient magnitude must scale as $1/r^2$. A derivative proportional to $1/r$ has confused cylindrical and spherical geometry. Two points anywhere in a connected conductor test the same condition. A measured nonzero potential difference in an alleged electrostatic equilibrium state indicates current, contact resistance in the measurement path, an unconnected piece of metal, or insufficient time for charge redistribution. For a field map around the same conductor, tangent field arrows at the surface indicate either a plotting error or a non-electrostatic condition. A closed grid loop is checked by summing signed voltage changes in one traversal direction. An electrostatic result is zero within uncertainty. A residual with random sign and size comparable with meter noise supports a measurement-limited interpretation. A residual that grows with time, tracks a changing current, or remains after reference calibration calls for an induction or systematic-error model. The distinction is observational and should be recorded with the potential map. **Units, signs, and reporting conventions.** Potential has units of joules per coulomb. A gradient has units of volts per meter, which equals newtons per coulomb. The equality follows from force per charge: $$ 1\ \mathrm{V/m}=1\ \mathrm{N/C}. $$ This unit relation is a check on every contour calculation. Dividing a voltage difference by a distance in centimeters without converting the distance produces a field value too small by a factor of one hundred. Dividing a potential by a distance and calling the result energy confuses voltage with the test-charge-dependent quantity $U=qV$. Coordinate signs must be stated before a component result is interpreted. In a one-dimensional map with positive $x$ to the right, a potential decreasing to the right has $\d V/\d x<0$ and therefore $E_x>0$. A positive charge accelerates rightward if no other forces act. An electron has $q<0$, so its force and acceleration are leftward. The potential map and field arrow remain unchanged when the test charge is switched; only force and potential energy reverse their sign relation. Potential difference notation carries endpoint order. The expression $V(B)-V(A)$ is the potential at $B$ relative to $A$. Reversing the labels changes the sign. The work done by an external agent in a quasistatic move is $q[V(B)-V(A)]$, while work done by the electric field has the opposite sign. A solution should label the moving charge and both endpoints before assigning positive or negative work. Contour labels should report the reference and the contour increment. A map marked ``0, 4, 8, 12 V'' communicates equal $4\ \mathrm V$ increments only if the labels refer to one common reference and one source state. A screenshot without coordinate scale, contour interval, or reference may show qualitative topology but cannot support a numerical field claim. The same requirement applies to simulation output: grid spacing, boundary values, material properties, and normalization must accompany any displayed potential field. A field vector reported from a potential fit needs components or magnitude and direction, the coordinate basis, the point or region where it applies, and the finite-difference or fitting scale used to obtain it. For a finite electrode, report field magnitude together with its location and direction. Potential, field, force, and energy have different units, sign conventions, and charge dependence. Keep their labels separate so that a scalar voltage is not used as a force vector or an energy without its charge factor. Before accepting a potential-gradient result, compare a derivative with an independent voltage difference over a small stated displacement. Then compare the field direction with the contour normal and the predicted force direction for the stated test charge. Agreement among these three checks ties together the scalar map, its local gradient, and the physical force interpretation. A disagreement usually identifies a reversed coordinate direction, an endpoint-order sign error, or a contour scale that has been read as a field scale. A gridded map should use symmetric point pairs about the reported location when the geometry permits it. Repeat the derivative with a smaller grid interval or a locally fitted potential surface. Stable field components across those choices support the stated spatial resolution; strong variation requires reporting an averaged field over the probe area rather than a point value. Retain the raw voltage readings with their timestamps and reference lead placement. That record permits a later check for source drift, probe loading, and inconsistent endpoint order. State the interpolation method whenever derivative values are reported from a sparse grid.