---
title: Electric Field Maps
module: Electric Fields
moduleNumber: 1
lessonNumber: 4
order: 104
summary: >
  A field is a vector at every point of space, and the quickest way to grasp one is
  to draw it. We build the two standard pictures — continuous field lines tangent to
  $\vec E$, and scaled vector arrows — and read direction, magnitude, and the location
  of nulls straight off them. We fix what a line drawing can and cannot say: density
  encodes magnitude only under a stated seeding rule, and integral curves never cross
  at a regular point. From there we work the topology near sources, sinks, and
  conductor surfaces, and state the step-size and interpolation checks a numerical
  map must pass.
topics: [Electric Fields]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 21 — The Electric Field I; §§21-4–21-5"
---

## Tangent curves of a vector field

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

$$
\frac{\d \vec r}{\d s}=\lambda(s)\,\vec E\!\left(\vec r(s)\right),
\qquad
\lambda(s)>0.
$$

The positive function $\lambda$ 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 | $\vec E$ at sampled positions | local direction; magnitude when an arrow scale is stated | behavior between unresolved samples |
| Field line | curve tangent to $\vec E$ | 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 $\vec E$ | 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.

$$
% caption: Vector samples, streamlines, and equipotential contours of one two-dimensional electric field.
% Vector samples retain local direction and a stated arrow scale; a streamline follows
% the tangent directions; equal-potential contours are normal to the field recovered
% from a measured potential surface.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% panel A: sampled vectors
\draw[black] (0.35,0.55) rectangle (3.05,2.55);
\foreach \x in {0.85,1.45,2.05,2.65} {\foreach \y in {0.95,1.55,2.15} {
  \filldraw[draw=black,fill=black!8] (\x,\y) circle (1pt);
  \draw[->,acc,thick] (\x,\y)--++(0.34,0.16);}}
\node[below] at (1.70,0.55) {vector samples};
% panel B: one streamline over a faint grid
\draw[black] (3.55,0.55) rectangle (6.25,2.55);
\draw[black,dashed,xstep=0.55,ystep=0.5] (3.85,0.85) grid (5.95,2.35);
\draw[acc,very thick] plot[smooth] coordinates {(3.75,0.80)(4.25,1.05)(4.75,1.55)(5.30,2.05)(5.95,2.30)};
\foreach \x/\y/\a in {4.08/0.94/28,4.60/1.42/44,5.16/1.96/32} {\draw[->,black,thick] (\x,\y)--++({0.30*cos(\a)},{0.30*sin(\a)});}
\node[below] at (4.90,0.55) {streamline};
% panel C: equal-potential contours crossed by the field
\draw[black] (6.75,0.55) rectangle (9.45,2.55);
\foreach \x in {7.20,7.75,8.30,8.85} {\draw[black] (\x,0.80).. controls ({\x+0.22},1.45) and ({\x+0.22},1.70)..(\x,2.30);}
\foreach \y in {1.05,1.55,2.05} {\draw[->,acc,thick] (6.95,\y)--(9.20,\y);}
\node[below] at (8.10,0.55) {equal V};
\end{tikzpicture}
$$

At any point where $\vec E\ne\vec 0$, 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.

$$
% caption: Tangent construction for an electric field line. Each blue arrow gives
% the local direction of the prescribed electric field, and the gray curve has a
% tangent parallel to those arrows at its sampled points; a crossing would require
% two directions at one location.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0.5,0.6) .. controls (1.7,1.2) and (2.6,1.0) .. (3.5,1.85)
  .. controls (4.1,2.4) and (4.7,2.25) .. (5.4,3.0);
\foreach \x/\y/\a in {1.0/0.90/26,1.9/1.10/6,2.75/1.20/22,3.5/1.85/50,4.25/2.32/30,5.0/2.62/44} {
  \draw[->,acc,thick] (\x,\y)--++({0.5*cos(\a)},{0.5*sin(\a)});}
\filldraw[draw=black,fill=black!8] (2.75,1.20) circle (0.05);
\node[below] at (2.95,1.12) {sample site};
\node at (4.25,3.05) {tangent direction};
\end{tikzpicture}
$$

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 $\vec F=q\vec E$
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

$$
\vec E(\vec r)
=\frac{1}{4\pi\varepsilon_0}
\frac{q}{r^2}\,\hat r.
$$

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.

$$
% caption: Radial electric-field map of a positive point source. Arrows point
% outward at every sample position, while the shorter outer arrows decrease with
% inverse distance squared; the source marker itself is excluded from the
% point-charge approximation.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\filldraw[draw=black,fill=black!8] (0,0) circle (0.26);
\node[black!75] at (0,0) {+};
\foreach \a in {0,45,...,315} {
  \draw[->,acc,thick] ({0.36*cos(\a)},{0.36*sin(\a)}) -- ({1.05*cos(\a)},{1.05*sin(\a)});
  \draw[->,black] ({1.45*cos(\a)},{1.45*sin(\a)}) -- ({1.85*cos(\a)},{1.85*sin(\a)});
}
\node[acc,above] at (0.70,0.02) {near};
\node[black,above] at (1.65,0.02) {far};
\end{tikzpicture}
$$

**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

$$
\frac{\d N}{\d A}=C\,|\vec E|,
$$

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 $2q$ 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 $dA=r^2\,d\Omega$. 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 $\vec E$, 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,

$$
\oint\vec E\cdot\d\vec\ell=0,
$$

so a line that returned to its starting point with a fixed positive tangent would
imply a nonzero component of $\vec E$ 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:

$$
\vec E_{\rm left}+\vec E_{\rm right}=\vec 0.
$$

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.

> **Worked example (field on the bisector of two equal charges).** Two charges
> $q=+5.0\ \mathrm{nC}$ sit at $(\pm0.20,0)\ \mathrm m$. Find $\vec E$ at $P=(0,0.30)\ \mathrm m$
> on the perpendicular bisector. Each source is a distance
> $r=\sqrt{0.20^2+0.30^2}\ \mathrm m=0.361\ \mathrm m$ from $P$, giving field magnitude
>
> $$
> E_1=E_2=k\frac{q}{r^2}=(8.988\times10^9)\frac{5.0\times10^{-9}}{0.13}=346\ \mathrm{N/C}.
> $$
>
> The two $x$-components cancel by symmetry; the $y$-components add. Each makes angle
> with the vertical whose cosine is $0.30/0.361=0.832$, so
>
> $$
> E_y=2E_1\frac{0.30}{r}=2(346)(0.832)=5.8\times10^2\ \mathrm{N/C},
> $$
>
> directed straight up ($+y$). On the bisector the field is purely vertical — the
> horizontal cancellation the midpoint null makes exact holds by symmetry everywhere on
> that line.

$$
% caption: Superposed field near two equal positive charges. The midpoint has zero
% electric field because equal horizontal components oppose one another, whereas
% points on the vertical symmetry line retain upward or downward components from
% both sources.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\filldraw[draw=black,fill=black!8] (0,0) circle (0.22);
\filldraw[draw=black,fill=black!8] (4,0) circle (0.22);
\node[black!75] at (0,0) {+};
\node[black!75] at (4,0) {+};
\filldraw[draw=black,fill=black!10] (2,0) circle (0.05);
\draw[->,acc,thick] (1.72,0)--(1.28,0);
\draw[->,acc,thick] (2.28,0)--(2.72,0);
\draw[->,acc,thick] (2,0.60)--(2,1.30);
\draw[->,acc,thick] (2,-0.60)--(2,-1.30);
\node[below] at (1.55,-0.08) {zero site};
\node[above] at (2,1.30) {sum upward};
\node[below] at (2,-1.30) {sum downward};
\end{tikzpicture}
$$

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,

$$
\vec E_{\rm total}(\vec r)
=\sum_i\vec E_i(\vec r).
$$

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.

$$
% caption: Superposition map for an equal positive-negative source pair. Vectors
% in the region between sources point from plus to minus, while off-axis vectors
% tilt because both source contributions have horizontal and vertical components.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\filldraw[draw=black,fill=black!8] (0.8,1.7) circle (0.22);
\filldraw[draw=black,fill=black!10] (5.2,1.7) circle (0.22);
\node[black!75] at (0.8,1.7) {+};
\node[black!75] at (5.2,1.7) {-};
\foreach \x/\y/\a in {1.7/2.5/-22,2.6/2.55/-8,3.5/2.45/12, 1.7/1.7/0,2.6/1.7/0,3.5/1.7/0, 1.7/0.9/22,2.6/0.85/8,3.5/0.95/-12} {
  \draw[->,acc,thick] (\x,\y)--++({0.5*cos(\a)},{0.5*sin(\a)});}
\draw[black,dashed] (3.0,0.35)--(3.0,3.05);
\node[black,above] at (3.0,3.05) {symmetry axis};
\end{tikzpicture}
$$

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

$$
\vec E_{\rm net}=\sum_i\vec E_i,
\qquad
E_x=\sum_i E_{i,x},
\qquad
E_y=\sum_i E_{i,y}.
$$

- **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

$$
\vec E=E_0\,\hat n,
$$

where both $E_0$ and the unit direction $\hat n$ 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.

$$
% caption: Parallel-plate field map. Central arrows are parallel and evenly
% separated, marking the uniform-region approximation; edge arrows bend outward
% through the fringing region, where field direction and magnitude vary near the
% finite plate ends.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,very thick] (0.6,2.7)--(5.4,2.7);
\draw[black,very thick] (0.6,0.6)--(5.4,0.6);
\node[black,left] at (0.6,2.7) {+};
\node[black,left] at (0.6,0.6) {-};
\foreach \x in {1.3,2.1,2.9,3.7,4.5} {\draw[->,acc,thick] (\x,2.45)--(\x,0.85);}
\draw[->,black] (0.85,2.4).. controls (0.20,1.95) and (0.25,1.30)..(0.85,0.90);
\draw[->,black] (5.15,2.4).. controls (5.80,1.95) and (5.75,1.30)..(5.15,0.90);
\node[above] at (3.0,2.7) {upper plate};
\node[below] at (3.0,0.6) {lower plate};
\node[black] at (5.35,0.25) {fringing};
\end{tikzpicture}
$$

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.

$$
% caption: Field directions beside a conducting surface in electrostatic
% equilibrium. Arrows meet the surface normally because a tangential electric-field
% component would drive mobile surface charge; closer normal arrows indicate a
% larger local field magnitude in the mapped exterior region.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\fill[black] (0.7,0.40) rectangle (5.5,0.90);
\draw[black,thick] (0.7,0.90)--(5.5,0.90);
\foreach \x/\l in {1.2/0.70,1.85/0.95,2.5/1.15,3.15/1.30,3.8/1.15,4.45/0.95,5.1/0.70} {
  \draw[->,acc,thick] (\x,1.00)--(\x,{1.00+\l});}
\node[below] at (3.1,0.40) {conductor};
\node[above] at (3.1,2.45) {normal arrows};
\end{tikzpicture}
$$

## 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.

$$
% caption: Qualitative induced-charge map for a positive charge inside a cavity of an initially neutral conductor. Cavity lines end on negative induced inner surface charge; positive induced outer-surface charge produces exterior field lines, while the metal itself contains no electrostatic field lines.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[black] (0.4,0.35) rectangle (5.8,3.15);
  \fill[white] (2.5,1.75) circle (0.85);
  \draw[black,thick] (0.4,0.35) rectangle (5.8,3.15);
  \draw[black,thick] (2.5,1.75) circle (0.85);
  \filldraw[draw=black,fill=black!8] (2.5,1.75) circle (0.16);
  \node[black!75] at (2.5,1.75) {+};
  \foreach \a in {0,60,...,300} {
    \draw[->,acc,thick] ({2.5+0.30*cos(\a)},{1.75+0.30*sin(\a)})--({2.5+0.74*cos(\a)},{1.75+0.74*sin(\a)});}
  \foreach \a in {30,90,...,330} {\node[black] at ({2.5+0.80*cos(\a)},{1.75+0.80*sin(\a)}) {-};}
  \foreach \x/\y in {1.2/2.90,4.6/2.90,1.2/0.55,4.6/0.55} {\node[black] at (\x,\y) {+};}
  \draw[->,acc,thick] (5.8,2.40)--(6.35,2.40);
  \draw[->,acc,thick] (5.8,1.75)--(6.35,1.75);
  \draw[->,acc,thick] (5.8,1.10)--(6.35,1.10);
  \node[black] at (4.30,1.35) {metal};
  \node[black,anchor=south] at (5.75,2.72) {exterior};
\end{tikzpicture}
$$

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 $\vec E=\vec 0$,
so its enclosed charge must vanish:

$$
\oint_S\vec E\cdot\d\vec A=0
\quad\Longrightarrow\quad
Q_{\rm inner}=-Q_{\rm cavity}.
$$

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 $Q_{\rm inner}$ |
| 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

$$
\vec E=-\nabla V.
$$

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.

$$
% caption: Field lines and equipotential curves. Electric-field directions cross equal-potential curves normally, while closer equipotential spacing corresponds to a larger local field magnitude for the same potential interval.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\foreach \x in {0.7,1.7,2.7,3.5,4.0,4.5,5.0} {\draw[black] (\x,0.50)--(\x,2.70);}
\draw[->,acc,very thick] (0.35,1.00)--(5.55,1.00);
\draw[->,acc,very thick] (0.35,2.10)--(5.55,2.10);
\node[black,above] at (0.70,2.72) {equal V};
\node[below] at (1.70,0.50) {wide spacing};
\node[below] at (4.25,0.50) {close spacing};
\node[acc,right] at (5.60,1.55) {E lines};
\end{tikzpicture}
$$

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:

$$
E_x(x_i,y_j)\approx-\frac{V_{i+1,j}-V_{i-1,j}}{2\Delta x},
\qquad
E_y(x_i,y_j)\approx-\frac{V_{i,j+1}-V_{i,j-1}}{2\Delta y}.
$$

For independent equal voltage uncertainties $\sigma_V$, the first estimate has
the approximate random uncertainty

$$
\sigma_{E_x}\approx\frac{\sigma_V}{\sqrt2\,\Delta x}.
$$

Halving $\Delta x$ 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 $\Delta x,\Delta y$ and the model domain | compare a reference separation with the survey scale |
| repeated voltages | estimates $\sigma_V$ 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 $\vec E$ | 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 $\vec r_0$, the streamline tangent is the local field
direction. A simple forward step is

$$
\vec r_{j+1}=\vec r_j+h\frac{\vec E(\vec r_j)}{|\vec E(\vec r_j)|},
$$

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.

> **Worked example (net field by superposition).** Place $q_1=+2.0\ \mathrm{nC}$ at
> $(0,0)$ and $q_2=-1.0\ \mathrm{nC}$ at $(0.30\ \mathrm m,0)$; find $\vec E$ at
> $P=(0.15,0.20)\ \mathrm m$. Each source lies $0.25\ \mathrm m$ from $P$, but along
> different directions: $\vec r_1=(0.15,0.20)\ \mathrm m$ and
> $\vec r_2=(-0.15,0.20)\ \mathrm m$. Using $\vec E_i=k\,q_i\vec r_i/|\vec r_i|^3$,
>
> $$
> \vec E_1=(173\,\hat x+230\,\hat y),\qquad
> \vec E_2=(86.3\,\hat x-115\,\hat y)\quad(\mathrm{N/C}).
> $$
>
> Adding component by component,
>
> $$
> \vec E_{\rm net}=(259\,\hat x+115\,\hat y)\ \mathrm{N/C},
> \qquad
> \lvert\vec E_{\rm net}\rvert=283\ \mathrm{N/C},
> \qquad
> \theta=\atanTwo(115,259)=24^\circ.
> $$
>
> The resultant sets the plotted arrow and the tangent of a nearby streamline. Adding
> the two magnitudes would discard the directions and give a wrong result.

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

$$
\vec r_i=\vec E_{\rm map,i}-\vec E_{\rm ref,i},
\qquad
z_i=\frac{|\vec r_i|}{\sigma_{E,i}},
$$

where $\vec E_{\rm ref,i}$ is a direct Coulomb or potential-gradient estimate
and $\sigma_{E,i}$ 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 | $\vec E_{\rm map,i}$ and $\vec E_{\rm ref,i}$ | 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 $\vec E/|\vec E|$ 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.

> **Worked example (null point between two positive charges).** Charges
> $q_1=+4.0\ \mathrm{nC}$ at $x=0$ and $q_2=+1.0\ \mathrm{nC}$ at $x=0.30\ \mathrm m$
> both push a positive test charge outward, so their fields can cancel only between
> them. At a point $x$ on the segment, $\vec E_1$ points $+x$ and $\vec E_2$ points
> $-x$; setting magnitudes equal,
>
> $$
> \frac{kq_1}{x^2}=\frac{kq_2}{(0.30-x)^2}
> \quad\Rightarrow\quad
> \frac{x}{0.30-x}=\sqrt{\frac{q_1}{q_2}}=\sqrt{4}=2.
> $$
>
> Solving $x=2(0.30-x)$ gives $x=0.20\ \mathrm m$. The null lies between the charges,
> closer to the weaker one, as it must. A grid map should show arrows shrinking toward
> this point, but a single short arrow is not proof — the location must survive grid
> refinement and small perturbations of the charge values.

Map resolution should be reported as an error bound rather than as a visual density
choice. A grid with spacing $\Delta x$ 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.