Electric Field and Force

An electric field is defined operationally through the force on a sufficiently small positive test charge. At position $r$ ,

E (r) = q_{0} \to 0 lim \frac{F ( r )}{q _{0}} .

The limiting condition prevents the probe from becoming part of the source configuration. A large test charge can alter the distribution on nearby conductors or polarize nearby matter, producing a force that no longer measures the original source configuration. A source configuration determines a field vector at each position independently of a later particle placed there.

A particle with charge $q$ responds according to

F = q E, m a = q E .

Positive charge accelerates with the field. A negative charge experiences force and acceleration opposite to the field, even though the field itself has not reversed. The field has SI unit $N/C$ , equivalent to $V/m$ . Its direction is defined by a positive test charge, not by electron motion in a conductor.

At $P$ , the source configuration sets $E$ , and a probe experiences $F = q E$ .

A positive source sets up a radial field. At

P

the field

E

points away from the source; a test charge placed there feels

F = q E

, whose size scales with

q

but leaves the source field unchanged.

Probe limit and source-response separation

The definition uses a limiting probe because every charged probe exerts a reciprocal force on the sources. A finite probe charge can move a light source particle, redistribute charge on a conductor, or polarize an insulating body. The ratio $F / q_{0}$ then describes the combined source-plus-probe configuration rather than the field prepared before the probe was introduced. The field at a point is defined by the limiting configuration:

E (r) = q_{0} \to 0 lim \frac{F _{sources \to probe} ( r )}{q _{0}} .

The limit specifies the regime in which reducing the probe charge leaves the inferred ratio unchanged within measurement uncertainty. A probe of charge $q_{0}$ measures a force $F = q_{0} E$ ; halving $q_{0}$ should halve the force while preserving $F / q_{0}$ if source perturbation is negligible.

The field and the probe response are different physical quantities:

Field $E$ : determined by the source configuration and the field point; its unit is $N/C$ .
Force $F$ : determined by the field and the particular probe charge; its unit is $N$ .
Acceleration $a$ : determined by force and probe mass; its unit is $m s^{- 2}$ .

Changing the sign of the probe reverses the force and acceleration while leaving the field unchanged. Changing the mass alters acceleration only. These distinctions prevent a common error in which a field arrow is reversed when an electron is placed at the point.

The probe's sign flips its response, not the field. Along the same field

E

, a positive charge is pushed with the field while an electron is pushed against it; the source-fixed field arrow is identical in both panels.

Field of one point charge

Let a source charge $q_{i}$ occupy position $r_{i}$ and let $P$ be a field point at $r$ . Coulomb's law gives the force on a positive test charge $q_{0}$ :

F_{i \to 0} = k q_{i} q_{0} \frac{r - r _{i}}{∣ r - r _{i} ∣ ^{3}} .

Dividing by $q_{0}$ and taking the probe limit yields the point-charge field:

E_{i} (r) = k q_{i} \frac{r - r _{i}}{∣ r - r _{i} ∣ ^{3}} = k \frac{q _{i}}{r _{i P}^{2}} \overset{r}{^}_{i P} .

The unit vector points from the source location toward the field point. A positive source gives a field in that direction; a negative source reverses it. The field magnitude is

E_{i} = k \frac{∣ q _{i} ∣}{r _{i P}^{2}} .

The denominator has third power in the first vector expression because the displacement vector contains one factor of distance. Its magnitude therefore reduces to the inverse-square scalar form. The same algebra appeared in the vector form of Coulomb's law, with the test-charge factor removed.

At a fixed distance, doubling the source charge doubles the field magnitude. At a fixed source charge, doubling distance reduces the field magnitude to one quarter:

\frac{E ( 2 r )}{E ( r )} = \frac{1}{4} .

The field diverges at the mathematical location of a point source. A real charged object has finite size and charge distribution, so the point-source expression is used only outside the scale where its geometry is unresolved.

Field measurement and units.

The operational definition gives a direct measurement procedure. Place a small positive probe at the selected point, measure the force vector, and divide by its charge. Repeating the measurement with a different small positive probe checks whether the ratio is independent of the probe. The field unit follows directly:

[E] = \frac{N}{C} .

The equivalent unit $V/m$ follows from the later definition of electric potential. The two unit forms represent the same dimensions, but $N/C$ keeps the force-per-charge meaning visible at this stage.

A negative test charge of the same magnitude at that point would feel $- 2.0 \times 1 0^{- 4} N \overset{}{^}$ , yet the inferred field is unchanged, $4.0 \times 1 0^{4} N/C \overset{}{^}$ .

Force and acceleration in a prescribed field

Once a field has been specified, the motion of a particle follows from

m a = q E .

In a uniform field $E = E \overset{x}{^}$ , acceleration is constant:

a_{x} = \frac{q E}{m}, x (t) = x_{0} + v_{0 x} t + \frac{1}{2} \frac{q E}{m} t^{2} .

The same kinematic equations used for uniform gravitational acceleration apply after replacing $g$ with $q E / m$ and retaining the charge sign. A positive particle in a positive $x$ field accelerates toward positive $x$ . An electron has $q = - e$ and accelerates toward negative $x$ . Its acceleration magnitude can be large because the electron mass is small.

The field is a property of the source arrangement, whereas particle trajectories also depend on charge, mass, initial velocity, and boundaries. Two particles at the same point can experience forces of opposite direction and accelerations of different magnitude without implying different fields.

The point-charge field and the force-response equation describe electrostatic configurations. A rapidly changing source arrangement requires the time-dependent electromagnetic field description. For stationary or slowly varying source charges, the electrostatic field gives the local force relation used throughout the following lessons.

Components of a point-charge field.

Coordinate components follow from the source-to-field-point displacement before any magnitude is rounded. For a source at the origin and a field point $P = (x, y)$ ,

r = x \overset{}{^} + y \overset{}{^}, r = x^{2} + y^{2} .

Substitution into the point-charge field gives

E = k q \frac{x  ^ + y  ^}{( x ^{2} + y ^{2} ) ^{3/2}},

with components

E_{x} = k q \frac{x}{( x ^{2} + y ^{2} ) ^{3/2}}, E_{y} = k q \frac{y}{( x ^{2} + y ^{2} ) ^{3/2}} .

The sign of $q$ remains in both components. A positive source produces component signs that match the coordinate signs of the field point. A negative source reverses both. This form retains direction algebraically throughout the calculation.

Components of a point-charge field. The field point

P

lies

0.30 m

x

and

0.40 m

y

from the source, so

E

has positive

E_{x}

and

E_{y}

and a

0.50 m

separation.

With an arbitrary source location $r_{s}$ , use

R = r_{P} - r_{s}, E (P) = k q \frac{R}{∣ R ∣ ^{3}} .

The subtraction order is source to field point. Reversing it gives the opposite vector and would draw a positive-source field toward the source.

Uniform fields and trajectory response

A uniform field has the same magnitude and direction throughout the region of interest. Parallel plates with a large central region provide an approximate example; edge regions have fringing fields and require a more detailed model. In a uniform field $E = E \overset{y}{^}$ , a particle has constant acceleration

a_{y} = \frac{q E}{m} .

If the particle enters with horizontal speed $v_{x}$ and zero initial vertical speed, then

x (t) = v_{x} t, y (t) = y_{0} + \frac{1}{2} \frac{q E}{m} t^{2} .

Eliminating time yields a parabolic trajectory:

y (x) = y_{0} + \frac{q E}{2 m v _{x}^{2}} x^{2} .

The sign of $q E$ determines the direction of curvature. A positive particle curves with the field. An electron curves against it. The field remains the same in both cases; the charge changes the force relation.

A charged particle crossing a uniform field. Between the plates the field is uniform (parallel arrows); a positive charge entering horizontally curves with the field, an electron against it, at the same entry speed.

A particle deflected by plates of length $L$ has transit time $t = L / v_{x}$ . The vertical exit speed is

v_{y} = \frac{q E}{m} \frac{L}{v _{x}},

and the deflection at the exit is

y_{exit} = \frac{q E L ^{2}}{2 m v _{x}^{2}} .

The dependence on $v_{x}^{- 2}$ gives a strong speed-selection effect. Doubling the entry speed reduces the deflection to one quarter. A measured deflection therefore cannot be interpreted as a field value unless the charge, mass, entry speed, and field-region length are also known.

Field magnitude scales and source changes.

Point-charge fields span a wide range. A field of order $1 0^{2} N/C$ occurs in ordinary atmospheric conditions, while fields in an atomic environment can be many orders of magnitude larger. The magnitude alone does not determine the force on a particle: a $1 nC$ probe in a field of $1 0^{5} N/C$ experiences $1 0^{- 4} N$ , whereas an electron in the same field experiences a force magnitude $e E$ .

The inverse-square dependence gives a local scale estimate. A source charge of $1.0 nC$ at $0.10 m$ produces

E \approx (9.0 \times 1 0^{9}) \frac{1.0 \times 1 0 ^{- 9}}{( 0.10 ) ^{2}} = 9.0 \times 1 0^{2} N/C .

Moving to $0.20 m$ reduces the field to $2.25 \times 1 0^{2} N/C$ . Such estimates identify whether a stated instrument sensitivity or particle deflection is plausible before detailed numerical work begins.

The electrostatic field model assumes a source arrangement that is fixed over the observation interval. If a source charge is moved, the field at distant points does not change instantaneously. Electromagnetic changes propagate at finite speed. The static point-charge formula remains the appropriate local result once the new source configuration has settled and the observation time is long compared with the propagation time across the apparatus.

Direction, sign, and local vector measurements

An electric field is a vector at each observation point, so a magnitude without a direction is incomplete information. For one positive point source, the field points away from the source at every surrounding point. For one negative point source, the field points toward the source. The reversal belongs to the source charge in

E = k q_{s} \frac{R}{R ^{3}}, R = r_{P} - r_{s},

not to a later test charge. The sign of a test charge enters only after the field is known, through $F = q_{0} E$ . Keeping those two signs in separate steps prevents source--probe sign reversals in diagrams that contain both a source and a particle whose force is being found.

For example, place a negative source at the origin and examine a point on the positive $x$ axis. The displacement vector $R$ points in the positive $x$ direction, whereas the factor $q_{s} < 0$ reverses the field. Thus $E_{x} < 0$ at that point. A positive test charge placed there feels a negative- $x$ force, directed toward the source. An electron placed at the same point feels a positive- $x$ force, directed away from the source. The source field has not changed between the two experiments.

Source sign and probe sign are separate choices. To the right of a negative source the field points back toward the source; a positive probe is pushed that way, a negative probe the opposite way — the field itself does not change.

The local vector can be measured by resolving force into coordinate components. Suppose a small positive probe of charge $q_{0}$ experiences

F = (F_{x} \overset{}{^} + F_{y} \overset{}{^} + F_{z} \hat{k}) .

Then

E_{x} = \frac{F _{x}}{q _{0}}, E_{y} = \frac{F _{y}}{q _{0}}, E_{z} = \frac{F _{z}}{q _{0}} .

Component signs are obtained from the force components on the positive probe. A negative $F_{y}$ produces a negative $E_{y}$ ; it does not mean that the magnitude of the field is negative. The magnitude remains

E = E_{x}^{2} + E_{y}^{2} + E_{z}^{2} .

At a point where $E = (300 \overset{}{^} - 400 \overset{}{^}) N/C$ , the magnitude is $500 N/C$ . A $+ 2.0 nC$ probe has force

F = (6.0 \overset{}{^} - 8.0 \overset{}{^}) \times 1 0^{- 7} N,

and an equal-magnitude negative probe has the opposite force vector. The calculation also checks units: multiplying charge in coulombs by field in newtons per coulomb leaves newtons.

Component form retains signs when a field point lies left of, below, or behind the source. The signs follow directly from the displacement vector and the source charge, whereas a scalar inverse-square calculation returns only a magnitude. In three dimensions, retain all three components until the final magnitude or direction is needed; projecting a result onto a drawing plane can hide a nonzero third component.

Distance scaling and the point-source approximation.

Holding source charge fixed eliminates Coulomb's constant from a field-magnitude comparison. If the field is measured at distances $r_{1}$ and $r_{2}$ ,

\frac{E _{2}}{E _{1}} = (\frac{r _{1}}{r _{2}})^{2} .

Moving from $r$ to $3 r$ reduces the magnitude to $E /9$ ; moving from $r$ to $r /2$ increases it to $4 E$ . Direction at corresponding radial locations remains unchanged for a single source. The comparison is valid only when the same source can be treated as a point at both observation distances.

Consider a $+ 4.0 nC$ charge. At $0.20 m$ the point-charge model gives

E_{1} = (8.988 \times 1 0^{9}) \frac{4.0 \times 1 0 ^{- 9}}{( 0.20 ) ^{2}} = 8.99 \times 1 0^{2} N/C .

At $0.60 m$ , the ratio calculation gives $E_{2} = E_{1} /9$ , or $9.99 \times 1 0^{1} N/C$ . Substitution into the inverse-square expression gives the same value. A ratio calculation exposes an incorrect distance exponent because that error changes the result by a large factor.

The point-source model requires the source size to be small compared with the source-to-observer distance and with the spatial resolution of the calculation. A charged metal sphere with radius $a$ has the exact external field of a point charge at its center when the charge distribution is spherically symmetric and the field point lies outside the sphere. A charged rod, disk, or irregular object generally needs a distributed-charge calculation when the observer is near enough to distinguish different parts of the source. Those calculations are developed in the next module.

Far from any bounded charge distribution, the total charge provides the leading field term. Suppose several charges occupy a region of size $a$ and the field point is at distance $R$ with $R ≫ a$ . Replacing the collection by one point charge equal to its total charge gives the dominant radial field. The approximation loses the internal geometry: two equal and opposite charges have total charge zero, so their far field is much smaller than either individual field but is not identically zero. That residual structure belongs to the dipole treatment rather than the one-point source formula.

Near and far views of a finite charged object. From a far point the rays from its parts are nearly parallel and the total charge suffices; from a near point they arrive from visibly different directions and the distribution matters.

Use the point-source expression only for an exterior point sufficiently far from a compact, effectively symmetric source. Other geometries require continuous-charge or symmetry methods.

Finite uniform-field regions.

The uniform-field model applies to a specified region of an apparatus. Between two large oppositely charged parallel conducting plates, the central region is well approximated by parallel field vectors normal to the surfaces. Near an edge, field lines spread outward and the field develops a horizontal component. A charged particle can be treated with constant acceleration only while it remains in the central region and while its displacement is small enough that the field variation is negligible.

Let the particle enter the central region at $x = 0$ , $y = y_{0}$ with velocity components $v_{0 x}$ and $v_{0 y}$ . With a field $E = E \overset{y}{^}$ ,

x (t) = v_{0 x} t, y (t) = y_{0} + v_{0 y} t + \frac{q E}{2 m} t^{2},

and

v_{x} (t) = v_{0 x}, v_{y} (t) = v_{0 y} + \frac{q E}{m} t .

The horizontal velocity is constant because the field has no $x$ component in this model. A nonzero entry value $v_{0 y}$ changes both the exit height and the exit angle; the familiar simple parabola with horizontal entry is the special case $v_{0 y} = 0$ . With plate length $L$ and $v_{0 x} > 0$ , the particle spends

t_{L} = \frac{L}{v _{0 x}}

inside the modeled field. Its exit height and vertical velocity are therefore

y_{L} = y_{0} + \frac{v _{0 y} L}{v _{0 x}} + \frac{q E L ^{2}}{2 m v _{0 x}^{2}}, v_{y L} = v_{0 y} + \frac{q E L}{m v _{0 x}} .

These equations give two separate observables. The position $y_{L}$ depends on the full transit history, whereas $v_{y L}$ records the accumulated impulse. A detector placed beyond the plates receives a particle moving in a straight line at the exit velocity if no other force acts in that later region.

The geometry imposes a further condition. If the plate separation is $d$ and the midplane is $y = 0$ , the particle reaches a plate before the exit whenever

∣ y (t) ∣ \geq \frac{d}{2}

for some $0 < t < t_{L}$ . The formula for $y_{L}$ alone is insufficient when the trajectory first moves toward a plate and then turns around, because the maximum displacement can occur before the exit. Solve $v_{y} (t) = 0$ for a turning time when $q E$ and $v_{0 y}$ have opposite signs, then evaluate $y (t)$ there. The same check appears in a beam deflection tube, an electrostatic steering element, or a charged-particle analyzer.

For horizontal entry, the curvature has the sign of $q E$ . A positive field in the negative $y$ direction bends a positive ion downward and an electron upward. Reversing the plate charge reverses $E$ and therefore reverses both trajectories. The initial horizontal speed also matters strongly: a particle traveling twice as fast spends half as long in the field and acquires one quarter of the vertical displacement.

An electric field calculation must identify the source region and particle observation point. Extending a uniform field beyond the plate edges is an idealization that changes the predicted flight time, exit direction, and detector position. Keeping the boundaries in the diagram avoids that hidden assumption.

Electric force together with gravity.

The equation $F = q E$ gives one force contribution. A particle in a laboratory also has weight, contact forces, drag, or forces from other fields. Newton's second law uses their vector sum. For a particle of mass $m$ in a vertical electric field, take upward as the positive $y$ direction. If the field is upward, then

\sum F_{y} = q E - m g .

A positive charge experiences an upward electric force; a negative charge experiences a downward one. A suspended positive charge requires $q E = m g$ , so the field must point upward. A suspended negative charge requires a downward field. The field direction can therefore be inferred from a known charge sign and an observed force balance.

The balance condition determines a ratio rather than either property alone:

\frac{q}{m} = \frac{g}{E} .

Vertical balance of a charged droplet. In an upward field the electric force

q E

acts up and the weight

m g

down; equal magnitudes give zero net force and the droplet floats.

For many elementary charged-particle calculations, gravity is negligible. The comparison is quantitative. An electron in a field of only $1 N/C$ has electric-force magnitude $e E = 1.60 \times 1 0^{- 19} N$ , while its weight is about $8.9 \times 1 0^{- 30} N$ . Their ratio is approximately $1.8 \times 1 0^{10}$ . For a macroscopic charged bead with a much larger mass and a small net charge, the ratio may be near unity instead. Dropping gravity without a scale comparison is justified for some particles and wrong for others.

The vector equation also distinguishes static equilibrium from motion at constant speed. When the net force vanishes, acceleration is zero; the particle may remain at rest or continue with whatever velocity it already has. A particle launched upward in an exact electric-gravitational balance travels upward at constant speed until another force or boundary changes its motion. Conversely, a particle at rest in an imbalance immediately begins accelerating even though its initial velocity is zero.

When the field is horizontal and gravity is vertical, the motions separate into two components:

a_{x} = \frac{q E}{m}, a_{y} = - g .

The resulting trajectory is determined by both accelerations. A calculation that uses $q E / m$ as the entire acceleration implicitly assumes that all other forces are either negligible or included in a separately stated direction. That assumption must be tied to the physical setup, particularly for slowly moving charged droplets and charged grains.

Impulse, momentum change, and charge-to-mass ratio.

During a finite interval, a prescribed electric field changes momentum through the impulse relation

Δ p = t_{i} \int t_{f} q E d t .

If both $q$ and $E$ are constant over an interval $Δ t$ , this reduces to

Δ p = q E Δ t .

The momentum form compares equal impulses before particle mass is specified. Two particles with the same charge passing through the same field for the same time receive equal electric impulses. Their velocity changes differ because $Δ v = Δ p / m$ . A light particle therefore responds with a larger acceleration than a heavy particle under identical field conditions.

In a uniform field of length $L$ traversed with constant horizontal speed $v_{x}$ , the time in the field is $L / v_{x}$ . With the field along $y$ ,

Δ p_{y} = q E \frac{L}{v _{x}}, Δ v_{y} = \frac{q E L}{m v _{x}} .

The ratio $q / m$ controls the velocity deflection. Suppose two singly charged positive ions enter the same horizontal field region at the same speed. The smaller-mass ion has a greater vertical exit velocity and a larger deflection. The sign of $q$ fixes the direction; the mass affects only the magnitude of the acceleration for a fixed charge.

Equal impulse, unequal deflection. Two positive charges spend the same time in the same downward field, so both gain the same downward momentum

q E Δ t

; the lighter particle gains more speed and curves more sharply.

A measured deflection can determine $q / m$ when the field, geometry, and initial velocity are independently known. For horizontal entry with $v_{0 y} = 0$ ,

y_{L} = \frac{q E L ^{2}}{2 m v _{x}^{2}}

gives

\frac{q}{m} = \frac{2 y _{L} v _{x}^{2}}{E L ^{2}} .

The sign of $y_{L}$ must be retained. When the positive $y$ axis is chosen along the field, a positive measured $y_{L}$ indicates a positive $q / m$ ; a negative deflection indicates a negative charge. Using only the absolute displacement yields the magnitude $∣ q ∣/ m$ and discards that information.

A numerical check can use a particle entering a $5.0 \times 1 0^{3} N/C$ field at $v_{x} = 2.0 \times 1 0^{6} m/s$ through plates of length $L = 4.0 \times 1 0^{- 2} m$ . If its measured exit displacement is $y_{L} = 1.0 \times 1 0^{- 3} m$ , then

\frac{q}{m} = \frac{2 ( 1.0 \times 1 0 ^{- 3} ) ( 2.0 \times 1 0 ^{6} ) ^{2}}{( 5.0 \times 1 0 ^{3} ) ( 4.0 \times 1 0 ^{- 2} ) ^{2}} = 1.0 \times 1 0^{9} C/kg .

The value is positive because the deflection has been taken in the field direction. Before trusting a result like this, verify that the displacement is much smaller than the plate separation, that the trajectory did not strike a plate, and that the horizontal speed remains approximately constant in the central field. Each condition belongs in the model before the numerical result is interpreted.

The momentum form also helps compare a short, strong field pulse with a weak field acting for a long time. Equal values of $E Δ t$ give equal impulse per unit charge when the field direction is the same. The detailed trajectory differs if the particle moves through a nonuniform region during the interval, because the field must then be integrated along the actual path rather than replaced by one constant vector.

Measuring a field without changing its source

The test-charge definition requires a probe small enough to measure the prepared field without significantly altering the source configuration. Its charge must also produce a force above the sensor's resolution. The measurement range balances two conditions:

source disturbance is negligible, ∣ q_{0} ∣ E is measurable .

Repeated readings with progressively smaller probes provide a practical test. If the inferred ratio $F / q_{0}$ remains unchanged within uncertainty, the probe is in the test-charge regime. If the ratio drifts as the charge is reduced, the larger probes were polarizing, displacing, or otherwise modifying the source arrangement. A charged conducting source is particularly sensitive because an external probe redistributes its surface charge.

At one location, record the force components on a calibrated positive probe. For example,

q_{0} = (2.00 \pm 0.02) nC, F_{x} = (6.00 \pm 0.06) \times 1 0^{- 6} N .

The inferred horizontal field is

E_{x} = \frac{F _{x}}{q _{0}} = 3.00 \times 1 0^{3} N/C .

For independent small relative uncertainties, the fractional uncertainty estimate is

\frac{δ E _{x}}{∣ E _{x} ∣} \approx (\frac{δ F _{x}}{∣ F _{x} ∣})^{2} + (\frac{δ q _{0}}{∣ q _{0} ∣})^{2} .

Here each relative input uncertainty is $1%$ , so the combined estimate is about $1.4%$ . The field should be reported with an uncertainty consistent with that precision, such as $(3.00 \pm 0.04) \times 1 0^{3} N/C$ . The sign of $E_{x}$ follows from the measured force direction on the positive probe; it is independent of the uncertainty convention.

Mapping a field requires repeating this component measurement at a series of fixed positions. The resulting data set is a list of vectors, not a single scalar profile. A point source gives equal field magnitudes at samples on a circle of constant radius within experimental uncertainty, while their directions rotate with the radius. Sample points at increasing radius should follow the inverse-square trend when the source is sufficiently compact. These comparisons test the model using measured geometry rather than a drawing alone.

An experimental map also distinguishes a field measurement from the subsequent motion of one freely released particle. A held probe permits force components to be measured at a specified location. A released particle has a changing position, and it samples different field vectors as it moves through a nonuniform region. Treating its initial force as a constant force requires an additional uniform-field approximation.

Systematic checks can expose a sign or calibration error. Reversing the probe charge reverses the measured force and leaves $F / q_{0}$ unchanged. Rotating the apparatus through $180$ degrees reverses the coordinate components assigned to a fixed physical field. Repeating measurements at the same marked position checks drift; a different mark changes the observation point and constitutes a different observation. These controls keep the measured field tied to a particular source configuration and coordinate system.

Complete vector calculation at one field point

A point-field calculation keeps four quantities distinct: source position, field position, displacement vector, and probe response. Combining them too early often reverses a direction or incorrectly includes the probe charge in the field. Let a source charge $q_{s}$ be at $(x_{s}, y_{s}, z_{s})$ and a field point be at $(x_{P}, y_{P}, z_{P})$ . The required displacement is

R = (x_{P} - x_{s}) \overset{}{^} + (y_{P} - y_{s}) \overset{}{^} + (z_{P} - z_{s}) \hat{k} .

Its length is

R = (x_{P} - x_{s})^{2} + (y_{P} - y_{s})^{2} + (z_{P} - z_{s})^{2} .

The source field follows directly:

E (P) = k q_{s} \frac{R}{R ^{3}} .

Only after that step is a probe charge introduced through $F = q_{0} E$ . The component formula preserves all signs without a separate verbal direction rule.

Consider a source $q_{s} = - 3.0 nC$ at $(0.10, - 0.20) m$ and a field point $P = (0.40, 0.20) m$ . The source-to-point displacement is

R = (0.30 \overset{}{^} + 0.40 \overset{}{^}) m, R = 0.50 m .

The field is

E (P) = (8.988 \times 1 0^{9}) (- 3.0 \times 1 0^{- 9}) \frac{0.30  ^ + 0.40  ^}{( 0.50 ) ^{3}} = (- 64.7 \overset{}{^} - 86.3 \overset{}{^}) N/C .

Its magnitude is $108 N/C$ , directed down and left toward the negative source. A positive $2.0 nC$ probe at $P$ experiences

F = (- 1.29 \overset{}{^} - 1.73 \overset{}{^}) \times 1 0^{- 7} N .

The force direction agrees with the field direction because the probe charge is positive. Replacing that probe with a $- 2.0 nC$ electron-scale charge reverses the force components while preserving the field components just calculated.

Check that $R > 0$ and that each field component has units of newtons per coulomb. The expression $k q_{s} R / R^{3}$ has units

(\frac{N m ^{2}}{C ^{2}}) \frac{C m}{m ^{3}} = \frac{N}{C} .

The field must point away from a positive source and toward a negative source. Its magnitude must decrease if the field point is moved farther away along the same ray. For the example, the ratio $∣ E_{y} ∣/∣ E_{x} ∣$ equals $0.40/0.30$ , so the component ratio matches the displacement geometry. A component sign that breaks any of these checks usually indicates a reversed displacement or an omitted source-charge sign.

Coordinates can be translated without changing the physical field. Shifting both source and field point by the same vector leaves $R$ unchanged, hence leaves $E$ unchanged. Rotating the coordinate axes changes the numerical components but not the field magnitude or its direction in physical space. These invariances separate a true physical prediction from a bookkeeping choice.

Spatial variation and the local-field approximation.

The inverse-square field changes continuously with position. Along a radial line from a positive point source,

E (r) = \frac{k q _{s}}{r ^{2}}, \frac{d E}{d r} = - \frac{2 k q _{s}}{r ^{3}} = - \frac{2 E}{r} .

The derivative gives the local rate at which field magnitude changes with radial distance. A small outward displacement $Δ r$ produces the approximation

Δ E \approx - \frac{2 E}{r} Δ r, \frac{Δ E}{E} \approx - 2 \frac{Δ r}{r} .

At $r = 1.0 m$ , a radial sensor motion of $1.0 cm$ changes the field by about $2%$ . At $r = 0.10 m$ , the same motion changes the field by about $20%$ . Thus the phrase field at a point is an approximation whenever a probe has finite size or a particle travels through an extended region. The variation must be small over the relevant distance before a single constant vector can represent the field there.

The exact comparison across a finite radial interval is

\frac{E ( r + Δ r )}{E ( r )} = (\frac{r}{r + Δ r})^{2} .

For $r = 0.50 m$ and $Δ r = 0.050 m$ , the exact ratio is $(0.50/0.55)^{2} = 0.826$ . The field falls by $17.4%$ . The derivative estimate predicts $- 2 (0.050/0.50) = - 20%$ , close enough to show the scale but not exact because the displacement is no longer extremely small compared with $r$ . The exact form should be used for a stated finite interval; the derivative is a local approximation.

A small test charge has a finite spatial extent, and a force sensor can hold it only over a finite position tolerance. The reported vector is therefore an average over a small region. When the field changes little across that region, the average agrees closely with the field at its center. When the field changes rapidly, the result depends on the probe shape, orientation, and exact center location. Finite probe size makes a point-charge model delicate close to a source even if the source itself is approximately compact.

Particle motion through a nonuniform field requires the same caution. The force is

F (t) = q E (r (t)),

so the acceleration changes as the particle changes location. Replacing it by $q E (r_{0}) / m$ gives a local, short-time approximation around the starting point $r_{0}$ . A finite trajectory requires either a field that is uniform over the path or a solution of the position-dependent equation of motion. The inverse-square field of a point charge is a standard case in which acceleration is radial and changes strongly with radius.

The local approximation has a clear geometric criterion. If a particle travels a distance $ℓ$ near radius $r$ in a point field, then $2 ℓ / r$ estimates the fractional field change along a radial path. Values much smaller than one support a constant-field approximation over that segment. Values comparable with one require the position dependence to remain in the force law. Tangential motion changes field direction even when the radius remains nearly constant, so a vector-field check must consider both magnitude and direction.

Domain of the electrostatic point-field model

The point-charge expression is an electrostatic model. Its source charge is treated as fixed while the field is evaluated, and the observation point is distinct from the source location. At the source location itself, the expression contains $R = 0$ and is undefined. The divergence belongs to the ideal mathematical point charge. Finite charged objects have size, structure, and charge distributions that replace the singular model in their immediate vicinity.

An isolated conducting sphere at electrostatic equilibrium has an exterior field that is the same as that of a point charge equal to the sphere's total charge located at its center. Only exterior points and spherical symmetry meet those conditions. A charged rod observed close to one end requires its actual charge distribution because its field depends strongly on the positions of individual charge elements. Those cases need the field integrated over a continuous charge distribution, together with symmetry arguments.

The model also assumes that the sources do not change appreciably during the observation interval. Moving a source charge, closing a circuit, or redistributing charge on a conductor creates time-dependent electromagnetic fields. The effect of that change reaches a distant observation point after a finite propagation time. A static formula describes the field before the change arrives and after the new arrangement has settled. Time-dependent fields are developed later with induction and electromagnetic waves.

The distinction between source field and response remains valid in every regime. A prescribed field at $r$ determines the force $q_{0} E (r)$ on a small probe located there. A probe large enough to alter the source distribution requires a coupled calculation in which its own field is part of the configuration. A statement of “electric field due to the source” must specify which charges are held fixed and which material responses are included.

Several limiting behaviors provide compact checks on a result:

Vanishing source charge. As $q_{s} \to 0$ , every component of the source field tends to zero.
Large distance. As $R \to \infty$ , the point-source field tends to zero as $1/ R^{2}$ .
Positive versus negative source. Replacing $q_{s}$ by $- q_{s}$ reverses every field component at the same field point.
Positive versus negative probe. Replacing $q_{0}$ by $- q_{0}$ leaves the field unchanged and reverses the force.
Mass change. Changing probe mass leaves field and force unchanged, while changing acceleration through $a = q_{0} E / m$ .

Each limit follows from a different physical part of the model. Source charge sets the field; field-point geometry sets its spatial dependence; probe charge converts a field into force; probe mass converts force into acceleration. Keeping these roles separate is the basis for the superposition, continuous-distribution, and particle motion calculations that follow.

Units, numerical scale, and reporting a vector field.

The SI unit $N/C$ follows directly from the operational definition. A field of $1 N/C$ exerts a $1 N$ force on a $1 C$ positive test charge. Coulomb is a large laboratory charge, so field-force calculations often combine smaller prefixes. A field of $2.5 \times 1 0^{4} N/C$ acting on a $4.0 nC$ probe produces

F = q E = (4.0 \times 1 0^{- 9}) (2.5 \times 1 0^{4}) = 1.0 \times 1 0^{- 4} N .

The conversion of nanocoulombs to coulombs introduces nine powers of ten. Omitting that conversion changes the force by a factor of $1 0^{9}$ , a common scale error in otherwise correct algebra. Scientific notation keeps the charge, field, and force exponents visible until the final rounding step.

Vector reports require both components and a coordinate convention. The statement

E = (- 2.6 \overset{}{^} + 1.4 \overset{}{^}) \times 1 0^{3} N/C

specifies a field completely in a two-dimensional coordinate system. Its magnitude is $3.0 \times 1 0^{3} N/C$ to two significant figures, but the magnitude alone does not specify the force direction on a charged particle. A diagram should state which axis is positive whenever components are read from geometry; a different axis orientation changes the signs of the components while leaving the physical vector unchanged.

Precision should follow the data. If a source-field distance is measured to three significant figures, the inverse-square calculation has roughly twice the fractional distance uncertainty. For a distance $r \pm δ r$ ,

\frac{δ E}{E} \approx 2 \frac{δ r}{r}

when source-charge uncertainty is negligible. A $1%$ position uncertainty therefore creates about a $2%$ field uncertainty for a point-source measurement. Reporting six digits from Coulomb's constant cannot compensate for a poorly known source-to-point distance. Reported precision belongs to the complete physical measurement, not to one constant in the formula.

Use the reciprocal relationship between field and force as a final numerical check. Dividing a force by the known test charge must recover the original field vector; multiplying that field by a negative test charge must reverse every force component. Those two calculations test signs, units, and powers of ten without introducing a new model.

From a field measurement to a particle calculation

A measured or calculated field becomes a motion model only after a particle and a region of validity are specified. Along one coordinate, the governing equation is

m \frac{d ^{2} x}{d t ^{2}} = q E_{x} (x, t) .

In a uniform static field, $E_{x}$ is constant and the familiar constant-acceleration equations follow. For a point source, $E_{x}$ varies with position, and a particle moving toward or away from the source experiences changing acceleration. At the particle position, evaluate the source field and use $a = q E / m$ .

A short numerical approximation keeps that interface explicit. At time $t_{n}$ , with position $r_{n}$ and velocity $v_{n}$ , evaluate

a_{n} = \frac{q}{m} E (r_{n}),

then update over a sufficiently short interval $Δ t$ by

v_{n + 1} \approx v_{n} + a_{n} Δ t, r_{n + 1} \approx r_{n} + v_{n} Δ t .

Reducing $Δ t$ should make the calculated trajectory converge. A large time step can move the particle across a region where the field direction or magnitude changes substantially, replacing a varying force with one outdated value. The fractional field-change estimate from the preceding section sets a physical step-size criterion near a point source.

The charge sign remains in every update. A negative particle in a prescribed field has acceleration opposite the local field vector at each step. Its path does not reverse the field arrows; it samples the same vector map with the opposite force law. Mass enters only through $q / m$ , so particles with equal charge magnitude but different masses trace different paths through the same field. These distinctions remain necessary when later lessons add superposed fields, continuous charge distributions, magnetic forces, and time-dependent sources.