Superposition Principle for Electric Fields

Here is a fact so familiar that you could study electromagnetism for years without noticing it is a fact at all. Two charges sit near a test point. Each produces its Coulomb field as if it were alone in the universe; the actual field is simply the two vectors added tip-to-tail. The third charge you bring in changes nothing about how the first two contribute — no charge screens, distorts, or renegotiates the field-producing power of another. Sources act in parallel, and space itself does the adding. That is the superposition principle, and nothing in Coulomb's law forces it to be true. You can write down perfectly self-consistent theories where fields combine nonlinearly — where two overlapping fields interfere with each other's sources — and nature almost chose one: in quantum electrodynamics, at fields around $10^{18}$ V/m, the vacuum itself becomes nonlinear and light scatters off light. Everywhere below that ferocious scale, though, the vacuum behaves as a perfectly linear medium, and superposition holds to every decimal place ever measured — inside atoms, across capacitors, in the fields of thunderstorms. This article is short on new formulas and long on consequences, because linearity is the single greatest labor-saving device in the subject. It is why hard problems decompose into easy symmetric pieces, why the continuous distributions of the next article are just integrals of point-charge fields, why flux adds charge-by-charge into Gauss's law, and why the method of images and the multipole expansion exist at all. Learn to wield linearity here, and half the chapter ahead becomes bookkeeping.

Key takeaways

The total electric field of many sources is the vector sum of their individual fields: $\mathbf{E}=\sum_i\mathbf{E}_i$ , each computed as if the others did not exist.
Superposition is an independent experimental fact about nature, not a consequence of Coulomb's law — it says the field is linear in its sources.
It adds vectors, never magnitudes: resolve into components first, always.
Linearity survives every upgrade in this chapter: potentials add as scalars, fluxes add surface-by-surface, and the field equations $\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$ , $\nabla\times\mathbf{E}=\mathbf{0}$ are linear, so solutions add too.
Strategy dividend: decompose an ugly source into symmetric pieces, solve each, add — the engine behind continuous distributions, images, and multipoles.
Superposition fails only at the quantum frontier: near the Schwinger field $E_c\approx1.3\times10^{18}$ V/m, vacuum polarization makes electrodynamics nonlinear.
Key idea: the field is a linear functional of its sources — each charge contributes its lone-universe Coulomb field, and the vector sum is exact.

The core insight

The electric field is a linear functional of its sources: $\mathbf{E}\left[\sum_i q_i\right] = \sum_i \mathbf{E}[q_i]$ . Each charge contributes its Coulomb field independently, and the total field is their vector sum.

Where this lives today

Superposition is the load-bearing assumption of essentially all electromagnetic computation. Finite-element solvers decompose sources into basis elements and add their fields; the "Green's function" methods that dominate antenna design, geophysics, and accelerator physics are superposition institutionalized — the field of any source as an integral of point-source responses. Molecular-dynamics engines add pairwise Coulomb fields over millions of atoms per femtosecond step, trusting linearity so deeply that the fast algorithms (Ewald sums, fast multipole methods) are pure mathematics of adding $1/r^2$ contributions efficiently. Meanwhile the frontier probes the principle's edge: petawatt laser facilities and the planned next generation push toward measurable vacuum birefringence — the Euler–Heisenberg nonlinearity — and X-ray free-electron lasers have begun hunting photon-photon scattering directly. Astrophysics already sees the breakdown at work: the magnetospheres of magnetars, with fields around $10^{11}$ T, are natural laboratories where the vacuum polarizes and superposition visibly fails. Between your phone's touchscreen and a magnetar's surface, there is one principle quietly doing all the addition.

The physical picture

Release a tiny test charge among a crowd of sources. Each source pushes or pulls on it with its own private Coulomb force, none of them consulting the others; the probe responds to the resultant of all those simultaneous pushes, exactly as a boat responds to the sum of wind and current. Divide by the probe charge and the statement transfers from forces to fields: at every point of space, the field vectors of the individual sources stack tip-to-tail into one resultant. The picture to hold is parallel action — a switchboard where every source has its own direct line to every point of space, with no crosstalk.

Figure

Parallel action — each source contributes its lone-universe field at P, and the vectors stack tip-to-tail

The second picture is subtler and more powerful: the vacuum is a linear medium. Double every source charge and the field doubles everywhere. Add a second population of charges and the new field is the old field plus the newcomers' field, untouched. Linearity is what lets you run every problem-solving strategy of this chapter: chop a rod into point-like slivers and integrate; replace a conductor by a fictitious image charge whose field, added to the real charge's, meets the boundary condition; expand a lumpy molecule as monopole plus dipole plus quadrupole and keep two terms. Each trick is one sentence of physics — fields add — plus bookkeeping. Keep one caution pinned throughout: superposition is a statement about vectors. Two fields of magnitude $100$ N/C can sum to anything between $0$ and $200$ N/C depending on the angle between them. The moment a problem has two or more sources, your first act is a components table, not an addition.

The principle, and what it licenses

For $N$ point charges at positions $\mathbf{r}_i$ , the field at $\mathbf{r}$ is the sum of lone-universe Coulomb fields:

\mathbf{E}(\mathbf{r}) = \sum_{i=1}^{N}\mathbf{E}_i(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^{N} q_i\,\frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3} \tag{1}

where: each term is the point-charge field of the previous article, aimed by the separation vector from source $i$ to the field point; the cube in the denominator just normalizes that direction. Nothing couples the terms — $q_3$ appears in exactly one summand no matter how large $q_1$ is. That decoupling is the principle; equation (1) is its notation. The same statement in the abstract, worth internalizing because it names what is being claimed:

\mathbf{E}[\alpha\rho_1+\beta\rho_2] \;=\; \alpha\,\mathbf{E}[\rho_1] + \beta\,\mathbf{E}[\rho_2] \tag{2}

where: the brackets read "the field produced by this source distribution," and $\alpha,\beta$ are any constants. Equation (2) says the map from sources to fields is linear — the defining property that mathematics calls a linear functional. Every scaling argument you will ever make ("double the plate charge, double the field") is (2) with $\beta=0$ ; every decomposition ("field of the slab = field of two half-slabs") is (2) with $\alpha=\beta=1$ . Linearity percolates through everything built on the field. The force on charge $q_j$ superposes over the other sources, $\mathbf{F}_j = q_j\mathbf{E}_{\neq j}(\mathbf{r}_j)$ — the self-field exclusion of the previous article survives the crowd. The flux of the total field through any closed surface splits into per-charge fluxes, $\oint\mathbf{E}\cdot\mathrm{d}\mathbf{a} = \sum_i\oint\mathbf{E}_i\cdot\mathrm{d}\mathbf{a}$ — the one-line lemma that will convert the point-charge flux result into the full Gauss's law two articles ahead. And when the potential $V$ arrives, it will superpose as a plain scalar sum — no components — which is much of why physicists adore it.

linearity dividend	statement	where it pays off
decompose and add	field of ugly source = sum of fields of easy pieces	continuous distributions (next)
scale	$\rho\to\alpha\rho$ gives $\mathbf{E}\to\alpha\mathbf{E}$ everywhere	capacitance, linear circuits
flux adds	$\Phi[\sum q_i]=\sum\Phi[q_i]$	Gauss's law
scalars add	$V=\frac{1}{4\pi\varepsilon_0}\sum_i q_i/\|\mathbf{r}-\mathbf{r}_i\|$	potential theory
solutions add	$(\mathbf{E}_1+\mathbf{E}_2)$ solves for $(\rho_1+\rho_2)$	images, multipoles, boundary-value problems

Derivation AWhere superposition comes from — force additivity plus the field definition

We want to locate exactly what is assumed and what is derived when we write $\mathbf{E}=\sum_i\mathbf{E}_i$ . The strategy is to start from the experimental input — forces on a test charge add vectorially, with each pair force given by Coulomb's law unmodified by bystanders — and push it through the definition of the field from the previous article. The point of the exercise is intellectual honesty: superposition of fields is not derivable from the pair law alone; it encodes a second, independent experimental fact, and this derivation shows precisely where that fact enters.

Step 1. State the experimental input. Place a probe $q_0$ at $\mathbf{r}$ among sources $q_1,\dots,q_N$ . Measurement shows the force on the probe is

\mathbf{F}_0 = \sum_{i=1}^{N}\frac{1}{4\pi\varepsilon_0}\,\frac{q_i q_0}{|\mathbf{r}-\mathbf{r}_i|^3}\,(\mathbf{r}-\mathbf{r}_i)

Why is this an assumption and not algebra? Because each summand is claimed to be the unmodified Coulomb pair force — the presence of $q_2$ does not alter the force $q_1$ exerts. A nonlinear theory would insert cross terms like $q_1q_2q_0$ ; experiment finds none, from ion traps to atomic spectra.

Step 2. Divide by the probe and take the test-charge limit:

\mathbf{E}(\mathbf{r}) \equiv \lim_{q_0\to0}\frac{\mathbf{F}_0}{q_0} = \sum_{i=1}^{N}\frac{1}{4\pi\varepsilon_0}\,\frac{q_i\,(\mathbf{r}-\mathbf{r}_i)}{|\mathbf{r}-\mathbf{r}_i|^3}

Why does the sum survive the limit? Each pair force is separately linear in $q_0$ , so the quotient is $q_0$ -free term by term; the limit only protects the source positions from probe back-reaction, as before.

Step 3. Recognize each summand as the lone-universe field: the $i$ -th term is exactly the point-charge field $\mathbf{E}_i(\mathbf{r})$ that $q_i$ would produce with all other sources removed. Hence $\mathbf{E}=\sum_i\mathbf{E}_i$ — equation (1).

Step 4. Audit the logic. Two experimental facts entered: Coulomb's pair law (fixes each $\mathbf{E}_i$ ) and force additivity without modification (fixes the plain sum). The second is independent of the first — which is why superposition can fail (strong-field QED) in a world where the pair law survives at low fields. When you quote "superposition," you are citing an experiment, not a theorem. Verified: structure matches Griffiths §2.1.2–2.1.4 and Purcell §1.7, both of which flag superposition as "an experimental fact" beyond the pair law; the limit argument is the previous article's Derivation A applied term by term.

Go deeper · Derivation B · 2 minLinear field equations guarantee superposition — the theory-side check

Derivation BLinear field equations guarantee superposition — the theory-side check

Having grounded superposition in experiment, we now verify that the local theory of electrostatics — the differential equations the field obeys — enforces it automatically, so the two formulations of the subject agree. The goal: show that if $\mathbf{E}_1$ solves the electrostatic equations with source $\rho_1$ , and $\mathbf{E}_2$ solves them with $\rho_2$ , then $\mathbf{E}_1+\mathbf{E}_2$ solves them with source $\rho_1+\rho_2$ . The strategy is one line per equation, because divergence and curl are linear operators — and that observation, small as it looks, is the entire structural reason the field theory is linear. Watch also the uniqueness clause at the end: adding solutions is only guaranteed to give the field if boundary conditions pin the answer down.

Step 1. Write the electrostatic field equations for each source separately:

\nabla\cdot\mathbf{E}_1=\frac{\rho_1}{\varepsilon_0},\quad \nabla\times\mathbf{E}_1=\mathbf{0}; \qquad \nabla\cdot\mathbf{E}_2=\frac{\rho_2}{\varepsilon_0},\quad \nabla\times\mathbf{E}_2=\mathbf{0}

Why start here? These two equations (derived from Coulomb's law at the end of this chapter) are the complete local content of electrostatics — by the Helmholtz theorem of tier E0, divergence plus curl plus decay determine the field.

Step 2. Add the divergence equations, using linearity of the derivative: $\nabla\cdot(\mathbf{E}_1+\mathbf{E}_2) = \nabla\cdot\mathbf{E}_1+\nabla\cdot\mathbf{E}_2 = (\rho_1+\rho_2)/\varepsilon_0$ . Why is $\nabla\cdot$ linear? It is built from partial derivatives, and $\partial_x(f+g)=\partial_x f+\partial_x g$ — the linearity of calculus itself. No property of electricity is used in this step.

Step 3. Add the curl equations the same way: $\nabla\times(\mathbf{E}_1+\mathbf{E}_2) = \mathbf{0}+\mathbf{0} = \mathbf{0}$ . So the sum field satisfies both electrostatic equations with the combined source $\rho_1+\rho_2$ .

Step 4. Close the loop with uniqueness. Both fields decay at infinity for localized sources, so their sum does; by the Helmholtz uniqueness theorem, a field with the right divergence, the right curl, and decay is the field of $\rho_1+\rho_2$ — not merely a solution. Contrast: if the governing equation contained a nonlinear term (say $\nabla\cdot\mathbf{E} \propto \rho + \gamma|\mathbf{E}|^2$ ), Step 2 would fail — cross terms $\mathbf{E}_1\cdot\mathbf{E}_2$ would appear, and the field of two charges would not be the sum of their fields. Linearity of the equations and superposition of solutions are the same property seen from two sides. Verified: this is the standard linearity/uniqueness argument of Jackson §1.5 and Zangwill §2.2; the failure mode sketched in Step 4 is precisely the structure of the Euler–Heisenberg correction in strong-field QED.

Limiting cases

case	result	meaning
classical vacuum, any lab field	superposition exact	no measured deviation, from V/m to GV/m
linear dielectric medium	superposition holds with $\varepsilon_0\to\varepsilon$	matter can stay linear — the basis of tier E3
nonlinear media (ferroelectrics, strong optics)	fails: $\mathbf{P}$ responds nonlinearly	frequency doubling in laser crystals lives here
strong-field QED, $E\sim E_c=1.3\times10^{18}$ V/m	vacuum itself nonlinear	light-by-light scattering; Euler–Heisenberg
one source only	(1) collapses to the point-charge field	consistency with the previous article
$\rho\to\alpha\rho$	$\mathbf{E}\to\alpha\mathbf{E}$	pure scaling — capacitance is charge-independent

Common trapAdding the magnitudes — the five-mark massacre

Superposition adds vectors. If $\mathbf{E}_1$ and $\mathbf{E}_2$ both have magnitude $100$ N/C, the total is $200$ N/C only if they are parallel; it is $141$ N/C at right angles, and $0$ if opposed. The reflex that kills exam scripts is computing two Coulomb magnitudes and adding them like grocery prices — worse, adding magnitudes when one field points toward a negative source. The immunizing habit: for every source, draw the field vector at the field point first (away from $+$ , toward $-$ ), then build a components table, then add columns. Only take the magnitude at the very end. A second, quieter trap from the same family: the field that pushes charge $q_j$ is the superposed field of everyone else — its own (infinite) self-field never joins the sum.

Worked examples

1 · Two charges, one apex (confidence builder)

Two charges $q = +3.0\ \mu\mathrm{C}$ sit at the base corners of an equilateral triangle of side $a = 0.50$ m. Find the field at the apex.

Step 1. One source at a time. Each charge is $0.50$ m from the apex, so each contributes magnitude $E_1 = k_e q/a^2 = (8.99\times10^9)(3.0\times10^{-6})/(0.25) = 1.08\times10^5$ N/C, pointing away from its own corner through the apex (both sources positive).

Step 2. Geometry. Each field vector tilts $30°$ from the vertical (half the $60°$ apex angle). Horizontal components: equal and opposite — they cancel by symmetry. Vertical components: each is $E_1\cos30° = (1.08\times10^5)(0.866) = 9.34\times10^4$ N/C, both pointing up.

Step 3. Add: $E = 2\times9.34\times10^4 = 1.87\times10^5$ N/C, straight up along the triangle's axis of symmetry.

Step 4. Read the shortcut you just used: symmetry told you the horizontal column of the components table would cancel before you computed anything — a linearity dividend. Note also the wrong answer avoided: adding magnitudes would give $2.16\times10^5$ N/C, off by 16%.

Answer: $E = 1.87\times10^5$ N/C, directed vertically away from the base — symmetry kills the horizontal components.

2 · Hunting the null point (exam level)

Charges $q_1 = +4.0\ \mu\mathrm{C}$ (at $x=0$ ) and $q_2 = +1.0\ \mu\mathrm{C}$ (at $x = 1.0$ m) sit on the $x$ -axis. Where is the total field zero, and why must that point be where it is?

Step 1. Reason before computing. Between the charges, $\mathbf{E}_1$ points right (away from $q_1$ ) and $\mathbf{E}_2$ points left (away from $q_2$ ) — opposition, so cancellation is possible there. Outside the pair, both fields point the same way and can never cancel. The null must also sit closer to the weaker charge: only proximity can let $1\ \mu$ C match $4\ \mu$ C.

Step 2. Set the magnitudes equal at position $x$ (in metres, $0<x<1$ ):

\frac{k_e\,(4.0\times10^{-6})}{x^2} = \frac{k_e\,(1.0\times10^{-6})}{(1-x)^2}

Step 3. Cancel $k_e$ and $10^{-6}$ , take the square root of $4/x^2 = 1/(1-x)^2$ : since both sides are positive in the gap, $2(1-x) = x$ , so $x = \tfrac{2}{3}$ m.

Step 4. Audit: at $x = 0.667$ m, $E_1 = (8.99\times10^9)(4.0\times10^{-6})/(0.667)^2 = 8.1\times10^4$ N/C rightward; $E_2 = (8.99\times10^9)(1.0\times10^{-6})/(0.333)^2 = 8.1\times10^4$ N/C leftward. ✓ The null sits twice as far from the $4\times$ charge as from the $1\times$ charge — exactly the $\sqrt{4}=2$ distance ratio the inverse-square demands.

Step 5. The physics of the point: a positive probe released just off the null feels a restoring push along the axis but a runaway push transversely (or vice versa) — field zeros in charge-free space are always saddle points, never traps. That "no electrostatic trap" observation is Earnshaw's theorem knocking; it returns with the potential.

Answer: $\mathbf{E}=\mathbf{0}$ at $x = 2/3$ m from the larger charge — between the charges, twice as far from the stronger one, and a saddle rather than a trap.

Figure

The null-point hunt — two rightward and leftward field profiles cross where magnitudes match, twice as far from the stronger charge

Origins · 1 minHenry Cavendish and how this was discovered

Henry Cavendish

1731–1810 · Clapham, London

The strangest great experimentalist in the history of the subject tested superposition's consequences decades before anyone else — and told nobody. Cavendish, a silent, pathologically shy aristocrat who communicated with his housekeeper by notes, built a laboratory in his London home and, in the 1770s, established that the electric force obeys the inverse square (via the null-field test inside a charged sphere — an argument that assumes fields of all the sphere's patches add), measured charge division between conductors, and anticipated Ohm's law by half a century using his own body as the galvanometer, calibrating current by the pain of the shock. Almost none of it was published. A century later Maxwell, first Cavendish Professor at the laboratory named for the family, spent years editing the unpublished manuscripts and discovered how much had been quietly known. Superposition's precision heritage runs straight through Cavendish's sphere: its modern descendants bound deviations at the $10^{-16}$ level.

History

1747

— Franklin's one-fluid theory treats charge as an additive, conserved quantity — arithmetic for charge before arithmetic for fields

1773

— Cavendish's unpublished null-field experiment inside a charged sphere implicitly tests that contributions from all surface patches add vectorially

1812

— Poisson's memoirs build potential theory on linearity: fields and potentials of charge elements superpose into integrals

1936

— Heisenberg and Euler compute the effective Lagrangian of QED: strong fields make the vacuum nonlinear, marking superposition's boundary

1997

— SLAC E-144 observes nonlinear light-by-light processes in strong laser fields — the failure of superposition, on schedule

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §2.1–2.2 · Jackson, Classical Electrodynamics, 3rd ed., §1.2 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §1.7 · Zangwill, Modern Electrodynamics, §2.2

Provenance. Equations cross-checked against Griffiths 4th ed. §2.1–2.2 (superposition, discrete and continuous forms, line-charge example), Jackson 3rd ed. §1.2 & §1.5 (linearity, uniqueness), Purcell 3rd ed. §1.7, and Zangwill §2.2. A verifier should reconfirm: the line-charge integral $\int_{-L}^{L}(x^2+z^2)^{-3/2}\mathrm{d}x = 2L/[z^2\sqrt{z^2+L^2}]$ and its $L\to\infty$ and $z\gg L$ limits — worked in the continuous-distributions article of this chapter; the Schwinger critical field $E_c=m_e^2c^3/(e\hbar)\approx1.32\times10^{18}\,\mathrm{V/m}$ ; and $\varepsilon_0=8.854\times10^{-12}\,\mathrm{F/m}$ . Worked numbers on this page (apex field $1.87\times10^5$ N/C from two $1.08\times10^5$ N/C contributions at $30°$ ; null point at $x=2/3$ m with both magnitudes $8.1\times10^4$ N/C) are illustrative arithmetic recomputed from equation (1).

Builds on: Coulomb's Law · The Electric Field Leads to: Continuous Charge Distributions · Gauss's Law in Integral Form · Electrostatic Potential · Poisson's Equation

Is superposition really not derivable from Coulomb's law? It feels like it should follow automatically.

It genuinely does not follow. Coulomb's law fixes the force between an isolated pair; it is silent about whether a third charge modifies that pair interaction. Imagine a theory where the force between $q_1$ and $q_2$ picks up a correction proportional to $q_1q_2q_3/r^4$ terms — pairwise Coulomb behavior survives whenever only two charges exist, yet fields would not add. Nature could have worked that way; experiment says it does not, to spectacular precision, at every field strength below the quantum-vacuum scale. That is why careful texts (Griffiths, Purcell) explicitly flag superposition as a second, independent postulate of electrostatics.

If fields from opposite charges cancel at a point, is there really "nothing" there?

The total field is genuinely zero at that point — a test charge placed exactly there feels no force, full stop. But the configuration differs physically from empty space in ways that show up off the point: move infinitesimally away and forces reappear (the null of worked example 2 is a saddle), and the energy density stored in the neighborhood is not what empty space would have. Cancellation is also frequency-fragile: wiggle one source and the zero moves. So "nothing there" is exactly true for the static force at the exact point, and false for essentially every other question — a useful precision to carry into interference physics later.

Does superposition mean charges don't interact with each other?

No — it means their field contributions don't interact. The sources themselves feel forces from one another (that is the whole content of Coulomb's law) and, if free to move, will rearrange: bring a charge near a conductor and the conductor's electrons genuinely redistribute. Superposition then applies to the final configuration: total field = field of your charge + field of the redistributed electrons, each computed by the plain Coulomb recipe. What never happens is one field warping another field's relation to its own source. Distinguish "sources push sources" (true, always) from "fields distort fields" (false, in vacuum, below the QED scale).

How can the field of each charge just pass through other charges' fields without any effect — don't they occupy the same space?

Fields are not substances that collide; they are vector-valued conditions of space, and conditions can coexist by adding. The same happens with small water waves or sound: two waves cross, the displacements add while overlapping, and each emerges unchanged — linearity is precisely the property of "passing through." The deeper answer is that the electromagnetic field's equations contain no term coupling the field to itself (photons carry no charge, in quantum language). Contrast gravity: the gravitational field carries energy, energy gravitates, so general relativity is nonlinear and gravitational fields do not superpose exactly — one reason it is a harder theory.

When I compute the force on one charge of a pair, do I include the field it produces itself?

Never. The force on $q_j$ is $q_j$ times the field of all other charges evaluated at $q_j$ 's location. Its own field diverges at its own position and, by symmetry, exerts no net push on it. In formulas this is the $i\neq j$ under the sum — easy to forget once distributions become continuous, where the rule takes a subtler form: for surface charges, the force per area uses the average of the fields on the two sides of the sheet, which is exactly "everything except the local patch's self-field." That average trick reappears in the conductors article; the principle behind it is planted here.

Since real problems have ~10²³ charges, is superposition actually usable or just formal?

Usable — it is the thing that makes $10^{23}$ charges tractable. Three routes. Symmetry: most of the $10^{23}$ contributions cancel in pairs, and superposition lets you argue that before computing (worked example 1 in miniature). Continuum: group the charges into infinitesimal cells, treat each as a point, and the sum becomes an integral — the entire next article. Fields from boundaries: for conductors you never track the electrons at all; uniqueness theorems plus linearity let a single fictitious image charge stand in for the whole redistributed swarm. Every one of these is superposition wielded as a weapon rather than checked as a fact.

What actually goes wrong at the Schwinger field — does Coulomb's law break too?

At fields approaching $E_c \approx 1.3\times10^{18}$ V/m, the energy density is high enough that virtual electron–positron pairs in the vacuum respond appreciably: the vacuum polarizes, like a dielectric made of nothing. Effective field theory (Euler–Heisenberg) adds small terms to Maxwell's equations proportional to higher powers of the fields — so two overlapping strong fields do modify each other, and light scatters off light. Coulomb's law also picks up tiny corrections at short distances (the Uehling potential, measurable in atomic spectra as part of the Lamb shift). Below these scales — meaning everywhere in this course — the corrections are fantastically negligible: superposition in your problems is exact to more decimal places than any instrument reads.