Coulomb's law and Gauss's law

Intuition [Beginner]

Matter carries a property called electric charge, measured in coulombs (C). Charge comes in two signs, positive and negative. Like charges repel; unlike charges attract. This is different from gravity, which only attracts.

Coulomb's law gives the force between two point charges $q_1$ and $q_2$ separated by a distance $r$:

$$F=k_e\frac{q_1{q}_2}{r^2},$$

where $k_e=1/(4\pi {ϵ}_0)\approx 9.0\times 10^9\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$. The force is along the line joining the charges. When $q_1{q}_2>0$ (same sign), $F>0$ and the force is repulsive; when $q_1{q}_2<0$ (opposite signs), $F<0$ and the force is attractive.

The electric field $\mathbf{E}$ is the force per unit charge that a test charge would feel at a given point. A point charge $q$ at the origin produces a field that points radially outward (if $q>0$) or inward (if $q<0$) with magnitude $E=k_e|q|/r^2$ at distance $r$.

The field picture replaces action-at-a-distance: charges create a field, and the field tells other charges what force to feel. The field is real — it carries energy and momentum — not just a bookkeeping device.

Visual [Beginner]

The density of field lines in a given region is proportional to the field strength there. This geometric picture is the bridge to Gauss's law: if you draw a closed surface around a charge, the number of field lines piercing that surface depends only on the enclosed charge, not on the shape of the surface.

Gauss's law says: the total electric flux through any closed surface equals the net charge inside, divided by ${ϵ}_0$. In words, if you surround a region of space with an imaginary bag, the electric field "flowing out" of that bag is determined entirely by the charge trapped inside. Charge outside the bag contributes zero net flux — its field lines enter and leave in equal numbers.

Gauss's law is powerful because symmetry lets you compute $\mathbf{E}$ without adding up contributions from every charge. Three standard symmetry classes:

• Spherical symmetry (e.g., a uniformly charged sphere or spherical shell): $\mathbf{E}$ points radially, with magnitude depending only on the distance from the centre. Choose the Gaussian surface to be a concentric sphere.
• Cylindrical symmetry (e.g., an infinite line of charge): $\mathbf{E}$ points radially outward from the line. Choose the Gaussian surface to be a coaxial cylinder.
• Planar symmetry (e.g., an infinite charged plane): $\mathbf{E}$ points perpendicular to the plane, constant in magnitude. Choose the Gaussian surface to be a pillbox straddling the plane.

Worked example [Beginner]

Two point charges sit on the $x$-axis: $q_1=+3\mu \mathrm{C}$ at $x=0$ and $q_2=-1\mu \mathrm{C}$ at $x=0.10\mathrm{m}$. Compute the force on $q_2$ due to $q_1$, and the electric field at the midpoint $x=0.05\mathrm{m}$.

Force on $q_2$. The separation is $r=0.10\mathrm{m}$. Coulomb's law gives

$$F=k_e\frac{q_1{q}_2}{r^2}=(9.0\times 10^9)\frac{(3\times 10^{-6})(-1\times 10^{-6})}{(0.10{)}^2}.$$

The product $q_1{q}_2=-3\times 10^{-12}{\mathrm{C}}^2$, so $F=-2.7\mathrm{N}$. The negative sign means the force on $q_2$ is toward $q_1$ (attractive, since the charges have opposite sign). In vector form: ${\mathbf{F}}_{21}=-2.7\hat{\mathbf{x}}\mathrm{N}$, pointing from $q_2$ toward $q_1$.

Electric field at the midpoint. The field at $x=0.05\mathrm{m}$ has contributions from both charges, both on the $x$-axis.

From $q_1$ (at distance $0.05\mathrm{m}$, pointing in $+\hat{\mathbf{x}}$ because $q_1>0$):

$$E_1=k_e\frac{|q_1|}{r_1^2}=(9.0\times 10^9)\frac{3\times 10^{-6}}{(0.05{)}^2}=1.08\times 10^{7}N/C\text{in }+\hat{\mathbf{x}}.$$

From $q_2$ (at distance $0.05\mathrm{m}$, pointing in $-\hat{\mathbf{x}}$ because $q_2<0$ and the field points toward negative charges):

$$E_2=k_e\frac{|q_2|}{r_2^2}=(9.0\times 10^9)\frac{1\times 10^{-6}}{(0.05{)}^2}=3.6\times 10^{6}N/C\text{in }+\hat{\mathbf{x}}.$$

Wait — $q_2$ is negative and sits to the right of the midpoint, so its field at the midpoint points toward $q_2$, which is the $+\hat{\mathbf{x}}$ direction. Both contributions point in $+\hat{\mathbf{x}}$. The total field is

$${\mathbf{E}}_{\mathrm{mid}}=(1.08\times 10^7+3.6\times 10^6)\hat{\mathbf{x}}=1.44\times 10^7\hat{\mathbf{x}}N/C.$$

Check: a test charge $q_0=+1\mu \mathrm{C}$ placed at the midpoint would feel a force $\mathbf{F}=q_0{\mathbf{E}}_{\mathrm{mid}}=14.4\hat{\mathbf{x}}\mathrm{N}$, pushing it toward $q_2$ (away from the repelling $q_1$ and toward the attracting $q_2$).

Check your understanding [Beginner]

Formal definition [Intermediate+]

Coulomb's law (vector form). A point charge $q_1$ at position ${\mathbf{r}}_1$ exerts a force on a point charge $q_2$ at position ${\mathbf{r}}_2$ given by

$${\mathbf{F}}_{12}=\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{|{\mathbf{r}}_2-{\mathbf{r}}_1{|}^3}({\mathbf{r}}_2-{\mathbf{r}}_1).$$

The force is along the line joining the charges, repulsive for like signs, attractive for opposite signs. The $1/4\pi {ϵ}_0$ prefactor ($\approx 9.0\times 10^9\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$) sets the SI scale; in Gaussian units it is replaced by 1.

Electric field. The electric field at position $\mathbf{r}$ due to a point charge $q$ at the origin is

$$\mathbf{E}(\mathbf{r})=\frac{1}{4\pi {ϵ}_0}\frac{q}{r^2}\hat{\mathbf{r}},$$

where $\hat{\mathbf{r}}=\mathbf{r}/r$. The force on a test charge $q_0$ placed at $\mathbf{r}$ is $\mathbf{F}=q_0\mathbf{E}(\mathbf{r})$. The principle of superposition extends this to multiple charges: the total field is the vector sum of the fields from each charge individually.

For a continuous charge distribution with volume charge density $\rho ({\mathbf{r}}^{\prime })$, the electric field is

$$\mathbf{E}(\mathbf{r})=\frac{1}{4\pi {ϵ}_0}\int \frac{\rho ({\mathbf{r}}^{\prime })(\mathbf{r}-{\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}{d}^3{\mathbf{r}}^{\prime }.$$

This is the superposition integral. Surface charge densities $\sigma$ and line charge densities $\lambda$ are handled by replacing $\rho d^3{\mathbf{r}}^{\prime }$ with $\sigma d{A}^{\prime }$ or $\lambda d{\ell }^{\prime }$ respectively.

Gauss's law (integral form). Let $S$ be a closed surface bounding a volume $V$. Then

$${\oint }_S\mathbf{E}\cdot d\mathbf{A}=\frac{Q_{\mathrm{enc}}}{{ϵ}_0},$$

where $Q_{\mathrm{enc}}={\int }_V\rho d^3{\mathbf{r}}^{\prime }$ is the total charge enclosed. The left side is the electric flux ${\Phi }_E$ through $S$.

Gauss's law (differential form). By the divergence theorem, ${\oint }_S\mathbf{E}\cdot d\mathbf{A}={\int }_V(\nabla \cdot \mathbf{E})d^3\mathbf{r}$. Equating with ${\int }_V(\rho /{ϵ}_0)d^3\mathbf{r}$ and using the arbitrariness of $V$:

$$\boxed{\nabla \cdot \mathbf{E}=\frac{\rho }{{ϵ}_0}.}$$

This is the first of Maxwell's equations in the electrostatic limit (time-independent fields, no currents). It is a local statement: the divergence of $\mathbf{E}$ at a point is proportional to the charge density at that point.

Electric potential. In electrostatics, $\nabla \times \mathbf{E}=0$ (the field is conservative). This guarantees the existence of a scalar potential $V$ such that

$$\mathbf{E}=-\nabla V.$$

For a point charge $q$ at the origin, $V(\mathbf{r})=q/(4\pi {ϵ}_{0}r)$. The potential satisfies Poisson's equation ${\nabla }^{2}V=-\rho /{ϵ}_0$, which in charge-free regions reduces to Laplace's equation ${\nabla }^{2}V=0$ — the subject of unit 10.01.02 pending.

Counterexamples to common slips

• Gauss's law is always true, but it is only useful for finding $\mathbf{E}$ when symmetry lets you pull $E$ out of the flux integral. For an arbitrary charge distribution with no symmetry, you must use the superposition integral instead. Gauss's law still holds, but the surface integral cannot be evaluated without already knowing $\mathbf{E}$.
• The charge in Gauss's law is the enclosed charge. A charge sitting outside the Gaussian surface contributes to $\mathbf{E}$ at every point on the surface, but its net flux through the surface is zero — every field line that enters also exits.
• A Gaussian surface is a mathematical tool, not a physical object. It can be placed anywhere, in any shape, even in vacuum. The law holds for every closed surface; the art is choosing one where symmetry simplifies the calculation.
• $\mathbf{E}$ inside a conductor in electrostatic equilibrium is zero, but this is a consequence of Gauss's law plus the property that free charges redistribute until the field vanishes — it is not an axiom. The Gaussian-surface argument for a conductor uses a surface just below the surface of the conductor; since $\mathbf{E}=0$ inside, the flux is zero, so the enclosed charge is zero, which means any excess charge on a conductor resides on its surface.

Key theorem with proof [Intermediate+]

Theorem (Gauss's law follows from Coulomb's law and the divergence theorem). Coulomb's law for a point charge $q$ at the origin, $\mathbf{E}(\mathbf{r})=(q/4\pi {ϵ}_0)\hat{\mathbf{r}}/r^2$, implies $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$.

Proof. For a point charge $q$ at the origin, compute $\nabla \cdot (\hat{\mathbf{r}}/r^2)$.

For $\mathbf{r}\ne 0$, a direct calculation gives $\nabla \cdot (\hat{\mathbf{r}}/r^2)=0$. The divergence vanishes everywhere except the origin. But the flux through a sphere of radius $R$ centred at the origin is

$${\oint }_S\frac{\hat{\mathbf{r}}}{r^2}\cdot d\mathbf{A}=\frac{1}{R^2}\cdot 4\pi R^2=4\pi .$$

Since the flux is $4\pi$ for any sphere surrounding the origin, and the divergence is zero everywhere else, the divergence must be a distribution concentrated at the origin. Writing $\nabla \cdot (\hat{\mathbf{r}}/r^2)=4\pi {\delta }^3(\mathbf{r})$ where ${\delta }^3$ is the three-dimensional Dirac delta, we recover

$$\nabla \cdot \mathbf{E}=\frac{q}{4\pi {ϵ}_0}\cdot 4\pi {\delta }^3(\mathbf{r})=\frac{q{\delta }^3(\mathbf{r})}{{ϵ}_0}=\frac{\rho (\mathbf{r})}{{ϵ}_0},$$

since for a point charge, $\rho (\mathbf{r})=q{\delta }^3(\mathbf{r})$. Superposition extends the result to arbitrary charge distributions. $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ follows. $■$

The Dirac delta is the rigorous way to express that $1/r^2$ has vanishing divergence away from the origin but nonzero flux through any surface enclosing it. Griffiths treats this in Ch. 1; Jackson makes the distributional statement explicit from the outset.

Worked example: infinite plane of charge

An infinite flat surface carries uniform surface charge density $\sigma$. Find $\mathbf{E}$.

By symmetry, $\mathbf{E}$ points perpendicular to the plane, with the same magnitude on both sides (reversing direction through the plane). Choose a Gaussian pillbox of cross-section area $A$ straddling the plane: the flat faces have area $A$ and the curved side contributes zero flux (the field is parallel to it). Gauss's law gives

$$E\cdot A+E\cdot A=\frac{\sigma A}{{ϵ}_0},$$

so $E=\sigma /(2{ϵ}_0)$. The field is uniform — it does not depend on distance from the plane. This is a consequence of the plane's infinite extent and translational symmetry; any real finite sheet deviates from this near its edges.

Exercises [Intermediate+]

Gauss's law as a Maxwell equation [Master]

Gauss's law $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ is the first of the four Maxwell equations. In the full time-dependent theory 10.04.01 pending, it remains unchanged — the divergence equation is exact even when fields vary in time. The companion equation $\nabla \cdot \mathbf{B}=0$ (no magnetic monopoles) is the magnetic analogue.

The electrostatic Maxwell system is two equations:

$$\nabla \cdot \mathbf{E}=\frac{\rho }{{ϵ}_0},\nabla \times \mathbf{E}=0.$$

The curl-free condition guarantees the existence of a scalar potential $V$ with $\mathbf{E}=-\nabla V$. Substituting into the divergence equation gives Poisson's equation ${\nabla }^{2}V=-\rho /{ϵ}_0$.

Helmholtz decomposition [Master]

Any sufficiently smooth vector field $\mathbf{F}$ on ${\mathbb{R}}^3$ that vanishes at infinity decomposes uniquely as

$$\mathbf{F}=-\nabla \Phi +\nabla \times \mathbf{A},$$

where $\Phi$ is a scalar potential and $\mathbf{A}$ is a vector potential. For the electrostatic field, the curl part is zero, so $\mathbf{E}=-\nabla V$ is the complete description. Gauss's law fixes the scalar potential via Poisson's equation. The Helmholtz theorem guarantees that specifying $\nabla \cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$ (with boundary conditions at infinity) determines $\mathbf{E}$ uniquely — which is why the two electrostatic Maxwell equations are sufficient.

The multipole expansion [Master]

For a localized charge distribution (charge confined to a finite region), the potential at large distances admits a series expansion in powers of $1/r$. Setting the origin inside the charge distribution and expanding $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$ for $r\gg r^{\prime }$:

$$V(\mathbf{r})=\frac{1}{4\pi {ϵ}_0}\left[\frac{Q}{r}+\frac{\mathbf{p}\cdot \hat{\mathbf{r}}}{r^2}+\frac{1}{2}\sum _{i,j}{Q}_{ij}\frac{{\hat{r}}_i{\hat{r}}_j}{r^3}+\cdots \right],$$

where $Q=\int \rho d^3{\mathbf{r}}^{\prime }$ is the total charge (monopole), $\mathbf{p}=\int {\mathbf{r}}^{\prime }\rho d^3{\mathbf{r}}^{\prime }$ is the dipole moment, and $Q_{ij}=\int (3r_i^{\prime }{r}_j^{\prime }-r^{\prime 2}{\delta }_{ij})\rho d^3{\mathbf{r}}^{\prime }$ is the quadrupole tensor. Each successive term falls off faster with distance.

If $Q\ne 0$, the monopole term dominates at large $r$ and the field is indistinguishable from a point charge. The dipole field ($\propto 1/r^3$) dominates when $Q=0$ — as for neutral molecules. The quadrupole and higher terms become important in nuclear physics and in the radiation fields of antennas.

Earnshaw's theorem [Master]

Theorem (Earnshaw, 1842). A collection of point charges cannot be maintained in stable static equilibrium by electrostatic forces alone.

Proof. In a charge-free region, $V$ satisfies Laplace's equation ${\nabla }^{2}V=0$. A function satisfying Laplace's equation can have no local maxima or minima in the interior of its domain (by the mean-value property: $V$ at any point equals the average of $V$ over any sphere centred at that point). Since $\mathbf{E}=-\nabla V$, a point of stable equilibrium for a positive test charge would require $V$ to have a local minimum, which is impossible. A negative test charge would require a local maximum, also impossible. Therefore, no stable equilibrium exists. $■$

Physical consequences: you cannot trap a charged particle using only static electric fields. Penning traps and Paul traps circumvent Earnshaw's theorem by using time-dependent fields or additional magnetic fields. The ion traps used in quantum computing 14.01.01 rely on this workaround.

Green's reciprocity theorem [Master]

Theorem. Let ${\rho }_1,{\rho }_2$ be two charge distributions, and $V_1,V_2$ the corresponding potentials (both satisfying ${\nabla }^{2}V=-\rho /{ϵ}_0$ with the same boundary conditions). Then

$$\int {\rho }_1{V}_2{d}^3\mathbf{r}=\int {\rho }_2{V}_1{d}^3\mathbf{r}.$$

Proof. Multiply Poisson's equation for $V_1$ by $V_2$ and integrate, then swap indices and subtract. The Laplacian terms yield $\int ({\nabla }^2{V}_1)V_2{d}^3\mathbf{r}-\int ({\nabla }^2{V}_2)V_1{d}^3\mathbf{r}=0$ by Green's second identity, provided the surface terms vanish (as they do with standard boundary conditions at infinity). The charge-density terms then give the stated identity. $■$

The reciprocity theorem is the electrostatic shadow of the self-adjointness of the Laplacian. It has powerful practical consequences: if you know the potential due to a charge $q$ at point $A$, you immediately know the potential at $A$ due to a charge at any other point $B$ — without solving a new boundary-value problem. It underpins the method of images for grounded conductors.

The Dirac delta and the divergence of $1/r^2$ [Master]

The identity $\nabla \cdot (\hat{\mathbf{r}}/r^2)=4\pi {\delta }^3(\mathbf{r})$ is the load-bearing piece connecting Coulomb's law to Gauss's law in differential form. The computation proceeds as follows.

For $\mathbf{r}\ne 0$, $\nabla \cdot (\hat{\mathbf{r}}/r^2)=0$ by direct calculation in spherical coordinates. The flux through any sphere centred at the origin is $4\pi$, computed as above. By the divergence theorem and the fact that the divergence is zero everywhere except the origin, the only way to reconcile these two facts is that $\nabla \cdot (\hat{\mathbf{r}}/r^2)$ is a distribution supported at the origin:

$$\nabla \cdot \left(\frac{\hat{\mathbf{r}}}{r^2}\right)=4\pi {\delta }^3(\mathbf{r}),$$

where ${\delta }^3(\mathbf{r})$ is the three-dimensional Dirac delta, defined by $\int {\delta }^3(\mathbf{r})d^3\mathbf{r}=1$ and ${\delta }^3(\mathbf{r})=0$ for $\mathbf{r}\ne 0$.

This is not merely a formal trick. Jackson (Ch. 1) treats it as the correct statement: the Coulomb field has a distributional divergence. The alternative, writing ${\nabla }^2(1/r)=-4\pi {\delta }^3(\mathbf{r})$, is the same statement in terms of the potential, since $\hat{\mathbf{r}}/r^2=-\nabla (1/r)$.

Connections [Master]

• Laplace equation and BVPs 10.01.02 pending is the direct continuation: in charge-free regions, $\nabla \cdot \mathbf{E}=0$ together with $\mathbf{E}=-\nabla V$ gives ${\nabla }^{2}V=0$. The method of images, separation of variables, and multipole expansions are the solution techniques for boundary-value problems that arise from Gauss's law.

• Biot-Savart and Ampere 10.02.01 pending is the magnetostatic analogue: $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$ plays the role that $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ plays here, and Ampere's law is the magnetic counterpart of Gauss's law.

• Full Maxwell equations 10.04.01 pending embed Gauss's law as the time-independent limit of the first Maxwell equation. In the full theory, $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ is exact for all times; what changes is that Faraday's law $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$ replaces the static $\nabla \times \mathbf{E}=0$.

• Atomic structure 14.01.01 applies the Coulomb potential $V=-e^2/(4\pi {ϵ}_{0}r)$ to the electron-nucleus system. The Bohr model computes orbital radii $r_n=n^2{a}_0$ and energies $E_n=-13.6\mathrm{eV}/n^2$ directly from the balance of Coulomb attraction and centripetal acceleration developed here. The quantum-mechanical hydrogen atom replaces classical orbits with wavefunctions but retains the same $1/r$ potential.

Historical & philosophical context [Master]

Coulomb reported his torsion-balance measurements of the electrostatic force law in 1785 (Mémoires de l'Académie Royale des Sciences), establishing the inverse-square law with the same $1/r^2$ dependence as Newton's law of gravitation (1687). The analogy between the two force laws — both central, both inverse-square, both additive by superposition — was noted immediately and drove much of 18th-century mathematical physics.

Gauss formulated the flux law in 1835 (published 1867 in the Werke), though the result was implicit in the superposition principle and the solid-angle arguments of Lagrange. The differential form $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ required the vector-calculus formalism developed by Heaviside, Gibbs, and Oliver Heaviside in the 1880s from the quaternionic framework of Hamilton.

Faraday's field concept (1830s–1850s) is the conceptual innovation that separates modern electromagnetism from the action-at-a-distance tradition of Coulomb and Ampere. Faraday thought in terms of lines of force — continuous curves filling space — rather than forces between distant particles. Maxwell translated Faraday's qualitative picture into quantitative field equations (1865), and the field concept became the foundation of classical field theory and, by extension, of general relativity and quantum field theory. Susskind's Special Relativity and Classical Field Theory (2017) uses the field concept as the starting point for the entire subject.

The experimental basis for Gauss's law has been tested to extraordinary precision. If Coulomb's law were $F\propto 1/r^{2+ϵ}$, then Gauss's law would fail. Laboratory experiments (Williams, Faller, and Hill, 1971) constrain $|ϵ|<10^{-16}$, making the inverse-square law one of the best-tested laws in physics. The null result is consistent with the photon being exactly massless; a massive photon would produce a Yukawa-type $e^{-\mu r}/r^2$ force law and violate Gauss's law at large distances.

Bibliography [Master]

Primary literature (cite when used; not all currently in reference/):

• Coulomb, C. A., "Premier mémoire sur l'électricité et le magnétisme," Mémoires de l'Académie Royale des Sciences (1785), 569–577; "Second mémoire…," 578–611. [Need to source — originator papers.]
• Gauss, C. F., Werke, Vol. 5 (Königliche Gesellschaft der Wissenschaften, Göttingen, 1867), §5. [Need to source.]
• Faraday, M., Experimental Researches in Electricity, 3 vols. (1839, 1844, 1855).
• Maxwell, J. C., "A dynamical theory of the electromagnetic field," Phil. Trans. Roy. Soc. 155 (1865), 459–512.
• Heaviside, O., Electromagnetic Theory, 3 vols. (1893, 1899, 1912).
• Earnshaw, S., "On the nature of the molecular forces which regulate the constitution of the luminiferous ether," Trans. Camb. Phil. Soc. 7 (1842), 97–112.
• Williams, E. R., Faller, J. E. & Hill, H. A., "New experimental test of Coulomb's law: a laboratory upper limit on the photon rest mass," Phys. Rev. Lett. 26 (1971), 721–724.
• Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge University Press, 2017).
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Zangwill, A., Modern Electrodynamics (Cambridge University Press, 2013).
• Susskind, L. & Friedman, A., Special Relativity and Classical Field Theory (Basic Books, 2017).
• Tong, D., Electromagnetism (DAMTP Cambridge lecture notes, §1 "Electrostatics").

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