Coulomb's law

Coulomb’s law gives the force between two stationary point charges $q_1$ and $q_2$ separated by a Distance vector ${\mathbf{R}}_{12}$:

${\mathbf{F}}_{12}=\frac{q_1{q}_2}{4\pi ϵ|{\mathbf{R}}_{12}{|}^2}{\hat{\mathbf{R}}}_{12}\text{(N)},$

where ${\hat{\mathbf{R}}}_{12}$ is the unit vector from $q_1$ to $q_2$. Like-sign charges repel (force along $+{\hat{\mathbf{R}}}_{12}$), opposite-sign attract. The force decays as $1/R^2$, an Inverse-square field — the same fall-off as gravity.

In free space the constant in the denominator is $4\pi {ϵ}_0$, with ${ϵ}_0=8.854\times 10^{-12}$ F/m. In a Dielectric medium replace ${ϵ}_0$ with $ϵ={ϵ}_r{ϵ}_0$, where ${ϵ}_r$ is the medium’s relative permittivity.

Field form

Dividing by the charge $q^{\prime }$ being acted on gives the Electric field of a single source charge $q$ at distance $R$:

$\mathbf{E}=\hat{\mathbf{R}}\frac{q}{4\pi ϵR^2}\text{(V/m)}.$

This is the “field-first” statement: $q$ sets up $\mathbf{E}$ in all of space; a test charge $q^{\prime }$ placed at any point feels $\mathbf{F}=q^{\prime }\mathbf{E}$.

Multiple charges: superposition

The field obeys superposition. For two charges $q_1,q_2$ at positions ${\mathbf{R}}_1,{\mathbf{R}}_2$, the field at observation point $\mathbf{R}$ is

$\mathbf{E}(\mathbf{R})=\frac{1}{4\pi ϵ}\left[\frac{q_1(\mathbf{R}-{\mathbf{R}}_1)}{|\mathbf{R}-{\mathbf{R}}_1{|}^3}+\frac{q_2(\mathbf{R}-{\mathbf{R}}_2)}{|\mathbf{R}-{\mathbf{R}}_2{|}^3}\right].$

The vector form $(\mathbf{R}-{\mathbf{R}}_i)/|\mathbf{R}-{\mathbf{R}}_i{|}^3$ packages the $\hat{\mathbf{R}}/R^2$ behavior — direction and inverse-square magnitude — into a single expression that handles the geometry automatically.

Worked example

Two point charges $q_1=2\times 10^{-5}$ C at $(1,3,-1)$ and $q_2=-4\times 10^{-5}$ C at $(-3,1,-2)$ in free space. Find $\mathbf{E}$ at $(3,1,-2)$.

Position and distance vectors:

• $\mathbf{R}-{\mathbf{R}}_1=(3-1)\hat{\mathbf{x}}+(1-3)\hat{\mathbf{y}}+(-2-(-1))\hat{\mathbf{z}}=2\hat{\mathbf{x}}-2\hat{\mathbf{y}}-\hat{\mathbf{z}}$. Magnitude $=\sqrt{4+4+1}=3$.
• $\mathbf{R}-{\mathbf{R}}_2=(3-(-3))\hat{\mathbf{x}}+0\hat{\mathbf{y}}+0\hat{\mathbf{z}}=6\hat{\mathbf{x}}$. Magnitude $=6$.

Substituting: $\mathbf{E}=\frac{1}{4\pi {ϵ}_0}\left[\frac{2\times 10^{-5}(2\hat{\mathbf{x}}-2\hat{\mathbf{y}}-\hat{\mathbf{z}})}{3^3}+\frac{-4\times 10^{-5}(6\hat{\mathbf{x}})}{6^3}\right].$

This evaluates numerically; the structure shown is the key takeaway: vector form makes multi-charge problems mechanical.

Why Coulomb’s law looks the way it does

The $1/R^2$ fall-off is a geometric consequence of “field lines spread out.” If the total electric flux $q/ϵ$ emanating from a point charge is uniformly distributed over a sphere of radius $R$, the flux density per unit area is $q/(4\pi ϵR^2)$ — exactly $|\mathbf{E}|$. This is the seed of Gauss’s law, which generalizes Coulomb’s law to arbitrary charge distributions.

In a region with dielectric polarization, the ${ϵ}_r$ factor reduces the field — bound charges in the medium partially cancel the source charge’s field. Same Coulomb’s law applies, just with $ϵ$ in place of ${ϵ}_0$.

From Gauss’s law to Coulomb’s law

Coulomb’s law and Gauss’s law are mathematically equivalent for a point charge plus spherical symmetry. The Gauss → Coulomb direction:

For a point charge $q$ in free space, spherical symmetry forces $\mathbf{E}=E_R(R)\hat{\mathbf{R}}$ — purely radial, depending only on $R$. Apply Gauss’s law to a sphere of radius $R$ centered on the charge:

${\oint }_S\mathbf{D}\cdot d\mathbf{s}=E_R{ϵ}_0\cdot 4\pi R^2=q.$

Solving for $E_R$:

$E_R=\frac{q}{4\pi {ϵ}_0{R}^2},$

which is exactly Coulomb’s law. The two formulations are different ways of writing the same physics: Coulomb’s law states the field of a single charge directly; Gauss’s law states the flux through any closed surface. Choose Coulomb when you have a few point charges and want the field at a specific point; choose Gauss when you have symmetry and want to avoid integration.

In context

Coulomb’s law is the static, point-charge starting point. From it everything else in electrostatics follows: Electric potential (integrate $\mathbf{E}$ along a path), Gauss’s law (the differential / integral restatement), Capacitance (geometry of two conductors). In dynamics, Coulomb’s force is replaced by the Lorentz force when charges move and magnetic fields are present.