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Coulomb's Law

Field: Electrostatics

A law stating that the force between two electric charges is proportional to the product of the charges and inversely proportional to the square of the distance between them.

Sequence of Expressions

Definition

Electric Charge

\text{The electric charge } q \text{ is defined as the source term in the differential form of Gauss's Law, relating the electric flux } \Phi_E \text{ through a closed surface } \partial V \text{ to the enclosed charge } Q \text{ within the volume } V. \text{Mathematically, this is expressed as:}\n\n∮∂VE⃗⋅dA⃗=Qϵ0\oint_{\partial V} \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0} \n\text{where } \vec{E} \text{ is the electric field, } \epsilon_0 \text{ is the permittivity of free space, and } Q \text{ is the total charge enclosed.}\n\n\text{Furthermore, the charge } q \text{ of a localized particle is quantized, meaning it must be an integer multiple of the elementary charge } e:\n\nq=nefor n∈Zq = n e \quad \text{for } n \in \mathbb{Z} \n\text{This quantization arises from the underlying symmetry structure of the vacuum and the coupling constant } e \text{ (the fundamental unit of charge).}
Definition

Point Charge

Let QQ be the total charge, and let r⃗0∈R3\vec{r}_0 \in \mathbb{R}^3 be the location of the charge. The point charge is mathematically modeled by the charge density ρ(r⃗)\rho(\vec{r}) defined as:\n\nρ(r⃗)=Qδ(r⃗−r⃗0)\rho(\vec{r}) = Q \delta(\vec{r} - \vec{r}_0)\n\nwhere δ(r⃗−r⃗0)\delta(\vec{r} - \vec{r}_0) is the three-dimensional Dirac delta function, satisfying ∫R3δ(r⃗−r⃗0)d3r=1\int_{\mathbb{R}^3} \delta(\vec{r} - \vec{r}_0) d^3r = 1.\n\nFrom this density, the electric potential V(r⃗)V(\vec{r}) at any point r⃗≠r⃗0\vec{r} \neq \vec{r}_0 is given by the integral of the potential kernel 14πϵ01∣r⃗−r⃗0∣\frac{1}{4\pi\epsilon_0} \frac{1}{|\vec{r} - \vec{r}_0|}:\n\nV(r⃗)=14πϵ0∫R3ρ(r⃗′)∣r⃗−r⃗′∣d3r′=14πϵ0Q∣r⃗−r⃗0∣V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathbb{R}^3} \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} d^3r' = \frac{1}{4\pi\epsilon_0} \frac{Q}{|\vec{r} - \vec{r}_0|}\n\nThe resulting electric field E⃗(r⃗)\vec{E}(\vec{r}) is the negative gradient of the potential:\n\nE⃗(r⃗)=−∇V(r⃗)=14πϵ0Q(r⃗−r⃗0)∣r⃗−r⃗0∣3\vec{E}(\vec{r}) = -\nabla V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{Q (\vec{r} - \vec{r}_0)}{|\vec{r} - \vec{r}_0|^3}\n\nThis formulation rigorously defines the point charge as a singular source term in the Poisson equation, ∇2V=1ϵ0ρ\nabla^2 V = \frac{1}{\epsilon_0} \rho, whose solution is the Coulomb potential.
Let q1q_1 and q2q_2 be two point charges located at positions r⃗1\vec{r}_1 and r⃗2\vec{r}_2 in a vacuum characterized by the permittivity ϵ0\epsilon_0. The electric field E⃗1\vec{E}_1 generated by q1q_1 at the position r⃗2\vec{r}_2 is given by the fundamental solution to Laplace's equation in electrostatics:\n\nE⃗1(r⃗2)=14πϵ0q1∣r⃗2−r⃗1∣2u^12\vec{E}_1(\vec{r}_2) = \frac{1}{4\pi\epsilon_0} \frac{q_1}{|\vec{r}_2 - \vec{r}_1|^2} \hat{u}_{12}\n\nwhere u^12=r⃗2−r⃗1∣r⃗2−r⃗1∣\hat{u}_{12} = \frac{\vec{r}_2 - \vec{r}_1}{|\vec{r}_2 - \vec{r}_1|} is the unit vector pointing from r⃗1\vec{r}_1 to r⃗2\vec{r}_2.\n\nThe electrostatic force F⃗21\vec{F}_{21} exerted on q2q_2 by q1q_1 is defined by F⃗21=q2E⃗1(r⃗2)\vec{F}_{21} = q_2 \vec{E}_1(\vec{r}_2). By comparing this definition with the standard form of Coulomb's Law, the Force Constant kk is rigorously defined as the proportionality factor relating the magnitude of the force to the product of the charges and the inverse square of the separation distance:\n\nF⃗21=kq1q2∣r⃗2−r⃗1∣2u^12\vec{F}_{21} = k \frac{q_1 q_2}{|\vec{r}_2 - \vec{r}_1|^2} \hat{u}_{12}\n\nEquating the two expressions yields the formal definition of kk:\n\nk=14πϵ0k = \frac{1}{4\pi\epsilon_0}
In the vacuum (r⃗∈R3\vec{r} \in \mathbb{R}^3, t=0t=0), the electric displacement field D⃗\vec{D} is defined by the linear constitutive relation D⃗=ϵ0E⃗\vec{D} = \epsilon_0 \vec{E}, where E⃗\vec{E} is the electric field. The constant ϵ0\epsilon_0 is rigorously determined by the requirement that the electric field energy density ueu_e must satisfy the Hamiltonian density formulation derived from the Lagrangian density L\mathcal{L}: ue=12D⃗⋅E⃗=12ϵ0E⃗2u_e = \frac{1}{2} \vec{D} \cdot \vec{E} = \frac{1}{2} \epsilon_0 \vec{E}^2 Furthermore, ϵ0\epsilon_0 is related to the vacuum permeability μ0\mu_0 and the speed of light cc by the fundamental identity derived from Maxwell's equations: ϵ0=1μ0c2\epsilon_0 = \frac{1}{\mu_0 c^2}
Consider a system of two point charges, q1q_1 and q2q_2, situated at positions r⃗1\vec{r}_1 and r⃗2\vec{r}_2, respectively, within a vacuum characterized by the permittivity ϵ0\epsilon_0. The separation vector is defined as r⃗=r⃗2−r⃗1\vec{r} = \vec{r}_2 - \vec{r}_1, and the magnitude of the separation is r=∣∣r⃗∣∣r = ||\vec{r}||.\n\nThe electric field E⃗1\vec{E}_1 generated by q1q_1 at the location r⃗2\vec{r}_2 is given by the fundamental solution to Poisson's equation for a point charge:\nE⃗1(r⃗2)=14πϵ0q1r2r^=14πϵ0q1(r⃗2−r⃗1)2(r⃗2−r⃗1)\vec{E}_1(\vec{r}_2) = \frac{1}{4\pi\epsilon_0} \frac{q_1}{r^2} \hat{r} = \frac{1}{4\pi\epsilon_0} \frac{q_1}{(\vec{r}_2 - \vec{r}_1)^2} (\vec{r}_2 - \vec{r}_1) \n\nThe electrostatic force F⃗12\vec{F}_{12} exerted by q1q_1 on q2q_2 is then determined by the interaction of q2q_2 with the field E⃗1\vec{E}_1: \nF⃗12=q2E⃗1(r⃗2)=14πϵ0q1q2r2r^=14πϵ0q1q2∣∣r⃗2−r⃗1∣∣2(r⃗2−r⃗1)\vec{F}_{12} = q_2 \vec{E}_1(\vec{r}_2) = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{||\vec{r}_2 - \vec{r}_1||^2} (\vec{r}_2 - \vec{r}_1) \n\nThis vector formulation encapsulates the scalar magnitude relationship, ∣F⃗12∣=k∣q1q2∣r2|\vec{F}_{12}| = k \frac{|q_1 q_2|}{r^2}, while maintaining directional rigor based on the separation vector r⃗\vec{r}.
Consider a system of NN point charges, qiq_i, located at positions r⃗i∈R3\vec{r}_i \in \mathbb{R}^3. The electrostatic force F⃗i\vec{F}_i acting on the charge qiq_i due to the presence of all other charges qjq_j (j≠ij \neq i) is defined by the principle of superposition. The force is given by the vector summation:\n\nF⃗i=∑j=1,j≠iNF⃗ji\vec{F}_i = \sum_{j=1, j \neq i}^{N} \vec{F}_{ji} \n\nwhere F⃗ji\vec{F}_{ji} is the force exerted by qjq_j on qiq_i. This pairwise force is rigorously defined using Coulomb's Law in vector form:\n\nF⃗ji=kqjqi∣r⃗i−r⃗j∣2(r⃗i−r⃗j)∣r⃗i−r⃗j∣=kqjqi∣r⃗i−r⃗j∣3(r⃗i−r⃗j)\vec{F}_{ji} = k \frac{q_j q_i}{|\vec{r}_i - \vec{r}_j|^2} \frac{(\vec{r}_i - \vec{r}_j)}{|\vec{r}_i - \vec{r}_j|} = k \frac{q_j q_i}{|\vec{r}_i - \vec{r}_j|^3} (\vec{r}_i - \vec{r}_j) \n\nHere, k=14πϵ0k = \frac{1}{4\pi\epsilon_0} is the Coulomb constant, ϵ0\epsilon_0 is the permittivity of free space, and r⃗i−r⃗j\vec{r}_i - \vec{r}_j is the displacement vector pointing from qjq_j to qiq_i.
Consider two point charges, q1q_1 and q2q_2, located at positions r⃗1\vec{r}_1 and r⃗2\vec{r}_2 in R3\mathbb{R}^3. Let the separation vector be r⃗=r⃗2−r⃗1\vec{r} = \vec{r}_2 - \vec{r}_1, and the distance be r=∣r⃗∣r = |\vec{r}|. The electrostatic force F⃗21\vec{F}_{21} exerted on q2q_2 by q1q_1 is rigorously defined by Coulomb's Law:\n\nF⃗21=kq1q2r2r^=kq1q2∣r⃗∣2r⃗∣r⃗∣\vec{F}_{21} = k \frac{q_1 q_2}{r^2} \hat{r} = k \frac{q_1 q_2}{|\vec{r}|^2} \frac{\vec{r}}{|\vec{r}|} \n\nwhere k=14πϵ0k = \frac{1}{4\pi\epsilon_0} is Coulomb's constant, and r^=r⃗r\hat{r} = \frac{\vec{r}}{r} is the unit vector pointing from q1q_1 to q2q_2. The scalar magnitude of this force, F=∣F⃗21∣F = |\vec{F}_{21}|, explicitly demonstrates the Inverse Square Law:\n\nF=k∣q1q2∣r2=Cr2F = k \frac{|q_1 q_2|}{r^2} = \frac{C}{r^2} \n\nwhere C=k∣q1q2∣C = k |q_1 q_2| is a constant determined by the product of the charges and the permittivity of free space.
Consider two point charges q1q_1 and q2q_2 located at positions r⃗1\vec{r}_1 and r⃗2\vec{r}_2 in R3\mathbb{R}^3. The electrostatic potential energy UU associated with this interaction is a scalar field defined by U(r⃗12)=kq1q2∣r⃗12∣U(\vec{r}_{12}) = k \frac{q_1 q_2}{|\vec{r}_{12}|}, where r⃗12=r⃗2−r⃗1\vec{r}_{12} = \vec{r}_2 - \vec{r}_1 and kk is Coulomb's constant. The force F⃗12\vec{F}_{12} exerted by q1q_1 on q2q_2 is a conservative force, and thus its vector nature is rigorously defined by the negative gradient of the potential energy function: F⃗12=−∇U(r⃗12)\vec{F}_{12} = -\nabla U(\vec{r}_{12})\nIn Cartesian coordinates, this yields the vector expression:\nF⃗12=−kq1q2r⃗12∣r⃗12∣3\vec{F}_{12} = -k q_1 q_2 \frac{\vec{r}_{12}}{|\vec{r}_{12}|^3}
Let S={r⃗1,r⃗2,…,r⃗N}S = \{\vec{r}_1, \vec{r}_2, \dots, \vec{r}_N\} be a finite set of positions in R3\mathbb{R}^3, and let qiq_i be the point charge located at r⃗i\vec{r}_i. We consider the electric field E⃗(r⃗)\vec{E}(\vec{r}) at an observation point r⃗∉S\vec{r} \notin S. The electric field E⃗i(r⃗)\vec{E}_i(\vec{r}) generated by the isolated charge qiq_i is given by Coulomb's Law: E⃗i(r⃗)=kqi∣r⃗−r⃗i∣2(r⃗−r⃗i)∣r⃗−r⃗i∣\vec{E}_i(\vec{r}) = k \frac{q_i}{|\vec{r} - \vec{r}_i|^2} \frac{(\vec{r} - \vec{r}_i)}{|\vec{r} - \vec{r}_i|}. The Superposition Principle asserts that the total electric field E⃗(r⃗)\vec{E}(\vec{r}) is the vector sum of the individual fields: \n\nE⃗(r⃗)=∑i=1NE⃗i(r⃗)=∑i=1Nkqi∣r⃗−r⃗i∣3(r⃗−r⃗i)\vec{E}(\vec{r}) = \sum_{i=1}^{N} \vec{E}_i(\vec{r}) = \sum_{i=1}^{N} k \frac{q_i}{|\vec{r} - \vec{r}_i|^3} (\vec{r} - \vec{r}_i) \n\nFurthermore, since the electric field is a conservative vector field, the total potential V(r⃗)V(\vec{r}) is also additive: \n\nV(r⃗)=∑i=1NVi(r⃗)=∑i=1Nkqi∣r⃗−r⃗i∣V(\vec{r}) = \sum_{i=1}^{N} V_i(\vec{r}) = \sum_{i=1}^{N} k \frac{q_i}{|\vec{r} - \vec{r}_i|}
Consider a system of NN point charges, q1,q2,…,qNq_1, q_2, \dots, q_N, situated at positions r⃗1,r⃗2,…,r⃗N\vec{r}_1, \vec{r}_2, \dots, \vec{r}_N in R3\mathbb{R}^3. The force F⃗i\vec{F}_i acting on the charge qiq_i due to all other charges qjq_j (j≠ij \neq i) is given by the superposition of Coulomb forces. Electrostatic Equilibrium is achieved if and only if the net force on every charge is zero. Mathematically, this condition is stated as:\n∑j=1,j≠iNF⃗ji=0⃗ for all i=1,2,…,N\sum_{j=1, j \neq i}^{N} \vec{F}_{ji} = \vec{0} \text{ for all } i = 1, 2, \dots, N \nwhere F⃗ji\vec{F}_{ji} is the force exerted by qjq_j on qiq_i, defined by:\nF⃗ji=keqiqj∣r⃗i−r⃗j∣3(r⃗i−r⃗j)\vec{F}_{ji} = k_e \frac{q_i q_j}{|\vec{r}_i - \vec{r}_j|^3} (\vec{r}_i - \vec{r}_j) \nHere, kek_e is the Coulomb constant, and the condition implies that the system configuration (r⃗1,…,r⃗N)(\vec{r}_1, \dots, \vec{r}_N) is a critical point of the potential energy function U(r⃗1,…,r⃗N)U(\vec{r}_1, \dots, \vec{r}_N) with respect to the coordinates.