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Electric Potential

Field: Electrostatics

The amount of work needed to move a unit of electric charge from a reference point to a specific point in an electric field.

Sequence of Expressions

In the context of a conservative force field F⃗=qE⃗\vec{F} = q\vec{E}, where qq is the test charge, the electric potential ϕ(r⃗)\phi(\vec{r}) at a point r⃗\vec{r} is defined as the negative line integral of the electric field E⃗\vec{E} along any differentiable path CC connecting a reference point r⃗0\vec{r}_0 to r⃗\vec{r}. Mathematically, this is expressed as:\n\nϕ(r⃗)=−∫CE⃗⋅dl⃗=−∫r⃗0r⃗E⃗⋅dl⃗\phi(\vec{r}) = -\int_{C} \vec{E} \cdot d\vec{l} = -\int_{\vec{r}_0}^{\vec{r}} \vec{E} \cdot d\vec{l} \n\nSince E⃗\vec{E} is conservative, the integral is path-independent. Furthermore, the electric field E⃗\vec{E} is related to the potential ϕ\phi via the negative gradient operator ∇\nabla: \n\nE⃗(r⃗)=−∇ϕ(r⃗)\vec{E}(\vec{r}) = -\nabla\phi(\vec{r})\n\nThis relationship implies that the potential ϕ\phi must satisfy Laplace's equation in charge-free regions (ρ=0\rho=0):\n\n∇2ϕ=0\nabla^2\phi = 0
The electric field E⃗(r⃗)\vec{E}(\vec{r}) at a point r⃗\vec{r} in a region of space containing a charge distribution ρ(r⃗′)\rho(\vec{r}') is formally defined as the limit of the force F⃗\vec{F} exerted by the source charges on an infinitesimally small test charge q0q_0 placed at r⃗\vec{r}, normalized by q0q_0. Mathematically, this is expressed as:\n\nE⃗(r⃗)=lim⁡q0→0F⃗source→q0q0\vec{E}(\vec{r}) = \lim_{q_0 \to 0} \frac{\vec{F}_{source \to q_0}}{q_0} \n\nIn the context of a conservative field derived from a scalar potential ϕ(r⃗)\phi(\vec{r}), the electric field is rigorously defined as the negative gradient of the potential energy per unit charge:\n\nE⃗(r⃗)=−∇ϕ(r⃗)\vec{E}(\vec{r}) = -\nabla \phi(\vec{r})\n\nWhere ϕ(r⃗)\phi(\vec{r}) is the electrostatic potential, which itself is defined by the volume integral over the charge density ρ(r⃗′)\rho(\vec{r}'):\n\nϕ(r⃗)=14πϵ0∫ρ(r⃗′)∣r⃗−r⃗′∣d3r′\phi(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vec{r}')}{\left|\vec{r} - \vec{r}'\right|} d^3r'
Let qq be a test charge moving along a path CC parameterized by r⃗(t)\vec{r}(t) from an initial point AA to a final point BB. The electric force acting on qq is F⃗E=qE⃗\vec{F}_E = q\vec{E}, where E⃗\vec{E} is the electric field. Since E⃗\vec{E} is derived from a scalar potential ϕ\phi such that E⃗=−∇ϕ\vec{E} = -\nabla\phi, the force is conservative: F⃗E=−q∇ϕ\vec{F}_E = -q\nabla\phi. The work done WABW_{AB} by the electric force is defined by the line integral:\n\nWAB=∫CF⃗E⋅dr⃗=∫C(−q∇ϕ)⋅dr⃗W_{AB} = \int_C \vec{F}_E \cdot d\vec{r} = \int_C (-q\nabla\phi) \cdot d\vec{r}\n\nBy the Fundamental Theorem of Calculus for Line Integrals, this simplifies to:\n\nWAB=−qϕ(B)+qϕ(A)=q(ϕ(A)−ϕ(B))W_{AB} = -q\phi(B) + q\phi(A) = q(\phi(A) - \phi(B))\n\nFurthermore, the change in potential energy ΔU\Delta U is defined as ΔU=UB−UA=q(ϕ(B)−ϕ(A))\Delta U = U_B - U_A = q(\phi(B) - \phi(A)). Equating the work done to the negative change in potential energy yields the Work-Energy Theorem in Electrostatics:\n\nWAB=−ΔU=q(ϕ(A)−ϕ(B))W_{AB} = -\Delta U = q(\phi(A) - \phi(B))
Let ϕ(r⃗)\phi(\vec{r}) be the scalar electric potential function defined over a region Ω⊂R3\Omega \subset \mathbb{R}^3, such that the electric field E⃗\vec{E} is derived from it via the gradient relationship E⃗=−∇ϕ\vec{E} = -\nabla \phi. The potential difference ΔV\Delta V between two points AA and BB is rigorously defined as the difference in the potential function evaluated at these points: ΔV=ϕ(B)−ϕ(A)\Delta V = \phi(B) - \phi(A). Alternatively, and equivalently, ΔV\Delta V is defined by the negative line integral of the electric field along any continuous path CC connecting AA to BB: \n\nΔV=−∫ABE⃗⋅dl⃗\Delta V = -\int_{A}^{B} \vec{E} \cdot d\vec{l} \n\nDue to the conservative nature of the electric field, this integral is path-independent, meaning that for any two paths C1C_1 and C2C_2 from AA to BB, the following equality holds:\n\n∫C1E⃗⋅dl⃗=∫C2E⃗⋅dl⃗\int_{C_1} \vec{E} \cdot d\vec{l} = \int_{C_2} \vec{E} \cdot d\vec{l}
In a domain Ω⊂R3\Omega \subset \mathbb{R}^3 where the electric potential ϕ(r⃗)\phi(\vec{r}) is defined, and assuming the medium is linear and isotropic, the governing relationship between the potential and the charge density ρ(r⃗)\rho(\vec{r}) is established by the Poisson equation. This equation is derived from Gauss's Law in differential form (∇⋅E⃗=ρ/ϵ0\nabla \cdot \vec{E} = \rho/\epsilon_0) and the definition of the electric field (E⃗=−∇ϕ\vec{E} = -\nabla \phi). The resulting partial differential equation is:\n\n∇2ϕ(r⃗)=−1ϵ0ρ(r⃗)\nabla^2 \phi(\vec{r}) = -\frac{1}{\epsilon_0} \rho(\vec{r})\n\nwhere ∇2\nabla^2 is the Laplacian operator, defined in Cartesian coordinates as ∇2=∂2∂x2+∂2∂y2+∂2∂z2\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}. The equation holds subject to appropriate boundary conditions on the boundary ∂Ω\partial \Omega, such as Dirichlet (ϕ∣∂Ω=f\phi|\partial \Omega = f) or Neumann (∂ϕ∂n∣∂Ω=g\frac{\partial \phi}{\partial n}|\partial \Omega = g). Here, ϵ0\epsilon_0 is the permittivity of free space, and ρ(r⃗)\rho(\vec{r}) is the volume charge density.
In the domain Ω⊂R3\Omega \subset \mathbb{R}^3, let V(r⃗)V(\vec{r}) be a scalar potential field, V:Ω→RV: \Omega \to \mathbb{R}, and let E⃗(r⃗)\vec{E}(\vec{r}) be the associated electric field vector field. The relationship is defined by the negative gradient operation, which quantifies the rate of change of the potential in the direction of steepest descent (the direction of the field). Formally, the electric field E⃗\vec{E} is the negative gradient of the electric potential VV: \n\nE⃗=−∇V\vec{E} = -\nabla V \n\nIn Cartesian coordinates, this relationship expands to:\n\nE⃗=−(∂V∂xi^+∂V∂yj^+∂V∂zk^)\vec{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \n\nThis implies that the field E⃗\vec{E} is a conservative vector field, satisfying the condition ∇×E⃗=0\nabla \times \vec{E} = 0, which is a direct consequence of VV being a scalar potential function.
Let Ω⊂R3\Omega \subset \mathbb{R}^3 be a simply connected domain. A vector field E⃗:Ω→R3\vec{E}: \Omega \to \mathbb{R}^3 is defined as a conservative electric field if and only if there exists a scalar potential function ϕ:Ω→R\phi: \Omega \to \mathbb{R} (the electric potential) such that E⃗=−∇ϕ\vec{E} = -\nabla\phi. Equivalently, this condition is characterized by the vanishing of the curl of E⃗\vec{E}: \n\n∇×E⃗=0⃗in Ω\nabla \times \vec{E} = \vec{0} \quad \text{in } \Omega \n\nFurthermore, if E⃗\vec{E} is sufficiently smooth (i.e., E⃗∈C1(Ω)\vec{E} \in C^1(\Omega)), the potential ϕ\phi can be found by integrating the line integral along any path CC from a reference point AA to a point BB: \n\nϕ(B)−ϕ(A)=−∫ABE⃗⋅dl⃗\phi(B) - \phi(A) = -\int_{A}^{B} \vec{E} \cdot d\vec{l} \n\nThis implies that the line integral is path-independent, which is the fundamental physical definition of a conservative field.
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 (ϵ0\epsilon_0). The electric potential V1V_1 generated by q1q_1 at the location of q2q_2 is defined by the scalar field:\n\nV1(r⃗2)=14πϵ0q1∣∣r⃗2−r⃗1∣∣V_1(\vec{r}_2) = \frac{1}{4\pi\epsilon_0} \frac{q_1}{||\vec{r}_2 - \vec{r}_1||}\n\nThe potential energy UU of the system is the product of the charges and the potential difference: U=q2V1(r⃗2)U = q_2 V_1(\vec{r}_2).\n\nCoulomb's Law is then expressed as the negative gradient of the potential energy, yielding the force F⃗12\vec{F}_{12} exerted by q1q_1 on q2q_2:\n\nF⃗12=−∇U=−q2∇V1(r⃗2)=14πϵ0q1q2∣∣r⃗2−r⃗1∣∣3(r⃗2−r⃗1)\vec{F}_{12} = -\nabla U = -q_2 \nabla V_1(\vec{r}_2) = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{||\vec{r}_2 - \vec{r}_1||^3} (\vec{r}_2 - \vec{r}_1)\n\nAlternatively, defining the separation vector r⃗=r⃗2−r⃗1\vec{r} = \vec{r}_2 - \vec{r}_1 and the distance r=∣∣r⃗∣∣r = ||\vec{r}||, the force vector is:\n\nF⃗12=14πϵ0q1q2r2r^=14πϵ0q1q2r3r⃗\vec{F}_{12} = \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}{r^3} \vec{r}
Let E⃗\vec{E} be the electric field vector field defined over a region V⊂R3V \subset \mathbb{R}^3, and let ρe\rho_e be the volume charge density. The fundamental statement of Gauss's Law in differential form is:\n\n∇⋅E⃗=ρeϵ0\nabla \cdot \vec{E} = \frac{\rho_e}{\epsilon_0} \n\nThis relationship holds everywhere in vacuum or linear dielectric media, where ∇⋅\nabla \cdot is the divergence operator, ρe\rho_e is the charge density, and ϵ0\epsilon_0 is the permittivity of free space. \n\nAlternatively, using the Divergence Theorem, the flux ΦE\Phi_E through any closed surface S=∂VS = \partial V enclosing a volume VV is given by:\n\nΦE=∮SE⃗⋅dA⃗=Qencϵ0\Phi_E = \oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0} \n\nwhere dA⃗d\vec{A} is the differential area vector pointing outward from SS, and QencQ_{enc} is the total enclosed charge, defined by the volume integral: Qenc=∫VρedVQ_{enc} = \int_V \rho_e dV.
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, where qiq_i is the point charge located at r⃗i∈S\vec{r}_i \in S. The electric field E⃗i(r⃗)\vec{E}_i(\vec{r}) generated by the isolated charge qiq_i at the observation point r⃗\vec{r} is given by Coulomb's Law: E⃗i(r⃗)=kqi∣r⃗−r⃗i∣3(r⃗−r⃗i)\vec{E}_i(\vec{r}) = k \frac{q_i}{|\vec{r} - \vec{r}_i|^3} (\vec{r} - \vec{r}_i). The Electrostatic Principle of Superposition asserts that the total electric field E⃗total(r⃗)\vec{E}_{total}(\vec{r}) at r⃗\vec{r} is the vector sum of the fields generated by each charge: E⃗total(r⃗)=∑i=1NE⃗i(r⃗)=k∑i=1Nqi∣r⃗−r⃗i∣3(r⃗−r⃗i)\vec{E}_{total}(\vec{r}) = \sum_{i=1}^{N} \vec{E}_i(\vec{r}) = k \sum_{i=1}^{N} \frac{q_i}{|\vec{r} - \vec{r}_i|^3} (\vec{r} - \vec{r}_i). Furthermore, the total electric potential V(r⃗)V(\vec{r}) is the scalar sum: V(r⃗)=∑i=1NVi(r⃗)=∑i=1Nkqi∣r⃗−r⃗i∣V(\vec{r}) = \sum_{i=1}^{N} V_i(\vec{r}) = \sum_{i=1}^{N} \frac{k q_i}{|\vec{r} - \vec{r}_i|}. This linearity property holds due to the underlying structure of the electric field as a conservative vector field derived from a scalar potential.