Field–Potential Relation

Electric potential and electric field are two complementary ways of describing the same physical reality — the influence of electric charges on the space around them. While the electric field tells you the force per unit charge at a point, the electric potential tells you the potential energy per unit charge. These two quantities are deeply connected: the electric field is the negative gradient of the electric potential. This relationship is one of the most elegant results in electrostatics. Think of gravitational analogy: a ball on a hillside experiences a gravitational force that points downhill — in the direction of decreasing gravitational potential energy. The steeper the hill, the stronger the force. Electric fields behave in exactly the same way. They point from regions of high potential to regions of low potential, and their magnitude depends on how rapidly the potential changes with position. The 'steepness' of the potential landscape is what drives the field. Understanding the field–potential relation is crucial because it lets you calculate either quantity from the other. If you know the potential everywhere (often easier to calculate because it's a scalar), you can derive the vector field by differentiation. Conversely, integrating the field along a path gives you potential differences. This mathematical bridge between a scalar field and a vector field is a central theme in physics and engineering.

Key takeaways
  • Work and Potential Difference
  • Why the Negative Sign?
  • The Gradient in 3D
  • Equipotential Surfaces and Field Lines
  • Recovering Potential from the Field

Work and Potential Difference

When a charge q moves through an electric field, the work done by the field is W = qΔV, where ΔV is the potential difference between start and end points. But work is also W = F · d = qE · d (for a uniform field along displacement d). Equating these: qEd = qΔV, which gives E = ΔV/d. This already shows that the field equals the rate of change of potential with distance.

Why the Negative Sign?

The field points in the direction a positive charge naturally accelerates — from high potential to low potential. If potential decreases in the +x direction (ΔV is negative as x increases), then E_x is positive (pointing in +x). Mathematically, E_x = −dV/dx captures this: the field is the negative of the slope of V. The negative sign ensures field vectors point 'downhill' on the potential landscape.

The Gradient in 3D

In three dimensions, potential varies in all directions simultaneously. The gradient operator ∇ (del) computes the vector of partial derivatives: ∇V = (∂V/∂x)î + (∂V/∂y)ĵ + (∂V/∂z)k̂. This gradient vector points in the direction of steepest increase of V. The electric field is the negative of this gradient: E = −∇V. Each component satisfies E_x = −∂V/∂x, E_y = −∂V/∂y, E_z = −∂V/∂z.

Equipotential Surfaces and Field Lines

Surfaces on which V is constant are called equipotential surfaces. Since E = −∇V, and the gradient is always perpendicular to level surfaces, the electric field is always perpendicular to equipotential surfaces. This is why field lines cross equipotentials at right angles — a geometric consequence of the field–potential relation. No work is done moving a charge along an equipotential because E ⊥ displacement means E · ds = 0.

Recovering Potential from the Field

The inverse operation is integration: V(B) − V(A) = −∫_A^B E · ds. This line integral is path-independent in electrostatics (because the field is conservative), meaning it doesn't matter which route you take from A to B — the potential difference is always the same. This path-independence is guaranteed mathematically by the fact that ∇ × E = 0 in electrostatics.

The Key Equation

E⃗=−∇V=−(∂V∂xi^+∂V∂yj^+∂V∂zk^)\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)
  • \vec{E} — Electric Field (N/C or V/m): The vector field giving the force per unit positive charge at each point in space.
  • V — Electric Potential (Volt (V = J/C)): The electric potential energy per unit charge at a point; a scalar quantity.
  • \nabla — Gradient Operator (del) (m⁻¹): Vector differential operator; ∇V produces a vector pointing in the direction of steepest increase of V.
  • \partial V / \partial x — Partial Derivative of V (V/m): Rate of change of potential with respect to x, holding y and z constant.

Worked Example

Problem. The electric potential in a region of space is given by V(x, y) = 3x² − 2y (in volts, with x and y in metres). Find the electric field vector at the point (2 m, 1 m).

E⃗=−12 i^+2 j^  V/m,∣E⃗∣=148≈12.2  V/m\vec{E} = -12\,\hat{i} + 2\,\hat{j}\;\text{V/m}, \quad |\vec{E}| = \sqrt{148} \approx 12.2\;\text{V/m}

Answer. The electric field at (2 m, 1 m) is −12î + 2ĵ V/m, with magnitude √(144 + 4) ≈ 12.2 V/m.

Figure
Diagram
Electric Field Lines and Equipotential Surfaces (Point Charge)+V₁V₂V₃V₁ > V₂ > V₃E fieldlinesEquipotential surface (V = const)Electric field line (⊥ to equipotentials)Field lines always cross equipotential surfaces at right angles: E = −∇V
Origins · 1 minGeorge Green and how this was discovered
GG
George Green
1793–1841 · British

George Green was a self-taught mathematician and miller who introduced the concept of the 'potential function' in his 1828 essay on the theory of electricity and magnetism. He developed what are now called Green's theorems, which rigorously connect field quantities to their scalar potentials — laying the mathematical groundwork for the very relationship E = −∇V. His work was largely unrecognised until after his death, when Lord Kelvin republished it and recognised its profound importance. Green had almost no formal education and worked as a miller in Nottingham most of his life. He only enrolled at Cambridge University at age 40, just four years before his death, yet his self-published 1828 essay is considered a landmark in mathematical physics.

  • Electric Potential and Potential Energy — Understanding electric potential V is a prerequisite for the field–potential relation, since E = −∇V requires knowing what V represents physically.
  • Electric Field and Field Lines — The electric field E is the quantity produced by the gradient operation on V; understanding field lines helps visualise the field–potential relationship.
  • Gauss's Law — Gauss's Law is the complementary tool for finding E directly from charge distributions; combining it with E = −∇V leads to Poisson's equation, ∇²V = −ρ/ε₀.
  • Gradient and Directional Derivatives — The gradient operator ∇ is the mathematical engine of the field–potential relation; understanding it makes E = −∇V precise and calculable in 3D.