Electric Fields and Equipotentials

Voltage around charges is a function of position. Its level curves are equipotential lines, and the electric field always crosses them at right angles.

How it works

Level curves and gradients

In one sentence

The potential of point charges is $V = \sum kq_i/r_i$ ; equipotentials are its level curves, and the field $\mathbf E = -\nabla V$ crosses them at right angles.

General formula

{\htmlData{sym=0}{V}}(x, y) = \frac{{\htmlData{sym=1}{k}} {\htmlData{sym=2}{q_1}}}{{\htmlData{sym=4}{r_1}}} + \frac{{\htmlData{sym=1}{k}} {\htmlData{sym=3}{q_2}}}{{\htmlData{sym=5}{r_2}}}, \qquad {\htmlData{sym=6}{\mathbf{E}}} = -\nabla {\htmlData{sym=0}{V}}

$V$ adds each charge's influence, which falls off like $1/r$ . The field points downhill on $V$ , perpendicular to every level curve.

What the symbols mean

$V$
Electric potential
Energy per unit charge at a point, like height on a hill.
$k$
Coulomb's constant
Sets the strength of electric forces, about 8.99 × 10⁹ N·m²/C².
$q_1$
Charge 1
Size and sign of the first charge.
$q_2$
Charge 2
Size and sign of the second charge.
$r_1$
Distance to charge 1
How far the point is from the first charge.
$r_2$
Distance to charge 2
How far the point is from the second charge.
$\mathbf{E}$
Electric field
Force per unit charge. It points downhill on V, crossing the equipotential lines at right angles.

Each charge contributes $kq/r$ , so the voltage is a sum of simple terms.
Equipotential lines are the level curves $V(x, y) = c$ .
The gradient is perpendicular to level curves, so field lines cross equipotentials at right angles.

Learn the math

Functions of Several VariablesCalc III · Lesson 9, Three-Dimensional Space Higher Order Partial DerivativesCalc III · Lesson 21, Partial Derivatives Vector FieldsCalc III · Lesson 39, Line Integrals The Fundamental Theorem for Line IntegralsCalc III · Lesson 42, Line Integrals Curl and DivergenceCalc III · Lesson 45, Surface Integrals What Is a PDE?PDE · Lesson 1, Foundations Laplace's Equation on a RectanglePDE · Lesson 12, Laplace's Equation The Electric FieldE&M · Lesson 2, Charge and Electric Fields Potential Energy and Electric PotentialE&M · Lesson 9, Electric Potential Field from Potential: E = −∇VE&M · Lesson 10, Electric Potential Poisson's and Laplace's EquationsE&M · Lesson 12, Electric Potential Maxwell's EquationsE&M · Lesson 27, Maxwell's Equations and Light

Learn the math

Functions of Several VariablesCalc III · Lesson 9, Three-Dimensional Space

Higher Order Partial DerivativesCalc III · Lesson 21, Partial Derivatives

Vector FieldsCalc III · Lesson 39, Line Integrals

The Fundamental Theorem for Line IntegralsCalc III · Lesson 42, Line Integrals

Curl and DivergenceCalc III · Lesson 45, Surface Integrals

What Is a PDE?PDE · Lesson 1, Foundations

Laplace's Equation on a RectanglePDE · Lesson 12, Laplace's Equation

The Electric FieldE&M · Lesson 2, Charge and Electric Fields

Potential Energy and Electric PotentialE&M · Lesson 9, Electric Potential

Field from Potential: E = −∇VE&M · Lesson 10, Electric Potential

Poisson's and Laplace's EquationsE&M · Lesson 12, Electric Potential

Maxwell's EquationsE&M · Lesson 27, Maxwell's Equations and Light