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Electric Fields and Potential

Introduction

Electric Fields and Potential explains how electric charges influence the space around them and how energy is stored and transferred in electrostatic systems. This aiMOOC is designed for learners in Grades 11–13 and connects algebra, vectors, graphs, calculus ideas, laboratory work, simulations, and real-world applications.

You will learn to distinguish the vector quantity electric field from the scalar quantity electric potential, calculate fields and potentials for point charges, interpret field lines and equipotential surfaces, use conservation of energy, and connect the electric field to the spatial change of potential.

The diagram above compares field-line patterns around positive, negative, and neutral objects. Field lines are a visual model: they show direction and relative strength, but they are not physical strings in space.

Learning Goals

By the end of this aiMOOC, you should be able to explain and apply the following ideas:

Electric charge: Describe how positive and negative charges interact and how source charges create electric fields.
Electric field: Use the definition $\vec{E} = \vec{F} / q$ and calculate the field of point charges.
Superposition principle: Add electric fields as vectors and electric potentials as scalars.
Electric potential: Interpret $V = U / q$ and potential difference as energy transferred per unit charge.
Equipotential surface: Relate equipotential lines or surfaces to field direction and work.
Conservation of energy: Connect electric potential energy, kinetic energy, and work done by electric forces.
Gradient: Explain why electrostatic fields point toward decreasing potential and use $\vec{E} = - \nabla V$ .
Capacitor: Describe the approximately uniform field and potential change between large parallel plates.

Foundations: Charge, Force, and Fields

Electric Charge and Coulomb Interaction

Electric charge is a property of matter. Charges with the same sign repel, while charges with opposite signs attract. In an electrostatic model, the magnitude of the force between two point charges is described by Coulomb's law:

$F = k_{e} \frac{| q_{1} q_{2} |}{r^{2}}$

where $q_{1}$ and $q_{2}$ are the charges, $r$ is their separation, and $k_{e} \approx 8.99 \times 1 0^{9} N m^{2} / C^{2}$ in vacuum.

Force is a vector. Its direction lies along the line joining the charges. Coulomb's law is an inverse-square law, so doubling the distance reduces the force magnitude to one quarter.

The Electric Field

The electric field describes the force that a source-charge configuration would exert per unit positive test charge:

$\vec{E} = \frac{\vec{F}}{q_{t e s t}}$

Its SI unit is newton per coulomb, $N / C$ , which is equivalent to volt per meter, $V / m$ . The test charge is imagined to be small enough that it does not significantly disturb the source-charge arrangement.

For a point charge $Q$ in vacuum,

$\vec{E} = k_{e} \frac{Q}{r^{2}} \hat{r}$

where $\hat{r}$ points radially away from the source. A positive source produces an outward field; a negative source produces an inward field.

Force on a Charge in a Field

Once the field at a point is known, the electric force on a charge $q$ placed there is

$\vec{F} = q \vec{E}$ .

For a positive charge, the force points in the same direction as the field. For a negative charge, the force points in the opposite direction. This sign rule is one reason to separate the field, which belongs to the source configuration, from the force, which also depends on the charge placed in the field.

Superposition of Electric Fields

If several source charges are present, each contributes an electric field. The total field is the vector sum:

${\vec{E}}_{t o t a l} = \sum_{i} {\vec{E}}_{i}$ .

Because direction matters, you often need to resolve fields into components before adding them. Symmetry can simplify the calculation. At a point midway between two identical positive charges, for example, equal fields point in opposite directions and cancel.

Field Lines as a Visual Model

Electric field lines help you interpret a vector field without drawing an arrow at every point.

Field line direction is tangent to the electric field at each point.
Field lines start on positive charge and end on negative charge, or extend to or from infinity.
Closer line spacing represents a stronger field in the chosen drawing convention.
Field lines never cross because the field cannot have two directions at one point.
In electrostatic equilibrium, field lines meet the surface of a conductor perpendicularly.

A common mistake is to treat the number of lines as a physical count. The number of drawn lines is chosen by the illustrator; it is their direction and relative density that carry meaning.

Electric Potential and Potential Difference

Potential Energy per Unit Charge

Electric potential $V$ is the electric potential energy $U$ per unit charge:

$V = \frac{U}{q}$ .

Potential is a scalar. It has magnitude but no direction. Its SI unit is the volt:

$1 V = 1 J / C$ .

The zero of potential is a reference choice. For isolated point charges, it is convenient to choose $V = 0$ at infinity. In circuits and laboratory work, another reference such as ground may be chosen.

Potential Difference and Voltage

The potential difference between points A and B is

$Δ V = V_{B} - V_{A} = \frac{Δ U}{q}$ .

The everyday word voltage usually means potential difference. A battery labelled 9 V maintains a potential difference of approximately 9 joules per coulomb between its terminals under suitable conditions.

Potential difference matters because it links electric fields to energy transfer. If a charge moves between two points, its electric potential energy changes according to

$Δ U = q Δ V$ .

Potential of a Point Charge

For an isolated point charge $Q$ , with zero potential chosen at infinity,

$V = k_{e} \frac{Q}{r}$ .

Unlike field magnitude, which decreases as $1 / r^{2}$ , point-charge potential decreases as $1 / r$ . Positive source charges contribute positive potential and negative source charges contribute negative potential when the reference is infinity.

For several point charges,

$V_{t o t a l} = \sum_{i} k_{e} \frac{Q_{i}}{r_{i}}$ .

This is scalar addition, so you add signed values rather than vector components.

Electric Potential Energy of Two Point Charges

For two point charges $q_{1}$ and $q_{2}$ separated by distance $r$ , with zero potential energy at infinite separation,

$U = k_{e} \frac{q_{1} q_{2}}{r}$ .

Like charges have positive potential energy in this reference convention; opposite charges have negative potential energy. The sign tells you how the configuration compares energetically with the chosen reference.

Connecting Electric Field and Potential

Potential Difference from the Electric Field

In electrostatics, the electric field is conservative. The potential difference between two points can be obtained from the field:

$V_{B} - V_{A} = - \int_{A}^{B} \vec{E} \cdot d \vec{ℓ}$ .

The minus sign means that electric potential decreases in the direction of the electric field. If a positive test charge is released from rest and only electric forces act, it tends to accelerate toward lower electric potential while its kinetic energy increases.

For a uniform field and a displacement parallel to the field,

$Δ V = - E Δ x$ .

If you move in the direction opposite the field, the potential rises.

Electric Field from Potential

The field can also be found from how rapidly potential changes with position:

$\vec{E} = - \nabla V$ .

In one dimension,

$E_{x} = - \frac{d V}{d x}$ .

This relation is powerful because potential is a scalar and is often easier to calculate first. The field then follows from the spatial slope of the potential. Where the potential changes rapidly with distance, the field magnitude is large.

Equipotential Lines and Surfaces

An equipotential surface contains points that all have the same electric potential. Moving a charge along an equipotential causes no change in electric potential energy, so the electrostatic field does no net work along that path.

Electric field lines cross equipotential surfaces at right angles. If the field had a component along an equipotential, the potential would change along it, contradicting the definition.

The image shows field lines and equipotentials around an electron. The equipotential pattern is closely related to contour lines on a topographic map: closely spaced contours indicate a steep change.

Dipoles: Reading Field and Potential Together

An electric dipole consists of equal positive and negative charges separated by a distance. The field is a vector pattern, while the potential is a signed scalar pattern. At points on the perpendicular bisector of an ideal symmetric dipole, the contributions to the potential cancel, even though the electric field generally does not vanish there.

This distinction is important: zero potential does not necessarily mean zero electric field. Potential depends on the value of a scalar sum, while field depends on the gradient of potential and on vector superposition.

Work, Energy, and Motion of Charges

For electrostatic forces,

$W_{f i e l d} = - Δ U = - q Δ V$ .

If a positive charge moves to lower potential, its electric potential energy decreases. If no other energy transfers occur, the lost electric potential energy appears as kinetic energy:

$Δ K = - Δ U$ .

For a negative charge, the relationship $Δ U = q Δ V$ still applies, but the sign of $q$ reverses the connection between potential and potential energy. This is why electrons can gain kinetic energy while moving toward higher electric potential.

The electron-volt, abbreviated eV, is a unit of energy. One electron-volt is the energy change of a charge with magnitude equal to the elementary charge when it moves through a potential difference of one volt.

Conductors and Capacitors

Conductors in Electrostatic Equilibrium

In a conductor at electrostatic equilibrium, mobile charges have rearranged so that there is no continuing net motion caused by an internal electrostatic field. The idealized consequences are:

The electric field inside the conducting material is zero.
The conductor is an equipotential.
Any excess charge resides on the surface in the electrostatic model.
Just outside the surface, the electric field is perpendicular to the surface.

These results explain electrostatic shielding and help you reason about charge distributions on metal objects.

Parallel-Plate Capacitor

A parallel-plate capacitor stores separated charge and electric potential energy. Away from edge effects, large oppositely charged plates produce an approximately uniform field between them. In this region, the potential changes nearly linearly with distance.

For plate separation $d$ and potential difference magnitude $| Δ V |$ ,

$E \approx \frac{| Δ V |}{d}$ .

The diagram combines electric field and equipotential lines for a plate capacitor. Near the plate edges, fringing makes the field less uniform.

Worked Examples

Example: Field of a Point Charge

A positive charge $Q = 3.0 μ C$ is isolated in vacuum. Find the electric field magnitude 0.20 m away.

$E = k_{e} \frac{| Q |}{r^{2}}$

$E = (8.99 \times 1 0^{9}) \frac{3.0 \times 1 0^{- 6}}{(0.20)^{2}} \approx 6.7 \times 1 0^{5} N / C$ .

Because the source charge is positive, the field points radially outward.

Example: Potential of the Same Charge

At the same point,

$V = k_{e} \frac{Q}{r}$

$V = (8.99 \times 1 0^{9}) \frac{3.0 \times 1 0^{- 6}}{0.20} \approx 1.35 \times 1 0^{5} V$ .

Notice that the potential is a scalar. You report its sign and magnitude, not a direction.

Example: Energy Change Across a Voltage

A proton with charge $+ e$ moves through a potential drop of 200 V. Its potential-energy change is

$Δ U = q Δ V = (+ e) (- 200 V) = - 200 e V$ .

If the electric force is the only force doing work, its kinetic energy increases by 200 eV.

Example: Reading a Potential Graph

Suppose potential along the x-axis changes linearly from 12 V at $x = 0$ to 4 V at $x = 2.0 m$ . The slope is

$\frac{Δ V}{Δ x} = \frac{4 - 12}{2.0} = - 4 V / m$ .

Therefore,

$E_{x} = - \frac{Δ V}{Δ x} = + 4 V / m$ .

The field points in the positive x-direction because potential decreases as x increases.

Advanced Extension: When a Scalar Potential Is Enough

For static charge distributions, the electrostatic field is conservative and can be written as $\vec{E} = - \nabla V$ . This makes electrostatic potential single-valued up to an arbitrary additive constant.

In general electromagnetism, a time-varying magnetic field can produce a circulating electric field. In that case, the electric field is not purely the negative gradient of one scalar potential. This distinction becomes important in advanced studies of electromagnetic induction and Maxwell's equations.

Explore with Simulation

The University of Colorado Boulder PhET simulation Charges and Fields lets you place charges, display field vectors, measure voltage, and compare field and equipotential patterns. Use it to test predictions before calculating.

Open the PhET Charges and Fields simulation

Suggested investigation: place two equal positive charges symmetrically. Predict where the electric field is zero, then test your prediction. Next, identify whether the electric potential at that same point is zero or nonzero and explain why.

Common Misconceptions

Electric field: A field is not the same as force. Force depends on the test charge through $\vec{F} = q \vec{E}$ .
Electric potential: A potential of zero at a point does not guarantee that the electric field is zero there.
Potential difference: Voltage is always a difference between two potentials, even when one point is chosen as the zero reference.
Equipotential surface: A charge can move along an equipotential while the local electric field is nonzero; the field is perpendicular to the motion along the equipotential.
Superposition principle: Electric fields add as vectors, while electric potentials add as signed scalars.
Electron motion: Negative charges accelerate opposite the electric field direction when electric force is the only force considered.
Field line: Field lines are representations, not trajectories that every charged particle must follow.

Interactive Tasks

Quiz: Test Your Knowledge

How is the electric field at a point defined? (Force per unit positive test charge) (!Energy per unit distance) (!Charge per unit potential) (!Work per unit time)

Which SI unit can be used for electric field strength? (Newton per coulomb) (!Joule per coulomb) (!Coulomb per second) (!Newton per volt)

Which direction does the electric field of an isolated positive point charge have? (Radially outward) (!Radially inward) (!Clockwise around the charge) (!Parallel to every equipotential)

What kind of quantity is electric potential? (Scalar) (!Vector) (!Tensor) (!Direction only)

How does the potential of an isolated point charge vary with distance? (Inversely with distance) (!Inversely with distance squared) (!Directly with distance) (!Independently of distance)

What is the relationship between electrostatic field and potential? (The field points toward decreasing potential) (!The field points toward increasing potential) (!The field is always zero where potential changes) (!The field is parallel to equipotential surfaces)

How do electric field lines meet equipotential surfaces? (At right angles) (!Tangentially) (!At forty five degrees) (!They never meet)

How much electrostatic work is done when a charge moves entirely along one equipotential? (Zero) (!Always positive) (!Always negative) (!Equal to the charge magnitude)

What is the electric field inside an ideal conductor in electrostatic equilibrium? (Zero) (!Uniform and nonzero) (!Infinite) (!Always radial)

How are electric potentials from several point charges combined? (By signed scalar addition) (!By vector cross products) (!By multiplying all values) (!By adding only positive values)

Memory Game

Electric field	Force per unit positive test charge
Voltage	Electric potential difference between two points
Equipotential	Set of points with the same electric potential
Superposition	Rule for combining contributions from multiple sources
Test charge	Small hypothetical charge used to define a field
Conductor	Material with mobile charge carriers
Gradient	Spatial rate and direction of fastest increase of a scalar

Drag and Drop

Match the correct terms.	Topic
Vector quantity	Electric field
Scalar quantity	Electric potential
Perpendicular curves	Equipotential lines and field lines
Energy per charge	Electric potential
Force per charge	Electric field

...

Crossword Puzzle

Voltage	What common name is given to electric potential difference?
Coulomb	Which unit measures electric charge?
Gradient	What mathematical idea connects spatial change of potential to the field?
Equipotential	What word describes a set of points sharing the same electric potential?
Superposition	What principle lets you combine contributions from several source charges?
Conductor	What material allows charge carriers to move freely enough to redistribute?

LearningApps

Cloze Text

An electric field tells you the force per unit

. The SI unit of electric field can be written as

. Electric potential is electric potential energy per unit

. Potential difference is commonly called

. The potential of an isolated point charge varies inversely with

. Electric field lines point in the direction of decreasing

. Equipotential surfaces meet electric field lines at

. Moving a charge along one equipotential produces no change in electric

. In electrostatic equilibrium the electric field inside a conductor is

. For a uniform one-dimensional field, the potential slope has the opposite sign to the

.

Open-Ended Tasks

Easy

Field Line Sketch: Draw field-line patterns for one positive charge, one negative charge, and a dipole; add arrows and explain what line density represents.
Potential Analogy: Write a short comparison between electric potential and gravitational height, then identify one way the analogy is useful and one way it can mislead.
Voltage in Daily Life: Photograph or list three safe everyday devices with labelled voltages and explain what a voltage rating means as energy per unit charge.
Concept Video: Produce a one-minute video that explains why electric field is a vector but electric potential is a scalar.

Standard

PhET Field Investigation: Use the Charges and Fields simulation to create three charge arrangements, capture or sketch the field and equipotential patterns, and explain how symmetry affects each pattern.
Conductive Paper Experiment: With teacher supervision, map equipotential lines on conductive paper or another approved low-voltage apparatus and infer electric-field directions from your measurements.
Interview an Engineer: Interview an electrician, electrical engineer, physics teacher, or laboratory technician about how voltage and electric fields matter in their work; summarize two technical examples.
Data and Graphs: Calculate and graph both electric field magnitude and electric potential versus distance for a chosen point charge, then compare the shapes and their distance dependence.

Advanced

Numerical Superposition Project: Write a spreadsheet or short program that calculates total electric potential and electric-field components from several point charges on a two-dimensional grid.
Energy Transfer Investigation: Design a quantitative problem about a charged particle accelerated through a potential difference, solve it using conservation of energy, and analyze how changing the sign of the particle changes the motion.
Field Mapping Project: Create a contour map of electric potential for a dipole from calculations or simulation data, estimate the field from potential gradients, and compare the result with direct field calculations.
Electrostatics and Induction: Research why a scalar electrostatic potential is sufficient for static fields but not by itself for electric fields induced by changing magnetic flux; present your findings as a poster, essay, or narrated video.

Learning Assessment

Field and Potential Comparison: Given a diagram containing source charges, predict both the direction of the electric field and the sign of the electric potential at selected points, then justify every prediction.
Two-Charge Transfer Problem: Solve a multi-step problem in which a charged particle moves between two points in the field of two fixed charges; calculate potential difference, energy change, and final speed under clearly stated assumptions.
Graph-to-Field Reasoning: Interpret a graph of potential versus position, identify regions of strongest and weakest field, and determine field direction from the slope.
Experimental Evaluation: Analyze measured equipotential data, construct approximate field lines, identify sources of uncertainty, and explain whether the measurements support the theoretical model.
Model Critique: Compare field-line diagrams, equipotential maps, and vector plots as representations of the same electrostatic situation; discuss what each representation reveals and what it hides.
Transfer to Capacitors: Use the field-potential relationship to explain why the potential between large parallel capacitor plates changes nearly linearly with distance and why edge regions differ.
Advanced Transfer: Explain how the relation between electrostatic field and scalar potential changes when time-varying magnetic fields are introduced, using Faraday's law conceptually.

Evidence of Learning

Strong evidence of learning includes accurate knowledge, reliable problem-solving methods, clear representations, practical investigation, and transfer to unfamiliar situations. You should be able to demonstrate:

Conceptual knowledge: Explain charge, field, force, potential, potential difference, potential energy, equipotentials, conductors, and superposition in your own words.
Mathematical skill: Use inverse-square and inverse-distance relationships, vectors, signed scalar sums, energy equations, graphs, and basic derivatives or gradients where appropriate.
Representation skill: Translate among charge diagrams, field vectors, field lines, equipotential maps, potential graphs, and equations.
Experimental skill: Plan or carry out a safe low-voltage field-mapping investigation, organize measurements, and discuss uncertainty.
Digital modelling: Use simulation, spreadsheet, or code to investigate multi-charge systems and compare numerical results with physical reasoning.
Communication: Present a coherent explanation in writing, diagrams, spoken presentation, poster, or video using correct physics vocabulary and units.
Transfer: Apply the field-potential relationship to capacitors, particle acceleration, electrostatic shielding, and introductory electromagnetic induction.