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Fields, Work & Potential Energy

Key idea: Define gravitational and electric fields, read field lines and equipotentials, and relate work done by a field to potential-energy change.

  • GCE A-Level H2 Physics 2027
On this page

Learning objectives

  • Represent fields and relate work done by a field to potential-energy change.
  • Draw field-line representations of uniform and radial gravitational and electric fields.
  • Use force–extension graphs to determine elastic potential energy.

1. Field concepts

A field is a region of space in which a suitable body may experience a force associated with that field.

Gravitational field strength at a point is force per unit mass on a small test mass:

vector g = (vector F_g)/m

Electric field strength at a point is force per unit positive charge on a small positive test charge:

vector E = (vector Fₑ)/q

Both are vector quantities. The direction of vector E is defined using a positive test charge; a negative charge experiences force opposite to vector E.

2. Field lines and equipotentials

Field lines show the direction of force on a positive test object: a mass for a gravitational field and a positive charge for an electric field.

  • The tangent to a field line gives the field direction.
  • Closer line spacing represents greater field strength in a qualitative diagram.
  • Uniform fields have parallel, equally spaced straight lines.
  • Radial fields have lines directed towards or away from a central source.
  • Field lines do not cross because a field has one direction at each point.

An equipotential surface joins points of equal potential. Moving along it gives no potential-energy change, so the field does no work. Equipotentials meet field lines at right angles.

3. Work done by a field

For gravitational or electric interactions:

W_field = -Δ U

where U is the corresponding potential energy.

  • If the field does positive work, Δ U < 0: potential energy decreases.
  • If an external agent moves the object slowly against the field, the agent does positive work and potential energy increases.
  • If motion is along an equipotential, Δ U = 0 and W_field = 0.
Work and potential-energy change in gravitational and electric fieldsA mass moves downward along uniform gravitational field lines and a positive charge moves right along uniform electric field lines. In both cases the field does positive work and potential energy decreases. Equipotential lines are perpendicular to the field lines.Gravitational fieldElectric fieldmmotionfield and motion downward+qmotionfield and positive-charge motion rightWfield > 0, so ΔU < 0
Scroll diagram horizontally to read all labels.
For motion along either field, positive work done by the field corresponds to a negative potential-energy change: Wfield = −ΔU.
Three potential-energy stores

Gravitational potential energy belongs to a mass–source system, electric potential energy belongs to a charge–source system, and elastic potential energy belongs to a deformed material. Potential energy is a property of an interaction, not of an isolated object alone.

4. Sign reasoning

Before using an equation, ask whether the field assists or opposes the displacement.

For a positive charge moving along an electric field line, the electric force and displacement are aligned. The field does positive work, so electric potential energy decreases.

For a mass moving upward in a near-uniform gravitational field, the gravitational force is downward and the displacement is upward. Gravity does negative work, so gravitational potential energy increases.

5. Common mistakes

  • Drawing force on a negative charge in the same direction as the electric field.
  • Saying field lines are trajectories; they show field direction, not necessarily an object’s path.
  • Reversing the sign in W_field = -Δ U.
  • Treating potential energy as belonging to one object without its interacting source.
  • Drawing equipotentials parallel to field lines.

6. Worked Examples

Modelled example 1

Work done by an electric field

Core

Problem

An electric field does + 0.48 J of work on a charge. Find the change in the electric potential energy of the charge–source system.
Study the worked solution

  1. Use the field-work sign relation

    Method

    Δ U = -W_field.

    Reason

    Positive work by the field transfers energy out of the interaction’s potential-energy store.

    Working

    W_field = -Δ U

  2. Calculate and interpret

    Method

    Δ U = -0.48 J.

    Reason

    The negative change means the electric potential-energy store decreases by 0.48 J.

    Working

    Δ U = -(+0.48) = -0.48 J

Common misconception 2

Slow lifting against gravity

Find and correct the mistake

Learner claim

An external agent slowly lifts a mass, doing + 12 J of work while kinetic energy stays constant and air resistance is negligible. A learner says gravity also does + 12 J. Diagnose the sign, then state gravity’s work and the gravitational potential-energy change.

Try this before viewing the solution

Unit: J
Unit: J

View solution step by step

  1. Use the unchanged kinetic energy

    Method

    The net work is zero.

    Reason

    The mass moves slowly with no kinetic-energy change.

    Working

    Wₙₑₜ = Δ Eₖ = 0

  2. Find gravity's work

    Method

    W_g = -12 J.

    Reason

    The positive agent work is balanced by equal negative work from gravity.

    Working

    W_agent + W_g = 0 ⇒ W_g = -12 J

  3. Find the potential-energy change

    Method

    Δ U_g = +12 J.

    Reason

    Potential-energy change is the negative of work done by the gravitational field.

    Working

    Δ U_g = -W_g = +12 J

Challenge 3

Motion along an equipotential

Minimal support

Independent transfer

A small test charge moves between two points on the same equipotential surface. Determine its electric potential-energy change and the work done by the electric field.

Try this before viewing the solution

Unit: J
Unit: J

Hints

Hint 1: start from equal potential
A move along an equipotential has Δ V = 0, so Δ U = qΔ V.
View solution step by step

  1. Use the equipotential condition

    Method

    Δ U = 0.

    Reason

    Both endpoints have equal potential, so the charge–source interaction energy does not change.

    Working

    Δ U = qΔ V = q(0) = 0

  2. Find field work

    Method

    W_field = 0.

    Reason

    Field work is the negative of the potential-energy change.

    Working

    W_field = -Δ U = 0

7. Mind Stretchers

Mind stretcher 1: Negative charge moving along the fieldExtension

A negative charge is displaced in the direction of an electric field. Is the work done by the electric field positive or negative, and does the electric potential energy increase or decrease?

Show answer

The electric force on a negative charge points opposite to the electric field. Its force is therefore opposite to the stated displacement, so the field does negative work.

Since W_field = -Δ U, negative field work means Δ U is positive: the charge–source electric potential energy increases.

This lesson establishes the common language. Continue later to the Gravitational Fields hub and Electric Fields hub for source equations, potential and field-gradient calculations.

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027