Electric Fields

This guide covers Topic 18, Electric fields, in full — subtopics 18.1 Electric fields and field lines, 18.2 Uniform electric fields, 18.3 Electric force between point charges, 18.4 Electric field of a point charge and 18.5 Electric potential — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content, and parallels the treatment of gravitational fields from Topic 13.

Before studying this

This resource assumes gravitational fields and potential from Gravitational Fields, and electric current and potential difference from Electricity: Current, Potential Difference and Resistance.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 18

18.1 Electric fields and field lines — understanding the concept of an electric field as a field of force, and representing it with field lines.

18.2 Uniform electric fields — recalling and using E = V/d for the field between charged parallel plates; describing the effect of a uniform electric field on the motion of charged particles.

18.3 Electric force between point charges — understanding that, according to the electric force between two point charges is described by Coulomb’s law, recalling and using F = Qq/(4πε₀r²).

18.4 Electric field of a point charge — defining electric field strength at a point as the force per unit positive charge acting on a small test charge at that point; recalling and using E = Q/(4πε₀r²) for the field of a point charge; recalling and using F = Eq for the force on a charge placed in a field; treating an isolated charged sphere, for points outside it, as a point charge concentrated at its centre.

18.5 Electric potential — defining electric potential at a point as the work done per unit positive charge in bringing a small test charge from infinity to that point; recalling and using V = Q/(4πε₀r); deriving and using the electric potential energy of a pair of point charges, U = Qq/(4πε₀r); using the relationship between electric field strength and potential gradient, E = −dV/dr; recognising the analogy between the treatment of gravitational fields and electric fields.

Electric fields and field lines

An electric field is a region of space in which a charge experiences a force due to other charges. It is represented using field lines, which point in the direction of the force on a positive test charge, and are drawn closer together where the field is stronger.

Uniform electric fields

Between two oppositely charged parallel plates, separated by distance d and with potential difference V between them, the field is uniform and its strength is given by:

E = V/d

A charged particle entering this field experiences a constant force, producing motion analogous to projectile motion under gravity: constant velocity in the direction along the plates, constant acceleration perpendicular to them.

Coulomb’s law

The electric force between two point charges Q and q, separated by distance r, is given by Coulomb’s law:

F = Qq / (4πε₀r²)

where ε₀ is the permittivity of free space. Like gravitation, this is an inverse square law. Like charges repel; unlike charges attract — unlike gravity, which is always attractive.

Electric field of a point charge

Electric field strength E at a point is the force per unit positive charge acting on a small positive test charge placed there:

E = F/q

For a point charge Q, this gives:

E = Q / (4πε₀r²)

Worked example. Find the electric field strength 0.10 m from a point charge of +2.0 × 10⁻⁶ C (ε₀ = 8.85 × 10⁻¹² F m⁻¹):

E = Q/(4πε₀r²) = (2.0 × 10⁻⁶) / (4π × 8.85 × 10⁻¹² × 0.10²)
E ≈ 1.80 × 10⁶ N C⁻¹

Force on a charge in a field, and charged spheres

Whatever field it is placed in — uniform or that of a point charge — a charge Q experiences a force:

F = EQ

which is simply E = F/q rearranged, and applies generally, not only to the point-charge case. A uniformly charged sphere behaves, for any point outside it, exactly as if all its charge were concentrated at a point at its centre — so E = Q/(4πε₀r²) and V = Q/(4πε₀r) can both be applied to a charged conducting sphere using r measured from its centre, provided the point considered is outside the sphere.

Electric potential

Electric potential V at a point is the work done per unit positive charge in bringing a small positive test charge from infinity (defined as zero potential) to that point:

V = Q / (4πε₀r)

Unlike gravitational potential, which is always negative (since gravity is always attractive), electric potential can be positive (around a positive charge) or negative (around a negative charge), directly reflecting the sign of the source charge.

Electric potential energy

The electric potential energy of a pair of point charges Q and q, separated by distance r, is the work done bringing q from infinity to that separation:

U = Qq / (4πε₀r)

U is positive for like charges (work must be done against repulsion to bring them together) and negative for unlike charges (the field does work as they come together). This is directly analogous to U = Qq for electric potential energy compared with V = Q for potential — potential energy is potential multiplied by the charge placed in the field, U = qV.

Field strength as potential gradient

Electric field strength is related to the gradient of the potential against distance graph:

E = −dV/dr

The negative sign shows that the field points in the direction of decreasing potential — from high to low potential — which is consistent with field lines pointing away from positive charge (high potential) toward negative charge (low potential). On a graph of V against r, the field strength at a point is (minus) the gradient of the tangent at that point.

Comparing gravitational and electric fields

The mathematics of point-mass gravitational fields and point-charge electric fields are directly analogous: both are inverse square laws for force and field strength, and both define potential as work done per unit mass/charge bringing a test object from infinity. The key physical difference is that gravity is always attractive, giving a field strength formula with an implicit “always toward the mass” direction and a potential that is always negative, while electric forces can be attractive or repulsive depending on the sign of the charges involved.

Common mistakes

• Forgetting the inverse square relationship applies to both force and field strength, but not to potential (which is inverse, not inverse square) — check which quantity is being asked for.
• Assuming electric potential is always negative like gravitational potential — it depends on the sign of the source charge.
• Using E = V/d for a non-uniform field, e.g. around a point charge — this formula only applies to the uniform field between parallel plates.
• Mixing up the direction convention for field lines — they point in the direction of force on a positive test charge, away from positive charges and toward negative ones.
• Forgetting the minus sign in E = −dV/dr — field points toward decreasing potential, so a positive gradient corresponds to a field pointing in the negative r-direction.
• Applying E or V formulas for a point charge to a point inside a charged sphere — the point-charge treatment only holds for points outside the sphere.

Quick revision checklist

• Electric field lines and E = V/d for uniform fields between parallel plates
• F = Qq/(4πε₀r²) — Coulomb’s law
• E = Q/(4πε₀r²) for the field of a point charge, and F = Eq for the force on any charge in a field
• A charged sphere behaves as a point charge at its centre, for points outside it
• V = Q/(4πε₀r) for electric potential, and the gravitational-electric analogy
• U = Qq/(4πε₀r) for the electric potential energy of a pair of point charges
• E = −dV/dr relating field strength to the potential gradient

Related resources

• Oscillations — the previous A Level topic
• Capacitance — the next A Level topic, using uniform fields and potential difference
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.