Electric fields and capacitance (A Level)

18. Electric Fields

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18. Electric Fields — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Describe and sketch electric fields, use F = qE, E = V/d, Coulomb's law, the point-charge field, potential and potential energy equations, and compare them with…

Electric field strength E
Force on a charge
Drawing rules
Derivation
Limitation
Entering perpendicular to the field

Section 1 of 10Electric field and field strength

Electric field = a region of space in which a charge experiences a force.
Electric field strength E = the force per unit charge on a small stationary positive test charge, E = F/q, in N C⁻¹ (equivalent to V m⁻¹).
Direction
- E is a vector pointing away from positive charge and towards negative charge, because a positive test charge defines it.
Force on a charge
- F = qE.
- A positive charge feels a force along the field.
- A negative charge feels a force of the same size against the field.

Section 2 of 10Field line patterns

Field	Line pattern	Field strength
Uniform	Equally spaced parallel lines	Same size and direction everywhere
Radial (point charge)	Straight lines out of a positive charge or into a negative charge	Falls with distance; stronger where lines are closer
Charged conducting sphere	Same as a point charge at the centre; lines meet the surface at 90°	Treated like a point charge outside the sphere
Two opposite charges	Lines from the positive to the negative charge	Strong attraction when close
Two like charges	Lines leave (or enter) both and do not join them; a neutral point lies midway	Strong repulsion when close
Parallel plates	Uniform between the plates, curved and non-uniform beyond the edges	Constant between the plates

Drawing rules
- add arrows, make lines touch the source charge or plate, and never let lines cross.

Section 3 of 10Uniform field between parallel plates

Equation
- E = V/d, with V the potential difference between the plates and d their separation.
- A larger V gives a stronger field and a larger d a weaker field.
- The field points from the positive plate to the negative plate, and an earthed plate is at 0 V.
Derivation
- work on a charge moving between plates is W = Fd = qV, so E = F/q = V/d.
Limitation
- E = V/d applies only to parallel plates, not to the radial field of a point charge.

Section 4 of 10Charged particles in a uniform field

At rest
- a charge moves along the field line, in the field direction if positive and opposite if negative.
Entering perpendicular to the field
- it feels a constant force and follows a parabola, like a projectile in gravity.
- A positive charge curves towards the negative plate and a negative charge towards the positive plate.
- An uncharged particle such as a neutron passes through undeflected.

Factor	Effect on deflection
Greater charge	More deflection (larger force)
Greater mass	Less deflection (smaller acceleration)
Greater speed	Less deflection (less time between the plates)

Section 5 of 10Coulomb's law

Coulomb's law = the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of their separation.
- F = Q₁Q₂/(4πε₀r²), where r is the distance between centres.
Permittivity of free space
- ε₀ = 8.85 × 10⁻¹² F m⁻¹, and air has the same value.
- Other materials have a larger permittivity, which reduces the force.
Alternative form
- F = kQ₁Q₂/r² with k = 1/(4πε₀) = 8.99 × 10⁹ N m² C⁻².
Direction
- like charges repel and unlike charges attract (negative product).
  - (positive product)
Validity
- it applies to point charges or small charged spheres much smaller than their separation, not irregular charge distributions.
Spherical conductor
- outside it the charge acts as if concentrated at the centre, so the field lines match those of a point charge.

Section 6 of 10Electric field of a point charge

Equation
- E = Q/(4πε₀r²), from Coulomb's law divided by the test charge q.
Behaviour
- E follows an inverse square law and is not constant, and it is zero inside a charged conducting sphere.
Several charges
- field strengths add as vectors.
- Along a line, add if in the same direction and subtract if opposite.
- At right angles, use Pythagoras for the magnitude and trigonometry for the direction.
Method
- treat the point as a positive test charge, so the field points away from positive and towards negative sources.

Section 7 of 10Comparison with gravitational fields

Feature	Gravitational	Electric
Source	Mass	Charge
Force law	F = Gm₁m₂/r²	F = Q₁Q₂/(4πε₀r²)
Nature of force	Always attractive	Attractive or repulsive
Field strength of a point source	GM/r²	Q/(4πε₀r²)
Potential	−GM/r (always negative)	Q/(4πε₀r) (sign of the charge)

Section 8 of 10Electric potential

Electric potential = the work done per unit charge in bringing a small positive test charge from infinity to the point, in J C⁻¹ or V.
- It is a scalar, zero at infinity, with the sign of the source charge.
Point charge
- V = Q/(4πε₀r), and the same equation applies outside a charged sphere.
- Around a positive charge V is positive and rises as a positive test charge moves closer.
- Around a negative charge V is negative and falls as a positive test charge moves closer.
- Inside a charged conducting sphere the potential is constant.
Combining charges
- potentials add as scalars.
Direction
- potential decreases in the direction of the field lines.

Section 9 of 10Potential gradient

Potential gradient = the rate of change of potential with displacement in the direction of the field.
Field strength
- E = −ΔV/Δr, the negative of the potential gradient.
Graphs
- on a V–r graph the gradient at a point gives E there.
- On an E–r graph the area under the curve between two points gives the potential difference.
Shapes
- V against r follows 1/r, and E against r follows 1/r².
  - (and −1/r for a negative charge)

Section 10 of 10Electric potential energy

Work in a field
- W = qΔV for a charge q moved through a potential difference ΔV.
Potential energy of two charges
- E_p = Q₁Q₂/(4πε₀r), zero at infinity, with r not squared.
- It is positive for like charges and negative for unlike charges.
Change
- ΔE_p = qΔV = [Qq/(4πε₀)](1/r₂ − 1/r₁) when q moves from r₁ to r₂.
Work needed
- work must be done to bring like charges together or separate unlike charges, and the field does work when the opposite happens.

Last updated: 14 October 2026