Electric Fields and Coulomb's Law

An electric field is a region of space in which an electric charge experiences a force. Electric fields are created by charged objects and are central to AQA specification 3.7.3. The mathematical treatment of electric fields closely parallels that of gravitational fields, but with crucial differences.

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Electric Field Lines

Electric field lines represent the direction a positive test charge would move if placed in the field (by convention).

Described diagram — Field lines around a positive point charge: Lines radiate outward from the charge in all directions, with arrows pointing away from the charge. The lines are closer together near the charge (stronger field) and further apart at greater distances (weaker field).

Described diagram — Field lines around a negative point charge: Lines point radially inward towards the charge, converging at the centre. A positive test charge would be attracted towards the negative charge.

Described diagram — Field between two parallel plates: Uniform, equally spaced, parallel lines pointing from the positive plate to the negative plate. At the edges, the lines curve outward slightly (edge effects), but in the central region the field is uniform.

Key rules for electric field lines:

• Field lines go from positive to negative (or from positive to infinity, or from infinity to negative).
• They never cross.
• The density of lines indicates the field strength.
• Field lines are perpendicular to the surface of a conductor.
• Inside a conductor in electrostatic equilibrium, the field is zero.

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Electric Field Strength

Key Definition: The electric field strength (E) at a point is the force per unit positive charge acting on a small positive test charge placed at that point.

E = F/Q

where E is the electric field strength (N C⁻¹ or equivalently V m⁻¹), F is the force (N), and Q is the charge of the test charge (C).

Electric field strength is a vector quantity. Its direction is the direction of the force on a positive charge.

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Uniform Electric Fields

Between two parallel conducting plates separated by a distance d with a potential difference V across them, the electric field is uniform:

E = V/d

where E is the field strength (V m⁻¹), V is the potential difference (V), and d is the plate separation (m).

This is a crucial result: the field between parallel plates depends only on the voltage and separation, not on the plate area or the charge on the plates (as long as edge effects are neglected).

Worked Example 1 — Uniform Field Between Parallel Plates

Question: Two parallel plates are separated by 2.0 cm and have a potential difference of 500 V between them. Calculate: (a) the electric field strength between the plates; (b) the force on an electron placed between the plates.

Solution:

(a) E = V/d = 500 / (2.0 × 10⁻²) = 2.5 × 10⁴ V m⁻¹

(b) F = EQ = 2.5 × 10⁴ × 1.60 × 10⁻¹⁹ = 4.0 × 10⁻¹⁵ N

The force is directed from the positive plate towards the negative plate (opposite to the electron's motion, since the electron is negatively charged — it accelerates towards the positive plate).

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Coulomb's Law

Key Definition: Coulomb's law states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

F = kQ₁Q₂/r² = Q₁Q₂/(4πε₀r²)

where:

• F is the electrostatic force (N)
• Q₁ and Q₂ are the two charges (C)
• r is the distance between the centres of the charges (m)
• k = 1/(4πε₀) = 8.99 × 10⁹ N m² C⁻² (Coulomb's constant)
• ε₀ = 8.85 × 10⁻¹² F m⁻¹ (permittivity of free space)

Key features:

• Like charges repel (positive force); unlike charges attract (negative force, by convention).
• The force obeys an inverse square law.
• The force acts along the line joining the two charges.
• Newton's third law applies: Q₁ exerts a force on Q₂ equal in magnitude but opposite in direction to the force Q₂ exerts on Q₁.

Radial Electric Field Strength

For a point charge Q, combining E = F/Q_test with Coulomb's law:

E = F/Q_test = (kQQ_test/r²)/Q_test = kQ/r²