Electric Field Strength

We now leave capacitors for a while and spend four lessons on electric fields. A field is a region of space in which a charged object experiences a force. The central question of the topic is: "given a distribution of charges, what force would a small test charge experience at any point in space?" The answer is expressed as a vector field — the electric field strength — and just as with gravity, the field concept lets us separate the question of what a source does (produce a field) from the question of how matter responds (feel a force).

This lesson begins OCR H556 Module 6.2 — Electric fields, which parallels the Module 5.4 gravitational field content very closely. Many of the formulae look almost identical once you swap m for q and G for 1/(4πε₀).

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Defining Electric Field Strength

Take a small positive test charge +q and place it at a point in the field. It experiences an electric force F. The electric field strength E at that point is defined as the force per unit charge:

E = F / Q

Symbol	Meaning	Unit
E	Electric field strength	N C⁻¹ or V m⁻¹
F	Force on test charge	newton (N)
Q	Magnitude of test charge	coulomb (C)

Because force is a vector, the electric field is also a vector. Its direction is the direction of the force on a positive charge. A negative charge placed in the same field experiences a force in the opposite direction, but the field itself is unchanged.

Key idea: The field is a property of the region of space; the force depends on both the field and the charge placed in it.

The two units N C⁻¹ and V m⁻¹ are equivalent. Using V m⁻¹ is more common in practice and makes the uniform-field formula below look natural.

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Uniform Fields — Parallel Plates

Between two parallel plates held at potential difference V and separation d, the electric field is uniform: the same magnitude and direction everywhere between the plates (ignoring edge effects). It is given by

E = V / d

The field points from the high-potential plate to the low-potential plate, and its magnitude in V m⁻¹ is the voltage drop per metre.

graph LR
    A[+ plate<br/>+V] -->|E points right| B[− plate<br/>0 V]

Worked Example — Field Between Plates

Two parallel plates are 5.0 mm apart and have a potential difference of 200 V between them. Calculate the electric field strength and the force on a 2.0 nC charge placed between the plates.

E = V/d = 200 / 0.005 = 4.0 × 10⁴ V m⁻¹
F = EQ = 4.0 × 10⁴ × 2.0 × 10⁻⁹ = 8.0 × 10⁻⁵ N

The field is 40 kV m⁻¹ — substantial, but still well below the breakdown field of air (about 3 MV m⁻¹).

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Radial Fields — Point Charges

For a single point charge Q (or a spherical charge distribution), the field is radial: it points directly away from positive Q and directly towards negative Q. Its magnitude falls as 1/r²:

E = Q / (4πε₀r²)

This is the electric-field equivalent of Newton's g = GM/r². The constant 1/(4πε₀) plays the role of G.

Constant	Value
ε₀	8.85 × 10⁻¹² F m⁻¹
1/(4πε₀)	8.99 × 10⁹ N m² C⁻²

graph TD
    Q((+Q)) -->|"E₁ at r₁"| P1[Point 1]
    Q -->|"E₂ at r₂"| P2[Point 2]

Worked Example — Radial Field

Calculate the electric field 10 cm from a point charge of +5.0 nC.

E = Q / (4πε₀r²)
  = (5.0 × 10⁻⁹) / (4π × 8.85 × 10⁻¹² × 0.10²)
  = (5.0 × 10⁻⁹) × (8.99 × 10⁹) / 0.01
  = 4495 V m⁻¹
  ≈ 4500 V m⁻¹

Because the field is radial, this number is the same at every point on a sphere of radius 10 cm around the charge. Double the distance and the field becomes one quarter.

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Field Lines

Field lines are drawings that show the direction and (by their spacing) the magnitude of the electric field. Rules: