Electric Fields and Capacitors

Electric fields arise from electric charges and are responsible for the forces between charged particles. This topic draws many parallels with gravitational fields but includes some key differences — most notably that the electric force can be repulsive as well as attractive. The capacitor section builds on the principles of electric fields to explain charge storage, energy storage, and exponential charge/discharge behaviour. This material is tested in AQA Paper 2 (Section 7) and OCR Paper 1 (Module 5).

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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 k = 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻², Q₁ and Q₂ are the charges (C), r is the distance between them (m), and ε₀ is the permittivity of free space (8.85 × 10⁻¹² F m⁻¹).

Unlike gravity, the electric force can be either attractive (opposite charges, F is negative) or repulsive (like charges, F is positive). If Q₁ and Q₂ have the same sign, the force is positive (repulsive); if opposite signs, the force is negative (attractive).

Worked Example — Coulomb's Law Force Calculation

Question: Two point charges, Q₁ = +3.0 μC and Q₂ = −5.0 μC, are separated by a distance of 0.20 m in a vacuum. Calculate the magnitude and direction of the electrostatic force between them. (k = 8.99 × 10⁹ N m² C⁻²)

Solution:

Step 1: Convert charges to SI units. Q₁ = 3.0 × 10⁻⁶ C, Q₂ = 5.0 × 10⁻⁶ C

Step 2: Apply Coulomb's law. F = kQ₁Q₂/r² = 8.99 × 10⁹ × 3.0 × 10⁻⁶ × 5.0 × 10⁻⁶ / (0.20)²

F = 8.99 × 10⁹ × 1.5 × 10⁻¹¹ / 0.04

F = 0.13485 / 0.04 = 3.4 N

Step 3: Determine direction. The charges are opposite in sign, so the force is attractive — each charge is pulled towards the other.

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Electric Field Strength

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

E = F/Q

For a radial field around a point charge:

E = kQ/r² = Q/(4πε₀r²)

For a uniform field between parallel plates:

E = V/d

where V is the potential difference across the plates and d is the separation. The field between parallel plates is uniform — the field lines are parallel, equally spaced, and perpendicular to the plates. There is some fringing at the edges where the field becomes non-uniform, but this is usually neglected.

Electric Field Line Patterns

• Single positive charge: field lines radiate outwards in all directions (away from the charge).
• Single negative charge: field lines point radially inwards (towards the charge).
• Two opposite charges (dipole): field lines curve from the positive charge to the negative charge.
• Two like charges: field lines repel and there is a neutral point midway between them where E = 0.
• Parallel plates: field lines are straight, parallel, equally spaced, and perpendicular to the plates (from positive plate to negative plate).

Described diagram — Electric field lines between parallel plates and around point charges: For parallel plates, two horizontal rectangles represent the positive plate (top, marked +) and negative plate (bottom, marked −). Between them, straight vertical arrows point uniformly downward from positive to negative, all equally spaced and parallel. Near the edges, the field lines curve slightly outward (fringing), but this is often ignored. For a single positive point charge, arrows radiate outward in all directions from the charge. For a single negative point charge, arrows point radially inward towards the charge. For two opposite point charges (a dipole), curved field lines emerge from the positive charge and curve around to enter the negative charge, forming smooth arcs.

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Motion of Charged Particles in Uniform Electric Fields

When a charged particle enters a uniform electric field between parallel plates, it experiences a constant force in the direction of the field (for a positive charge, from positive plate to negative plate). This is analogous to a mass moving in a uniform gravitational field.

If the particle enters perpendicular to the field, it continues with constant velocity in that direction while accelerating uniformly in the direction of the field. The result is a parabolic path — identical in shape to projectile motion under gravity.

For a positive charge +Q entering horizontally between vertical plates with field E:

• Horizontal: constant velocity, x = vt
• Vertical: constant acceleration a = QE/m, so y = ½at² = QEt²/(2m)

Eliminating t gives y = QEx²/(2mv²), which is the equation of a parabola.

For a negative charge (e.g. an electron), the deflection is in the opposite direction — towards the positive plate.

Exam Tip: The parabolic path of a charged particle in a uniform electric field is directly analogous to projectile motion. The charge-to-mass ratio (Q/m) plays the same role as gravitational field strength (g). Questions may ask you to compare the deflection of different particles — remember that the deflection depends on Q/m, not Q alone.

Millikan's Oil Drop Experiment

Robert Millikan's experiment (1909) was crucial in determining the elementary charge (e = 1.60 × 10⁻¹⁹ C) and showing that charge is quantised.

Apparatus: Tiny charged oil drops fall between two horizontal parallel plates. The electric field between the plates can be adjusted.

Method:

1. A fine mist of oil drops is sprayed between the plates. Some drops become charged by friction or by exposure to X-rays.
2. The electric field is adjusted until a particular drop is suspended — hovering motionless.
3. At this point, the upward electric force exactly balances the downward weight (and buoyancy, though this is usually neglected):

QE = mg, so Q = mg/E = mgd/V

4. The mass of the drop is found from its terminal velocity when the field is switched off, using Stokes' law.

flowchart TD
    A["Spray oil drops between<br/>charged parallel plates"] --> B["Some drops become charged<br/>(friction or X-rays)"]
    B --> C["Adjust voltage until<br/>a drop hovers motionless"]
    C --> D["QE = mg<br/>Upward electric force = Weight"]
    D --> E["Switch off field<br/>Drop falls under gravity"]
    E --> F["Measure terminal velocity<br/>to find drop mass via Stokes law"]
    F --> G["Calculate Q = mg/E = mgd/V"]
    G --> H["Result: Q is always<br/>a whole-number multiple of e"]

Results: Millikan found that the charge on every drop was always a whole-number multiple of 1.60 × 10⁻¹⁹ C. This proved that charge is quantised — it comes in discrete packets of e.

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Electric Potential

Key Definition: The electric potential (V) at a point is the work done per unit positive charge in bringing a small positive test charge from infinity to that point.

For a point charge Q:

V = kQ/r = Q/(4πε₀r)

For a positive charge, the potential is positive (work must be done against the repulsive field to bring a positive test charge from infinity). For a negative charge, the potential is negative (the field does work on the test charge as it is brought in).

Equipotential Lines

Equipotential lines (or surfaces) connect points of equal potential. Key properties:

• Always perpendicular to the field lines.
• No work is done when a charge moves along an equipotential surface.
• For a point charge, equipotentials are concentric spheres.
• For parallel plates, equipotentials are planes parallel to the plates, equally spaced.
• Equipotentials are closer together where the field is stronger.

Relationship Between E and V

The electric field strength is the negative of the potential gradient:

E = −dV/dr

In a uniform field: E = V/d (the magnitude of the potential gradient is constant).

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Comparison of Gravitational and Electric Fields

Property	Gravitational Field	Electric Field
Source	Mass	Charge
Force law	F = GMm/r²	F = kQ₁Q₂/r²
Field strength definition	g = F/m (force per unit mass)	E = F/Q (force per unit positive charge)
Field strength formula	g = GM/r²	E = kQ/r²
Potential definition	Work done per unit mass from infinity	Work done per unit positive charge from infinity
Potential formula	V = −GM/r	V = kQ/r
Nature of force	Always attractive	Attractive or repulsive
Sign of potential	Always negative	Positive or negative
Field lines	Always point towards mass	Point away from +Q, towards −Q
Relative strength	Very weak (G ≈ 10⁻¹¹)	Very strong (k ≈ 10¹⁰)
Shielding	Cannot be shielded	Can be shielded (Faraday cage)