Electric Fields
Electric Fields
An electric field is a region of space where a charged particle experiences a force. Unlike gravity, electric forces can be attractive or repulsive.
Coulomb's Law
The force between two point charges:
F = kQ₁Q₂/r² = Q₁Q₂/(4πε₀r²)
where:
- F = force (N) — positive = repulsive, negative = attractive
- k = Coulomb constant = 8.99 × 10⁹ N m² C⁻²
- ε₀ = permittivity of free space = 8.854 × 10⁻¹² F m⁻¹
- Q₁, Q₂ = charges (C) — include the sign
- r = separation (m)
Like charges repel; unlike charges attract.
Electric Field Strength
Electric field strength, E, is the force per unit positive charge:
E = F/Q (units: N C⁻¹ or equivalently V m⁻¹)
Radial Fields (Point Charges)
For a point charge Q: E = kQ/r² = Q/(4πε₀r²)
- Field lines point radially outward from positive charges and radially inward toward negative charges
- E follows an inverse square law
Uniform Fields (Parallel Plates)
Between two parallel plates separated by distance d with potential difference V:
E = V/d
- Field lines are parallel and equally spaced
- E is constant everywhere between the plates (ignoring edge effects)
- Direction: from positive plate to negative plate
Worked Example — Parallel Plates
Two plates are separated by 2.0 cm with a p.d. of 500 V. Find the force on an electron between them.
E = V/d = 500/0.020 = 25,000 V m⁻¹
F = EQ = 25,000 × 1.6 × 10⁻¹⁹ = 4.0 × 10⁻¹⁵ N
Electric Potential
Electric potential, V, at a point is the work done per unit positive charge in bringing a small test charge from infinity to that point:
V = kQ/r = Q/(4πε₀r)
- V is positive around positive charges and negative around negative charges
- V = 0 at infinity
Electric Potential Energy
E_p = kQ₁Q₂/r = Q₁Q₂/(4πε₀r)
Positive E_p for like charges (energy stored in repulsion), negative for unlike charges (bound system).
Relationship Between E and V
E = −dV/dr
The electric field strength is the negative gradient of the potential. In a uniform field: E = ΔV/Δd.
Equipotential lines are perpendicular to field lines. No work is done moving a charge along an equipotential.
Comparing Gravitational and Electric Fields
| Property | Gravitational | Electric |
|---|---|---|
| Source | Mass | Charge |
| Force law | F = GMm/r² | F = kQq/r² |
| Field strength | g = GM/r² | E = kQ/r² |
| Potential | V = −GM/r | V = kQ/r |
| Nature | Always attractive | Attractive or repulsive |
| Shielding | Cannot be shielded | Can be shielded (Faraday cage) |
Motion of Charged Particles in Electric Fields
Parallel to field lines: The particle accelerates (or decelerates) uniformly.
Work done: W = QΔV → ½mv² = QV (for a charge accelerated from rest through p.d. V)
Perpendicular to uniform field (between parallel plates): The motion is analogous to projectile motion:
- Constant velocity parallel to the plates
- Constant acceleration toward one plate (a = EQ/m)
- Path is parabolic
Worked Example — Electron Acceleration
An electron is accelerated from rest through a p.d. of 2000 V. Find its final speed.
½mv² = eV → v = √(2eV/m) = √(2 × 1.6 × 10⁻¹⁹ × 2000 / 9.11 × 10⁻³¹)
v = √(7.02 × 10¹⁴) = 2.65 × 10⁷ m s⁻¹
(At this speed, ~9% of c, relativistic effects are small but beginning to matter.)
Millikan's Oil Drop Experiment
Millikan measured the charge on the electron by balancing the weight of charged oil droplets against the electric force between parallel plates.
At balance: QE = mg → Q = mg/E = mgd/V
By measuring many droplets, he found that charge always came in integer multiples of e = 1.6 × 10⁻¹⁹ C, demonstrating the quantisation of charge.
Electric Field Patterns
- Single positive charge: Radial field lines pointing outward
- Single negative charge: Radial field lines pointing inward
- Dipole (two opposite charges): Field lines from + to −, curving around
- Two like charges: Field lines repel; neutral point between them where E = 0
- Parallel plates: Uniform field between (parallel lines), fringing at edges