Electric Charges and Fields
In one line
Electric charges push or pull each other with a force that gets weaker with the square of the distance, and the space around a charge where this force acts is its electric field.
Properties of charge
- There are two kinds: positive and negative. Like charges repel; unlike charges attract.
- Quantisation: every charge is a whole-number multiple of the electron's charge. q = ne, with e = 1.6 × 10⁻¹⁹ C.
- Conservation: the total charge of an isolated system never changes. Charge is only moved from one body to another.
- Additivity: total charge is the algebraic sum of all charges: +3 C and −5 C give −2 C.
- Bodies can be charged by friction, conduction (touching) or induction (bringing a charge near, without touching).
Coulomb's law
F = k q₁q₂ / r², k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻²
ε₀ (permittivity of free space) = 8.854 × 10⁻¹² C² N⁻¹ m⁻². In a medium, the force is reduced by its dielectric constant K: F = kq₁q₂ / (K r²).
ExampleTwo charges of 2 μC and 3 μC are 0.3 m apart.
F = 9 × 10⁹ × (2 × 10⁻⁶)(3 × 10⁻⁶) / 0.09 = 0.6 N (repulsive).
F = 9 × 10⁹ × (2 × 10⁻⁶)(3 × 10⁻⁶) / 0.09 = 0.6 N (repulsive).
Superposition: with many charges, the force on one charge is the vector sum of the forces from each of the others.
Advertisement
Electric field
E = F / q₀ · for a point charge: E = kq / r² (unit N/C or V/m)
Field lines show the field's direction:
- They start on positive charges and end on negative charges.
- They never cross each other.
- Lines closer together mean a stronger field.
- They don't form closed loops in an electrostatic field.
Electric dipole
- Two equal and opposite charges, ±q, a small distance 2a apart.
- Dipole moment p = q × 2a, pointing from −q to +q. Unit: C m.
- Field on the axis (far away): E = 2kp / r³. On the equatorial line: E = kp / r³, in the opposite direction to p.
- Torque in a uniform field: τ = pE sin θ.
Electric flux and Gauss's law
Φ = E A cos θ · Gauss's law: Φ = qenclosed / ε₀
| Charge shape | Electric field |
|---|---|
| Infinite straight wire (λ = charge per length) | E = λ / (2πε₀ r) |
| Infinite flat sheet (σ = charge per area) | E = σ / (2ε₀) |
| Thin spherical shell, outside (r > R) | E = kq / r² |
| Thin spherical shell, inside (r < R) | E = 0 |
Common mistakes
- Forgetting to convert μC. 1 μC = 10⁻⁶ C; 1 nC = 10⁻⁹ C.
- Using r instead of r² for force and field; dipole fields use r³.
- Including outside charges in Gauss's law. Only charge inside the closed surface counts.
Quick check
Revise in 30 seconds
| Charge | Quantised q = ne · conserved · additive · e = 1.6 × 10⁻¹⁹ C |
|---|---|
| Coulomb | F = kq₁q₂/r², k = 9 × 10⁹ |
| Field | E = F/q₀ = kq/r² |
| Dipole | p = q·2a · axial 2kp/r³ · equatorial kp/r³ · τ = pE sin θ |
| Gauss | Φ = q/ε₀ · wire λ/2πε₀r · sheet σ/2ε₀ · inside shell 0 |