Electric charge and its properties
Rubbing a glass rod with silk, or a plastic comb through dry hair, transfers electrons from one object to the other. The objects become charged.
There are two kinds of charge: positive and negative. Like charges repel; unlike charges attract.
Additivity: the total charge of a system is the algebraic sum of its individual charges.
Conservation: charge can be transferred, but the total charge of an isolated system never changes.
Quantisation: every charge is a whole-number multiple of the elementary charge, q = ne, where e = 1.6 × 10⁻¹⁹ C.
Conductors (such as metals) let charge move freely. Insulators (such as glass and plastic) do not.
Coulomb's law
The force between two point charges q₁ and q₂ separated by a distance r is
F = k q₁q₂ / r², where k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻²
ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻² is the permittivity of free space. The force acts along the line joining the charges. In a medium of dielectric constant K, the force is reduced by a factor of K.
Superposition principle: the net force on a charge due to several other charges is the vector sum of the individual forces, each calculated as if the others were absent.
Electric field
The electric field E at a point is the force per unit positive test charge placed there:
E = F / q₀, measured in N/C (equivalently V/m)
For a point charge Q, the field at distance r is E = kQ/r². It points away from a positive charge and towards a negative one.
Electric field lines
Field lines are a way of picturing a field.
They start on positive charges and end on negative charges.
The tangent at any point gives the direction of E.
Lines are closer together where the field is stronger.
Two field lines never cross, because the field can have only one direction at a point.
Field lines are not closed loops (for electrostatic fields).
Electric dipole
A dipole is a pair of equal and opposite charges, +q and −q, separated by a distance 2a. Its dipole moment is
p = q × 2a, directed from −q to +q
On the axis of a short dipole, at a distance r: E = 2kp / r³.
On the equatorial line: E = kp / r³, directed opposite to p.
The field of a dipole falls off as 1/r³, faster than that of a single charge (1/r²).
In a uniform field E, a dipole feels no net force, but a torque τ = p × E (magnitude pE sin θ), which tends to align it with the field.
Electric flux
The electric flux through a small area ΔS is
Δφ = E · ΔS = E ΔS cos θ
where θ is the angle between E and the outward normal to the surface. Its unit is N m² C⁻¹.
Gauss's law
The total electric flux through any closed surface equals the net charge enclosed divided by ε₀: φ = q_enclosed / ε₀
Charges outside the surface contribute zero net flux. The closed surface used is called a Gaussian surface, and choosing it to match the symmetry of the problem makes the calculation easy.
Charge distribution | Field |
|---|---|
Infinitely long straight wire (charge λ per length) | E = λ / (2πε₀ r) |
Infinite plane sheet (charge σ per area) | E = σ / (2ε₀), independent of distance |
Thin spherical shell (total charge q), point outside (r > R) | E = q / (4πε₀ r²), as if all the charge were at the centre |
Thin spherical shell, point inside (r < R) | E = 0 |
The field inside a charged conducting shell is zero. This is the principle behind electrostatic shielding: a person inside a car is protected from lightning.