Charge, conductors and insulators
There are two kinds of electric charge: positive (+) and negative (−). Like charges repel; unlike charges attract. Charge is measured in coulombs (C). The smallest free charge is that of one electron, e = 1.6 × 10⁻¹⁹ C, so every charge is a whole number of e (quantisation): q = ne.
Charge is conserved: it is never made or destroyed, only moved. When you rub a glass rod with silk, electrons move from glass to silk; the glass becomes + and the silk becomes − by the same amount.
In a conductor (metals, the human body, the Earth) some electrons are free to move. In an insulator or dielectric (plastic, glass, dry paper, rubber) electrons are tied to their atoms.
Three ways to charge an object: friction (rubbing two different materials), conduction (touching a charged object, which gives the same sign) and induction (no touching, which gives the opposite sign).
Electrostatic induction: charging without touching
Induction is the separation of charges in a conductor caused by a nearby charged object.
- Bring a negative rod near a neutral metal sphere on an insulating stand.
- The rod repels free electrons to the far side. The near side is left with a shortage of electrons, so it is +. The sphere is still neutral overall, but it is now attracted to the rod, because the + side is closer than the − side.
- Touch the far side with an earthing wire (or your finger). Electrons flow into the ground, pushed by the rod.
- Remove the earth first, then the rod. The sphere is left with a net + charge, which spreads evenly over its surface.
The induced charge always has the opposite sign to the inducing charge. Order matters: if you remove the rod before the earth, the electrons simply come back and the sphere ends neutral.
A gold-leaf electroscope shows induction: bring a charged rod near its cap and the leaves spread, because the same kind of charge is pushed down to both leaves.
Inside a conductor at rest the electric field is zero; all excess charge sits on the outer surface. This is why a metal box (Faraday cage) shields what is inside.
Polarisation of insulators (dielectrics)
In an insulator, electrons cannot travel through the material. But an electric field can still shift them a tiny distance inside each atom or molecule.
- In non-polar molecules the electron cloud is pulled one way and the nucleus the other, so each molecule becomes a small dipole (a + end and a − end).
- In polar molecules such as water, the dipoles already exist but point randomly; the field turns them to line up.
The result: the surface facing a negative rod gets a thin layer of + (bound) charge, the far surface gets −. This is dielectric polarisation. The bound charges cannot be removed by earthing, unlike induced charges in a conductor.
Because the opposite charge is nearer, there is a net pull. That is why a charged comb attracts neutral paper and a charged balloon sticks to a wall. Inside the dielectric, the bound charges create a field that opposes the outside field, so the field is reduced by a factor called the dielectric constant κ (about 80 for water, about 2–3 for plastics).
Redistribution of charge between conductors
When two charged conductors are connected by a wire, charge flows from higher potential to lower potential until both have the same potential. Total charge stays the same.
For an isolated sphere of radius r with charge q: V = kq/r, with k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻².
For two spheres far apart joined by a thin wire: kq₁/r₁ = kq₂/r₂, so q₁/q₂ = r₁/r₂, with q₁ + q₂ = Q (the total). Hence q₁ = Q·r₁/(r₁ + r₂) and the common potential V = kQ/(r₁ + r₂).
Surface charge density σ = q/(4πr²). Since q ∝ r, σ ∝ 1/r: the smaller sphere has the larger σ and the stronger field at its surface (E = σ/ε₀). That is why charge leaks from sharp points (corona discharge) and why lightning rods are pointed.
Touching two identical spheres shares the charge equally. Earthing a conductor sets its potential to zero; for an isolated sphere far from other charges, this removes all its charge.
Energy is not conserved in this sharing: some electrical energy is lost as heat in the wire and as light or radio waves from sparks.
Try it at home
Rub a balloon on dry hair or a wool sweater. Hold it near tiny pieces of paper: they jump up (polarisation). Hold it near a thin, smooth stream of tap water: the stream bends towards it (water molecules are polar). Then hold it near an empty aluminium can lying on its side: the can rolls towards the balloon (induction in a conductor). Predict first, then test. In the 3D, step 4 shows what happens inside the paper.
Key formulas and definitions
- q = ne, e = 1.6 × 10⁻¹⁹ C (quantisation)
- Total charge is conserved: q₁ + q₂ = Q
- Potential of an isolated sphere: V = kq/r, k = 9 × 10⁹ N m² C⁻²
- Joined spheres: q₁/q₂ = r₁/r₂, V = kQ/(r₁ + r₂)
- Surface density: σ = q/(4πr²) ∝ 1/r for joined spheres
- Field just outside a conductor: E = σ/ε₀; inside a conductor E = 0
Worked examples
1. A rod gains 5 × 10¹⁰ extra electrons. What is its charge?
q = ne = 5 × 10¹⁰ × 1.6 × 10⁻¹⁹ C = 8 × 10⁻⁹ C = 8 nC, negative. So q = −8 nC.
2. Two identical metal spheres carry +10 μC and −4 μC. They touch and are then separated. Find the charge on each.
Total = +10 + (−4) = +6 μC. Identical spheres share equally: +3 μC each.
3. A sphere of radius 10 cm (q = 12 nC) is joined by a long thin wire to an uncharged sphere of radius 5 cm. Find the final charges.
q₁/q₂ = r₁/r₂ = 10/5 = 2 and q₁ + q₂ = 12 nC. So q₁ = 8 nC, q₂ = 4 nC.
4. Find the common potential in the previous example.
V = kq₁/r₁ = 9 × 10⁹ × 8 × 10⁻⁹ / 0.10 = 720 V. Check: kq₂/r₂ = 9 × 10⁹ × 4 × 10⁻⁹ / 0.05 = 720 V. Same, as expected.
5. Compare the surface charge densities of the two spheres above.
σ = q/(4πr²). σ₁/σ₂ = (q₁/q₂)(r₂/r₁)² = 2 × (1/2)² = 1/2. The small sphere has twice the surface charge density, and so twice the field at its surface.
6. A neutral sphere is charged by induction using a +20 nC rod. What sign does the sphere get, where does its charge come from, and does the rod lose charge?
The sphere becomes negative (opposite to the rod). The + rod attracts electrons up from the Earth through the earthing wire; after the earth is removed they are trapped on the sphere. The rod never touches the sphere, so it keeps its full +20 nC. Total charge of sphere + Earth is still conserved.
Common mistakes
- Thinking induction creates charge. It only separates charge that is already there; the sphere stays neutral until it is earthed.
- Removing the rod before the earth wire. The electrons flow back and the sphere ends neutral.
- Saying protons move in a metal. Only electrons move; a + region means a shortage of electrons.
- Assuming joined spheres share charge equally. They share potential equally; charge divides in the ratio of their radii.