How much work it takes to bring a charge somewhere, why conductors behave the way they do, and how capacitors store that work as usable energy.
Moving a charge against an electric field takes work, exactly like lifting a mass against gravity. That work gets stored as electrostatic potential energy — the charge's ability to do work by virtue of its position in the field. Move it back to where it started, and the field returns that energy in full; electrostatic force, like gravity, is conservative, so the work done is path-independent.
Electric potential at a point is the work done per unit charge in bringing a small positive test charge from infinity to that point, without any acceleration (i.e. quasi-statically).
Potential is defined relative to a reference point, conventionally taken as infinity, where V = 0.
For a system of several point charges, potential at any point is just the algebraic sum of the potentials due to each charge individually (potentials add as scalars, which makes this often easier than adding fields as vectors):
Unlike the field, the dipole's potential depends on direction as well as distance:
An equipotential surface is one on which every point has the same potential. Moving a charge along such a surface takes zero work, since ΔV = 0.
Field and potential are two sides of the same coin: the field is the (negative) rate at which potential changes with position.
For three or more charges, sum the potential energy of every unique pair. This is the work needed to assemble the configuration, bringing each charge in from infinity one at a time.
A handful of properties follow from the fact that charges in a conductor are free to move until equilibrium is reached:
Dielectrics are insulators where charge can't flow, but the molecules themselves can respond to an external field. Polar molecules (like water) already have a permanent dipole moment, which an external field tends to align. Non-polar molecules have no permanent dipole, but an external field induces one by slightly displacing their positive and negative charge centres.
Either way, the dielectric develops a net dipole moment per unit volume, called polarisation (P). This creates an internal field opposing the external one, which is why inserting a dielectric between capacitor plates always reduces the net field — and why it increases capacitance.
A capacitor is any arrangement of two conductors that can store charge (and hence energy) when a potential difference is applied across them. Capacitance measures how much charge a given potential difference can push onto the plates.
Charging a capacitor means doing work against the growing repulsion as more charge piles onto the plates — that work is stored as electrostatic potential energy.
Find the electric potential at a distance of 9 cm from a point charge of 2 µC.
Solution: V = kQ/r = (9×10⁹ × 2×10⁻⁶) / 0.09 = 1.8×10⁴ / 0.09 = 2×10⁵ V.
A parallel plate capacitor has plates of area 200 cm² separated by 2 mm of air. Find its capacitance, then find the new capacitance if a dielectric with K = 5 fills the gap.
Solution: C₀ = ε₀A/d = (8.854×10⁻¹² × 0.02) / 0.002 ≈ 8.85×10⁻¹¹ F ≈ 88.5 pF.
With the dielectric: C = K·C₀ = 5 × 88.5 pF = 442.5 pF.