1Electrostatic Potential 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.

2Electric Potential

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.

3Potential Due to a Point Charge

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):

4Potential Due to an Electric Dipole

Unlike the field, the dipole’s potential depends on direction as well as distance:

• On the axial line (θ = 0°): V = kp/r² — maximum positive value.
• On the equatorial line (θ = 90°): V = 0 — a result worth remembering, since the field there is not zero, only the potential is.

5Equipotential Surfaces

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.

• Equipotential surfaces are always perpendicular to field lines at every point.
• For a point charge, equipotential surfaces are concentric spheres.
• For a uniform field, they’re flat planes perpendicular to the field.
• No two equipotential surfaces can intersect — that would mean one point had two different potentials.

6Relation Between Electric Field and Potential

Field and potential are two sides of the same coin: the field is the (negative) rate at which potential changes with position.

7Potential Energy of a System of Charges

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.

Dipole in an External Field

8Electrostatics of Conductors

A handful of properties follow from the fact that charges in a conductor are free to move until equilibrium is reached:

1. The electric field inside a conductor, in electrostatic equilibrium, is always zero — if it weren’t, free charges would keep moving.
2. Just outside the surface, the field is always perpendicular to the surface; any tangential component would drive charge along the surface.
3. Any excess charge on a conductor resides entirely on its outer surface — never in the interior.
4. The entire conductor, surface included, is at one uniform potential (an equipotential body).
5. Field just outside the surface: E = σ/ε₀, where σ is the local surface charge density.
6. Electrostatic shielding: the field inside a cavity within a conductor is zero, regardless of external fields — the reasoning behind using metal enclosures (“Faraday cages”) to shield sensitive equipment.

9Dielectrics and Polarisation

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.

10Capacitors and 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.

Parallel Plate Capacitor

11Capacitors in Series and Parallel

Series Combination

Parallel Combination

12Energy Stored in a Capacitor

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.

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